From d8222e0838731da57e522a67739080b1f2d60c68 Mon Sep 17 00:00:00 2001 From: Samuel Schlesinger Date: Sun, 25 Oct 2026 01:25:16 +0000 Subject: [PATCH 1/5] feat(Interactive): layered arithmetic circuits and the GKR identity (#892) Add Cslib.Interactive.GKR.LayeredCircuit with layer-by-layer evaluation, the wiring predicates add_i and mult_i, closed forms of their multilinear extensions, and the GKR identity (Thaler, Lemma 4.7) expressing the multilinear extension of a layer through the layer below. Co-authored-by: Jetski (worker-jetski-17d3679e) --- Complexity.lean | 1 + .../Interactive/GKR/Layered.lean | 209 ++ ComplexityTests/GKRLayered.lean | 54 + ROADMAP.md | 5 + references.bib | 1927 +++++++++++++++++ 5 files changed, 2196 insertions(+) create mode 100644 Complexity/Computability/Interactive/GKR/Layered.lean create mode 100644 ComplexityTests/GKRLayered.lean diff --git a/Complexity.lean b/Complexity.lean index e61298bb50..37103319bb 100644 --- a/Complexity.lean +++ b/Complexity.lean @@ -834,6 +834,7 @@ public import Complexity.Computability.GenericRAM.Simulation.Turing public import Complexity.Computability.GenericRAM.SparseMemory.Basic public import Complexity.Computability.GenericRAM.SparseMemory.Turing public import Complexity.Computability.Interactive.Basic +public import Complexity.Computability.Interactive.GKR.Layered public import Complexity.Computability.Interactive.Plain public import Complexity.Computability.Interactive.Security public import Complexity.Computability.Interactive.Sequential diff --git a/Complexity/Computability/Interactive/GKR/Layered.lean b/Complexity/Computability/Interactive/GKR/Layered.lean new file mode 100644 index 0000000000..17307670f6 --- /dev/null +++ b/Complexity/Computability/Interactive/GKR/Layered.lean @@ -0,0 +1,209 @@ +/- +Copyright (c) 2026 Samuel Schlesinger. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Samuel Schlesinger +-/ + +module +-- upstream: candidate + +public import Complexity.Foundations.BooleanAnalysis.MultilinearExtension + +/-! +# Layered arithmetic circuits and the GKR identity + +A *layered arithmetic circuit* of fan-in two (`Cslib.Interactive.GKR.LayeredCircuit`) has its gates +arranged in layers, with the gates of layer `i` labelled by the bit strings of length `width i`. +Layer `0` is the input. Every gate of layer `i + 1` is an addition or a multiplication of two gates +of layer `i`, its left and right in-neighbours. Evaluating the layers in turn gives the gate values +`value x i` on an input `x` (`Cslib.Interactive.GKR.LayeredCircuit.value`). We number the layers +from the input, while [Thaler, §4.6][Thaler2022] numbers them from the output. + +The *wiring predicates* `add_i (a, b, c)` and `mult_i (a, b, c)` of layer `i + 1` are `1` exactly +when gate `a` of layer `i + 1` is an addition, respectively a multiplication, with in-neighbours +`b` and `c` (`addPred`, `multPred`). The identity at the heart of the GKR protocol expresses the +multilinear extension of the gate values of a layer through the multilinear extensions of the +wiring predicates and the gate values of the layer below +(`Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_step`) +[Thaler, Lemma 4.7][Thaler2022], [Goldwasser, Kalai, Rothblum][GoldwasserKalaiRothblum2015]: +`W̃_{i+1}(z) = ∑_{b, c} ãdd_i(z, b, c) (W_i(b) + W_i(c)) + m̃ult_i(z, b, c) W_i(b) W_i(c)`. + +## Main definitions + +* `Cslib.Interactive.GKR.LayeredCircuit`: a layered arithmetic circuit of fan-in two. +* `Cslib.Interactive.GKR.LayeredCircuit.step`, `Cslib.Interactive.GKR.LayeredCircuit.value`: + evaluation of one layer and of the circuit. +* `Cslib.Interactive.GKR.LayeredCircuit.addPred`, `Cslib.Interactive.GKR.LayeredCircuit.multPred`: + the wiring predicates. + +## Main results + +* `Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_addPred`, + `Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_multPred`: closed forms of the + multilinear extensions of the wiring predicates. +* `Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_step`: the GKR identity. + +## References + +* [S. Goldwasser, Y. T. Kalai, G. N. Rothblum, *Delegating computation: interactive proofs for + muggles*][GoldwasserKalaiRothblum2015] +* [J. Thaler, *Proofs, Arguments, and Zero-Knowledge*][Thaler2022], Section 4.6 +-/ + +@[expose] public section + +namespace Cslib.Interactive.GKR + +open MvPolynomial Cslib.BitString + +/-- The two kinds of gates of an arithmetic circuit of fan-in two. -/ +inductive Gate + /-- An addition gate. -/ + | add + /-- A multiplication gate. -/ + | mul + deriving DecidableEq + +/-- A **layered arithmetic circuit** of fan-in two. The gates of layer `i` are labelled by bit +strings of length `width i`; layer `0` is the input, and each gate of layer `i + 1` combines two +gates of layer `i`. -/ +structure LayeredCircuit where + /-- The number of layers above the input. -/ + depth : ℕ + /-- Layer `i` has `2 ^ width i` gates. -/ + width : ℕ → ℕ + /-- The kind of each gate of layer `i + 1`. -/ + gate : (i : ℕ) → (Fin (width (i + 1)) → Bool) → Gate + /-- The left in-neighbour, in layer `i`, of each gate of layer `i + 1`. -/ + left : (i : ℕ) → (Fin (width (i + 1)) → Bool) → Fin (width i) → Bool + /-- The right in-neighbour, in layer `i`, of each gate of layer `i + 1`. -/ + right : (i : ℕ) → (Fin (width (i + 1)) → Bool) → Fin (width i) → Bool + +namespace LayeredCircuit + +variable (C : LayeredCircuit) {R : Type*} [CommRing R] + +/-- The values of layer `i + 1`, given the values `W` of layer `i`. -/ +def step (i : ℕ) (W : (Fin (C.width i) → Bool) → R) (a : Fin (C.width (i + 1)) → Bool) : R := + match C.gate i a with + | .add => W (C.left i a) + W (C.right i a) + | .mul => W (C.left i a) * W (C.right i a) + +/-- The gate values of layer `i` on the input `x`. -/ +def value (x : (Fin (C.width 0) → Bool) → R) : (i : ℕ) → (Fin (C.width i) → Bool) → R + | 0 => x + | i + 1 => C.step i (value x i) + +/-- The output values of the circuit. -/ +def output (x : (Fin (C.width 0) → Bool) → R) : (Fin (C.width C.depth) → Bool) → R := + C.value x C.depth + +/-- The variables of a wiring predicate of layer `i + 1`: a gate `a` of layer `i + 1` and two gates +`b`, `c` of layer `i`. -/ +abbrev WiringVar (i : ℕ) : Type := Fin (C.width (i + 1)) ⊕ (Fin (C.width i) ⊕ Fin (C.width i)) + +/-- The wiring predicate of a gate kind: `1` exactly when gate `a` of layer `i + 1` has that kind +and in-neighbours `b` and `c`. -/ +def wiringPred (g : Gate) (i : ℕ) (v : C.WiringVar i → Bool) : R := + open Classical in + if C.gate i (v ∘ Sum.inl) = g ∧ C.left i (v ∘ Sum.inl) = v ∘ Sum.inr ∘ Sum.inl ∧ + C.right i (v ∘ Sum.inl) = v ∘ Sum.inr ∘ Sum.inr then 1 else 0 + +/-- The addition predicate `add_i`. -/ +abbrev addPred (i : ℕ) : (C.WiringVar i → Bool) → R := C.wiringPred .add i + +/-- The multiplication predicate `mult_i`. -/ +abbrev multPred (i : ℕ) : (C.WiringVar i → Bool) → R := C.wiringPred .mul i + +/-- The value at `z` of the indicator polynomial of the point `a` of the cube. -/ +def basis {ι : Type*} [Fintype ι] (a : ι → Bool) (z : ι → R) : R := + ∏ j, if a j then z j else 1 - z j + +theorem basis_bitVal {ι : Type*} [Fintype ι] (a b : ι → Bool) : + basis a (fun j => bitVal R (b j)) = if a = b then 1 else 0 := by + rw [basis, ← eval_indicatorPoly_apply, eval_indicatorPoly] + +/-- The multilinear extension of a wiring predicate at `(z, y, w)` is the sum, over the gates `a` +of that kind, of the product of the basis polynomials of `a`, its left and its right in-neighbour. +-/ +theorem eval_multilinearExtension_wiringPred (g : Gate) (i : ℕ) (z : Fin (C.width (i + 1)) → R) + (y w : Fin (C.width i) → R) : + eval (Sum.elim z (Sum.elim y w)) (multilinearExtension (C.wiringPred (R := R) g i)) = + ∑ a, if C.gate i a = g then basis a z * basis (C.left i a) y * basis (C.right i a) w + else 0 := by + simp only [multilinearExtension, map_sum, map_mul, eval_C, eval_indicatorPoly_apply] + rw [← (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.sum_comp, Fintype.sum_prod_type] + refine Finset.sum_congr rfl fun a _ => ?_ + rw [← (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.sum_comp, Fintype.sum_prod_type] + simp only [Equiv.sumArrowEquivProdArrow_symm_apply_inl, Fintype.prod_sum_type, + Sum.elim_inl, Sum.elim_inr, wiringPred] + by_cases hg : C.gate i a = g + · rw [ite_eq_left hg] + rw [Finset.sum_eq_single (C.left i a), Finset.sum_eq_single (C.right i a)] + · simp [hg, basis, Function.comp_def, mul_assoc] + congr 1 + · intro c _ hc + simp [Function.comp_def, Ne.symm hc] + · simp + · intro b _ hb + refine Finset.sum_eq_zero fun c _ => ?_ + simp [Function.comp_def, Ne.symm hb] + · simp + · rw [ite_eq_right hg] + refine Finset.sum_eq_zero fun b _ => Finset.sum_eq_zero fun c _ => ?_ + simp [Function.comp_def, hg] + +/-- The multilinear extension of `g` at `z` is the sum of the values of `g` weighted by the basis +polynomials. -/ +theorem eval_multilinearExtension_eq_sum {ι : Type*} [Fintype ι] [DecidableEq ι] + (g : (ι → Bool) → R) (z : ι → R) : + eval z (multilinearExtension g) = ∑ a, g a * basis a z := by + simp [multilinearExtension, eval_indicatorPoly_apply, basis] + +/-- The point of `R^k` with the coordinates of the bit string `b`. -/ +abbrev bitPoint {k : ℕ} (b : Fin k → Bool) : Fin k → R := fun j => bitVal R (b j) + +/-- **The GKR identity** [Thaler, Lemma 4.7][Thaler2022]. The multilinear extension of the values +of layer `i + 1` is a sum over the pairs of gates `b`, `c` of layer `i` of the multilinear +extensions of the wiring predicates times the sum, respectively the product, of the values of `b` +and `c`. -/ +theorem eval_multilinearExtension_step (i : ℕ) (W : (Fin (C.width i) → Bool) → R) + (z : Fin (C.width (i + 1)) → R) : + eval z (multilinearExtension (C.step i W)) = + ∑ b, ∑ c, + (eval (Sum.elim z (Sum.elim (bitPoint b) (bitPoint c))) + (multilinearExtension (C.addPred (R := R) i)) * (W b + W c) + + eval (Sum.elim z (Sum.elim (bitPoint b) (bitPoint c))) + (multilinearExtension (C.multPred (R := R) i)) * (W b * W c)) := by + rw [eval_multilinearExtension_eq_sum (C.step i W)] + simp only [eval_multilinearExtension_wiringPred, basis_bitVal] + have hinner : ∀ b c, (∑ a, if C.gate i a = .add then basis a z * + (if C.left i a = b then (1 : R) else 0) * (if C.right i a = c then 1 else 0) else 0) * + (W b + W c) + + (∑ a, if C.gate i a = .mul then basis a z * + (if C.left i a = b then (1 : R) else 0) * (if C.right i a = c then 1 else 0) else 0) * + (W b * W c) = + ∑ a, (if C.left i a = b ∧ C.right i a = c then basis a z * + (if C.gate i a = .add then W b + W c else W b * W c) else 0) := by + intro b c + rw [Finset.sum_mul, Finset.sum_mul, ← Finset.sum_add_distrib] + refine Finset.sum_congr rfl fun a _ => ?_ + by_cases hb : C.left i a = b <;> by_cases hc : C.right i a = c <;> + cases C.gate i a <;> simp [hb, hc] + simp only [hinner] + symm + refine (Finset.sum_congr rfl fun b _ => Finset.sum_comm).trans (Finset.sum_comm.trans ?_) + refine Finset.sum_congr rfl fun a _ => ?_ + rw [Finset.sum_eq_single (C.left i a), Finset.sum_eq_single (C.right i a)] + · simp only [and_self, ite_true, step] + cases C.gate i a <;> simp [mul_comm] + · intro c _ hc + simp [Ne.symm hc] + · simp + · intro b _ hb + exact Finset.sum_eq_zero fun c _ => by simp [Ne.symm hb] + · simp + +end LayeredCircuit + +end Cslib.Interactive.GKR diff --git a/ComplexityTests/GKRLayered.lean b/ComplexityTests/GKRLayered.lean new file mode 100644 index 0000000000..53a5cf501c --- /dev/null +++ b/ComplexityTests/GKRLayered.lean @@ -0,0 +1,54 @@ +/- +Copyright (c) 2026 Samuel Schlesinger. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Samuel Schlesinger +-/ + +import Complexity.Computability.Interactive.GKR.Layered + +/-! # Layered arithmetic circuits and the GKR identity + +A one-layer circuit multiplying its two inputs, its value, and an instance of the GKR identity. +-/ + +namespace ComplexityTests.GKRLayered + +open Cslib.Interactive.GKR MvPolynomial Cslib.BitString + +/-- One multiplication gate on two inputs: layer `0` has `2 ^ 1` gates, layer `1` has `2 ^ 0`. -/ +def mulTwo : LayeredCircuit where + depth := 1 + width := fun i => if i = 0 then 1 else 0 + gate _ _ := .mul + left _ _ := fun _ => false + right _ _ := fun _ => true + +-- The output is the product of the two inputs. +example (x : (Fin 1 → Bool) → ℤ) : + mulTwo.output x (fun _ => false) = x (fun _ => false) * x (fun _ => true) := rfl + +-- The GKR identity for the multiplication layer, at any point. +example (W : (Fin (mulTwo.width 0) → Bool) → ℤ) (z : Fin (mulTwo.width 1) → ℤ) : + eval z (multilinearExtension (mulTwo.step 0 W)) = + ∑ b, ∑ c, + (eval (Sum.elim z (Sum.elim (LayeredCircuit.bitPoint b) (LayeredCircuit.bitPoint c))) + (multilinearExtension (mulTwo.addPred (R := ℤ) 0)) * (W b + W c) + + eval (Sum.elim z (Sum.elim (LayeredCircuit.bitPoint b) (LayeredCircuit.bitPoint c))) + (multilinearExtension (mulTwo.multPred (R := ℤ) 0)) * (W b * W c)) := + mulTwo.eval_multilinearExtension_step 0 W z + +/-- +info: 'Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_step' depends on axioms: +[propext, Classical.choice, Quot.sound] +-/ +#guard_msgs (whitespace := lax) in +#print axioms LayeredCircuit.eval_multilinearExtension_step + +/-- +info: 'Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_wiringPred' depends on axioms: +[propext, Classical.choice, Quot.sound] +-/ +#guard_msgs (whitespace := lax) in +#print axioms LayeredCircuit.eval_multilinearExtension_wiringPred + +end ComplexityTests.GKRLayered diff --git a/ROADMAP.md b/ROADMAP.md index 892f65548f..c1883e20a1 100644 --- a/ROADMAP.md +++ b/ROADMAP.md @@ -1266,6 +1266,11 @@ Jukna coverage below is an additional target beyond the V1 port: components, stated as theory without concrete ciphersuites. The design is on #643, and the TQBF, PCP and GMW bridges are recorded there. +* **GKR delegation (#892).** [Layered](Complexity/Computability/Interactive/GKR/Layered.lean) + defines layered arithmetic circuits and proves the GKR identity expressing the multilinear + extension of a layer through the wiring predicates and the layer below. Open: the GKR protocol + itself (layer-by-layer sumcheck and the line reduction, with completeness and soundness), its + efficiency under uniformity, and the route to NEXP ⊆ MIP. * **Classic symmetric constructions (#640, track D).** Proved: PRF-based MACs (#677), keyed collision-resistant hash families and Merkle–Damgård (#678), universal hashing (#663), [Carter–Wegman MACs from Δ-universal hashing](Complexity/Crypto/Protocols/MAC/CarterWegman.lean) diff --git a/references.bib b/references.bib index 987485363f..aef760910e 100644 --- a/references.bib +++ b/references.bib @@ -3452,6 +3452,1933 @@ @article{ Shamir1979 doi = {10.1145/359168.359176} } +@article{BarringtonImmermanStraubing1990, + author = {Barrington, David A. 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{https://eccc.weizmann.ac.il/report/2012/070/} +} + +@article{FennerEtAl2003, + author = {Fenner, Stephen A. and Fortnow, Lance J. and Kurtz, Stuart A. and Li, Lide}, + title = {An Oracle Builder's Toolkit}, + journal = {Information and Computation}, + volume = {182}, + number = {2}, + pages = {95--136}, + year = {2003}, + doi = {10.1016/S0890-5401(03)00018-X}, + url = {https://lance.fortnow.com/papers/files/obt.pdf} +} + +@techreport{CookMertz2024, + author = {Cook, James and Mertz, Ian}, + title = {Tree Evaluation is in Space {$O(\log n \cdot \log\log n)$}}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR23-174}, + note = {Conference version in STOC 2024}, + year = {2023} +} + +@techreport{Goldreich2024TreeEval, + author = {Goldreich, Oded}, + title = {On the {Cook}--{Mertz} Tree Evaluation procedure}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR24-109}, + note = {Revision 2, 27 December 2024}, + year = {2024} +} + +@article{CMWBS2012, + author = {Cook, Stephen and McKenzie, Pierre and Wehr, Dustin and Braverman, Mark and + Santhanam, Rahul}, + title = {Pebbles and Branching Programs for Tree Evaluation}, + journal = {ACM Transactions on Computation Theory}, + volume = {3}, + number = {2}, + pages = {4:1--4:43}, + year = {2012} +} + +@article{SudanTrevisanVadhan2001, + author = {Sudan, Madhu and Trevisan, Luca and Vadhan, Salil}, + title = {Pseudorandom Generators without the {XOR} Lemma}, + journal = {Journal of Computer and System Sciences}, + volume = {62}, + number = {2}, + pages = {236--266}, + year = {2001}, + doi = {10.1006/jcss.2000.1730} +} + +@inproceedings{BabaiFortnowLevinSzegedy1991, + author = {Babai, L{\'a}szl{\'o} and Fortnow, Lance and Levin, Leonid A. and Szegedy, Mario}, + title = {Checking Computations in Polylogarithmic Time}, + booktitle = {Proceedings of the 23rd Annual ACM Symposium on Theory of Computing}, + series = {STOC 1991}, + pages = {21--32}, + year = {1991}, + doi = {10.1145/103418.103428} +} + +@inproceedings{Babai1985, + author = {Babai, L{\'a}szl{\'o}}, + title = {Trading Group Theory for Randomness}, + booktitle = {Proceedings of the 17th Annual ACM Symposium on Theory of Computing}, + pages = {421--429}, + year = {1985}, + doi = {10.1145/22145.22192} +} + +@article{BabaiMoran1988, + author = {Babai, L{\'a}szl{\'o} and Moran, Shlomo}, + title = {{Arthur}--{Merlin} Games: A Randomized Proof System, and a Hierarchy of Complexity + Classes}, + journal = {Journal of Computer and System Sciences}, + volume = {36}, + number = {2}, + pages = {254--276}, + year = {1988}, + doi = {10.1016/0022-0000(88)90028-1} +} + +@book{CyganEtAl2015, + author = {Cygan, Marek and Fomin, Fedor V. and Kowalik, {\L}ukasz and Lokshtanov, Daniel and + Marx, D{\'a}niel and Pilipczuk, Marcin and Pilipczuk, Micha{\l} and Saurabh, Saket}, + title = {Parameterized Algorithms}, + publisher = {Springer}, + address = {Cham}, + year = {2015}, + doi = {10.1007/978-3-319-21275-3} +} + +@book{DowneyFellows1999, + author = {Downey, Rodney G. and Fellows, Michael R.}, + title = {Parameterized Complexity}, + series = {Monographs in Computer Science}, + publisher = {Springer}, + address = {New York}, + year = {1999}, + doi = {10.1007/978-1-4612-0515-9} +} + +@book{FlumGrohe2006, + author = {Flum, J{\"o}rg and Grohe, Martin}, + title = {Parameterized Complexity Theory}, + series = {Texts in Theoretical Computer Science. An EATCS Series}, + publisher = {Springer}, + address = {Berlin, Heidelberg}, + year = {2006}, + doi = {10.1007/3-540-29953-X} +} + +@article{CaiChenDowneyFellows1997, + author = {Cai, Liming and Chen, Jianer and Downey, Rodney G. and Fellows, Michael R.}, + title = {Advice classes of parameterized tractability}, + journal = {Annals of Pure and Applied Logic}, + volume = {84}, + number = {1}, + pages = {119--138}, + year = {1997}, + doi = {10.1016/S0168-0072(96)00020-6} +} + +@article{LokshtanovMarxSaurabh2011, + author = {Lokshtanov, Daniel and Marx, D{\'a}niel and Saurabh, Saket}, + title = {Lower bounds based on the {Exponential Time Hypothesis}}, + journal = {Bulletin of the EATCS}, + volume = {105}, + pages = {41--72}, + year = {2011} +} + +@article{Reingold2008, + author = {Reingold, Omer}, + title = {Undirected Connectivity in Log-Space}, + journal = {Journal of the ACM}, + volume = {55}, + number = {4}, + pages = {17:1--17:24}, + year = {2008}, + doi = {10.1145/1391289.1391291} +} + +@article{LewisPapadimitriou1982, + author = {Lewis, Harry R. and Papadimitriou, Christos H.}, + title = {Symmetric Space-Bounded Computation}, + journal = {Theoretical Computer Science}, + volume = {19}, + number = {2}, + pages = {161--187}, + year = {1982}, + doi = {10.1016/0304-3975(82)90058-5} +} + +@inproceedings{KahnKalaiLinial1988, + author = {Kahn, Jeff and Kalai, Gil and Linial, Nathan}, + title = {The Influence of Variables on {Boolean} Functions}, + booktitle = {29th Annual Symposium on Foundations of Computer Science}, + publisher = {IEEE Computer Society}, + year = {1988}, + pages = {68--80}, + doi = {10.1109/SFCS.1988.21923} +} + +@article{Talagrand1994, + author = {Talagrand, Michel}, + title = {On {Russo}'s Approximate Zero-One Law}, + journal = {The Annals of Probability}, + volume = {22}, + number = {3}, + pages = {1576--1587}, + year = {1994}, + doi = {10.1214/aop/1176988612} +} + +@article{Friedgut1998, + author = {Friedgut, Ehud}, + title = {{Boolean} Functions With Low Average Sensitivity Depend On Few Coordinates}, + journal = {Combinatorica}, + volume = {18}, + number = {1}, + pages = {27--35}, + year = {1998}, + doi = {10.1007/PL00009809} +} + +@article{LundFortnowKarloffNisan1992, + author = {Lund, Carsten and Fortnow, Lance and Karloff, Howard and Nisan, Noam}, + title = {Algebraic Methods for Interactive Proof Systems}, + journal = {Journal of the ACM}, + volume = {39}, + number = {4}, + pages = {859--868}, + year = {1992}, + doi = {10.1145/146585.146605} +} + @article{Shamir1992, author = {Shamir, Adi}, title = {{IP} = {PSPACE}}, From 84650ffdf6aaef3fa3ac9fba0d4d666234916ec2 Mon Sep 17 00:00:00 2001 From: Samuel Schlesinger Date: Sun, 25 Oct 2026 01:52:09 +0000 Subject: [PATCH 2/5] refactor(Interactive): reuse indicatorPoly; move bitPoint and eval_multilinearExtension_eq_sum to MultilinearExtension (#892) --- .../Interactive/GKR/Layered.lean | 32 +++++-------------- .../BooleanAnalysis/MultilinearExtension.lean | 11 +++++++ ComplexityTests/GKRLayered.lean | 4 +-- 3 files changed, 21 insertions(+), 26 deletions(-) diff --git a/Complexity/Computability/Interactive/GKR/Layered.lean b/Complexity/Computability/Interactive/GKR/Layered.lean index 17307670f6..e152887ddc 100644 --- a/Complexity/Computability/Interactive/GKR/Layered.lean +++ b/Complexity/Computability/Interactive/GKR/Layered.lean @@ -115,21 +115,14 @@ abbrev addPred (i : ℕ) : (C.WiringVar i → Bool) → R := C.wiringPred .add i /-- The multiplication predicate `mult_i`. -/ abbrev multPred (i : ℕ) : (C.WiringVar i → Bool) → R := C.wiringPred .mul i -/-- The value at `z` of the indicator polynomial of the point `a` of the cube. -/ -def basis {ι : Type*} [Fintype ι] (a : ι → Bool) (z : ι → R) : R := - ∏ j, if a j then z j else 1 - z j - -theorem basis_bitVal {ι : Type*} [Fintype ι] (a b : ι → Bool) : - basis a (fun j => bitVal R (b j)) = if a = b then 1 else 0 := by - rw [basis, ← eval_indicatorPoly_apply, eval_indicatorPoly] - /-- The multilinear extension of a wiring predicate at `(z, y, w)` is the sum, over the gates `a` of that kind, of the product of the basis polynomials of `a`, its left and its right in-neighbour. -/ theorem eval_multilinearExtension_wiringPred (g : Gate) (i : ℕ) (z : Fin (C.width (i + 1)) → R) (y w : Fin (C.width i) → R) : eval (Sum.elim z (Sum.elim y w)) (multilinearExtension (C.wiringPred (R := R) g i)) = - ∑ a, if C.gate i a = g then basis a z * basis (C.left i a) y * basis (C.right i a) w + ∑ a, if C.gate i a = g then eval z (indicatorPoly a) * eval y (indicatorPoly (C.left i a)) * + eval w (indicatorPoly (C.right i a)) else 0 := by simp only [multilinearExtension, map_sum, map_mul, eval_C, eval_indicatorPoly_apply] rw [← (Equiv.sumArrowEquivProdArrow _ _ Bool).symm.sum_comp, Fintype.sum_prod_type] @@ -140,7 +133,8 @@ theorem eval_multilinearExtension_wiringPred (g : Gate) (i : ℕ) (z : Fin (C.wi by_cases hg : C.gate i a = g · rw [ite_eq_left hg] rw [Finset.sum_eq_single (C.left i a), Finset.sum_eq_single (C.right i a)] - · simp [hg, basis, Function.comp_def, mul_assoc] + · simp only [Function.comp_def, Equiv.sumArrowEquivProdArrow_symm_apply_inl, hg, + Equiv.sumArrowEquivProdArrow_symm_apply_inr, and_self, ↓reduceIte, one_mul, mul_assoc] congr 1 · intro c _ hc simp [Function.comp_def, Ne.symm hc] @@ -153,16 +147,6 @@ theorem eval_multilinearExtension_wiringPred (g : Gate) (i : ℕ) (z : Fin (C.wi refine Finset.sum_eq_zero fun b _ => Finset.sum_eq_zero fun c _ => ?_ simp [Function.comp_def, hg] -/-- The multilinear extension of `g` at `z` is the sum of the values of `g` weighted by the basis -polynomials. -/ -theorem eval_multilinearExtension_eq_sum {ι : Type*} [Fintype ι] [DecidableEq ι] - (g : (ι → Bool) → R) (z : ι → R) : - eval z (multilinearExtension g) = ∑ a, g a * basis a z := by - simp [multilinearExtension, eval_indicatorPoly_apply, basis] - -/-- The point of `R^k` with the coordinates of the bit string `b`. -/ -abbrev bitPoint {k : ℕ} (b : Fin k → Bool) : Fin k → R := fun j => bitVal R (b j) - /-- **The GKR identity** [Thaler, Lemma 4.7][Thaler2022]. The multilinear extension of the values of layer `i + 1` is a sum over the pairs of gates `b`, `c` of layer `i` of the multilinear extensions of the wiring predicates times the sum, respectively the product, of the values of `b` @@ -176,14 +160,14 @@ theorem eval_multilinearExtension_step (i : ℕ) (W : (Fin (C.width i) → Bool) eval (Sum.elim z (Sum.elim (bitPoint b) (bitPoint c))) (multilinearExtension (C.multPred (R := R) i)) * (W b * W c)) := by rw [eval_multilinearExtension_eq_sum (C.step i W)] - simp only [eval_multilinearExtension_wiringPred, basis_bitVal] - have hinner : ∀ b c, (∑ a, if C.gate i a = .add then basis a z * + simp only [eval_multilinearExtension_wiringPred, eval_indicatorPoly] + have hinner : ∀ b c, (∑ a, if C.gate i a = .add then eval z (indicatorPoly a) * (if C.left i a = b then (1 : R) else 0) * (if C.right i a = c then 1 else 0) else 0) * (W b + W c) + - (∑ a, if C.gate i a = .mul then basis a z * + (∑ a, if C.gate i a = .mul then eval z (indicatorPoly a) * (if C.left i a = b then (1 : R) else 0) * (if C.right i a = c then 1 else 0) else 0) * (W b * W c) = - ∑ a, (if C.left i a = b ∧ C.right i a = c then basis a z * + ∑ a, (if C.left i a = b ∧ C.right i a = c then eval z (indicatorPoly a) * (if C.gate i a = .add then W b + W c else W b * W c) else 0) := by intro b c rw [Finset.sum_mul, Finset.sum_mul, ← Finset.sum_add_distrib] diff --git a/Complexity/Foundations/BooleanAnalysis/MultilinearExtension.lean b/Complexity/Foundations/BooleanAnalysis/MultilinearExtension.lean index 5de569b8aa..56b616698d 100644 --- a/Complexity/Foundations/BooleanAnalysis/MultilinearExtension.lean +++ b/Complexity/Foundations/BooleanAnalysis/MultilinearExtension.lean @@ -28,6 +28,8 @@ Its total degree is at most the dimension. ## Main results * `Cslib.BitString.eval_multilinearExtension`: the extension agrees with the function on the cube. +* `Cslib.BitString.eval_multilinearExtension_eq_sum`: the extension at any point, in the indicator + basis. * `Cslib.BitString.totalDegree_multilinearExtension_le`: its total degree is at most the dimension. * `Cslib.BitString.rename_multilinearExtension`: renaming the coordinates along an equivalence @@ -74,6 +76,15 @@ theorem eval_indicatorPoly [Fintype ι] (a x : ι → Bool) : exact Finset.prod_eq_zero (Finset.mem_univ i) (by cases ha : a i <;> cases hx : x i <;> simp_all) +/-- The point of `R^ι` with the coordinates of the bit string `x`. -/ +abbrev bitPoint (x : ι → Bool) : ι → R := fun i => bitVal R (x i) + +/-- The multilinear extension at any point is the sum of the values of the function weighted by the +indicator polynomials. -/ +theorem eval_multilinearExtension_eq_sum [Fintype ι] [DecidableEq ι] (g : (ι → Bool) → R) + (z : ι → R) : eval z (multilinearExtension g) = ∑ a, g a * eval z (indicatorPoly a) := by + simp [multilinearExtension] + /-- The multilinear extension agrees with the function on the cube. -/ theorem eval_multilinearExtension [Fintype ι] [DecidableEq ι] (g : (ι → Bool) → R) (x : ι → Bool) : eval (fun i => bitVal R (x i)) (multilinearExtension g) = g x := by diff --git a/ComplexityTests/GKRLayered.lean b/ComplexityTests/GKRLayered.lean index 53a5cf501c..b166534c0a 100644 --- a/ComplexityTests/GKRLayered.lean +++ b/ComplexityTests/GKRLayered.lean @@ -31,9 +31,9 @@ example (x : (Fin 1 → Bool) → ℤ) : example (W : (Fin (mulTwo.width 0) → Bool) → ℤ) (z : Fin (mulTwo.width 1) → ℤ) : eval z (multilinearExtension (mulTwo.step 0 W)) = ∑ b, ∑ c, - (eval (Sum.elim z (Sum.elim (LayeredCircuit.bitPoint b) (LayeredCircuit.bitPoint c))) + (eval (Sum.elim z (Sum.elim (bitPoint b) (bitPoint c))) (multilinearExtension (mulTwo.addPred (R := ℤ) 0)) * (W b + W c) + - eval (Sum.elim z (Sum.elim (LayeredCircuit.bitPoint b) (LayeredCircuit.bitPoint c))) + eval (Sum.elim z (Sum.elim (bitPoint b) (bitPoint c))) (multilinearExtension (mulTwo.multPred (R := ℤ) 0)) * (W b * W c)) := mulTwo.eval_multilinearExtension_step 0 W z From 85a2dc2df181aab1d2f47adae28f8f6b4f3e9da9 Mon Sep 17 00:00:00 2001 From: Samuel Giuseppe Schlesinger Date: Sat, 24 Oct 2026 23:04:52 -0400 Subject: [PATCH 3/5] chore(bib): keep references.bib sorted by key (#953) --- references.bib | 4004 ++++++++++++++++++++++++------------------------ 1 file changed, 2002 insertions(+), 2002 deletions(-) diff --git a/references.bib b/references.bib index aef760910e..ff7ea22e21 100644 --- a/references.bib +++ b/references.bib @@ -49,6 +49,16 @@ @article{AFvMV2006 year = {2006} } +@article{AFvMV2006, + author = {Antunes, Lu{\'i}s and Fortnow, Lance and van Melkebeek, Dieter and + Vinodchandran, N. V.}, + title = {Computational Depth: Concept and Applications}, + journal = {Theoretical Computer Science}, + volume = {354}, + pages = {391--404}, + year = {2006} +} + @article{Ajtai1983, author = {Ajtai, Mikl{\'o}s}, title = {{$\Sigma^1_1$}-Formulae on Finite Structures}, @@ -125,6 +135,17 @@ @article{AllenderRubinstein1988 doi = {10.1137/0217075} } +@article{AllenderRubinstein1988, + author = {Allender, Eric W. and Rubinstein, Roy S.}, + title = {{P}-Printable Sets}, + journal = {SIAM Journal on Computing}, + volume = {17}, + number = {6}, + pages = {1193--1202}, + year = {1988}, + doi = {10.1137/0217075} +} + @article{AlonBoppana1987, author = {Alon, Noga and Boppana, Ravi B.}, title = {The Monotone Circuit Complexity of {B}oolean Functions}, @@ -147,6 +168,26 @@ @article{AlonGoldreichHastadPeralta1992 doi = {10.1002/rsa.3240030308} } +@article{AlonGoldreichHastadPeralta1992, + author = {Alon, Noga and Goldreich, Oded and H{\aa}stad, Johan and Peralta, Ren{\'e}}, + title = {Simple Constructions of Almost $k$-wise Independent Random Variables}, + journal = {Random Structures \& Algorithms}, + volume = {3}, + number = {3}, + pages = {289--304}, + year = {1992}, + doi = {10.1002/rsa.3240030308} +} + +@book{AlonSpencer2016, + author = {Alon, Noga and Spencer, Joel H.}, + title = {The Probabilistic Method}, + edition = {4}, + publisher = {Wiley}, + year = {2016}, + isbn = {978-1-119-06195-3} +} + @book{AlonSpencer2016, author = {Alon, Noga and Spencer, Joel H.}, title = {The Probabilistic Method}, @@ -204,6 +245,18 @@ @article{Andreev1989 note = {In Russian} } +@article{Andreev1989, + author = {Andreev, A. E.}, + title = {On the Complexity of the Realization of Partial {B}oolean Functions by Circuits of + Functional Elements}, + journal = {Diskretnaya Matematika}, + volume = {1}, + number = {4}, + pages = {36--45}, + year = {1989}, + note = {In Russian} +} + @article{Angluin1980, author = {Angluin, Dana}, title = {Inductive Inference of Formal Languages from Positive Data}, @@ -257,6 +310,29 @@ @article{AroraLundMotwaniSudanSzegedy1998 doi = {10.1145/278298.278306} } +@article{AroraLundMotwaniSudanSzegedy1998, + author = {Arora, Sanjeev and Lund, Carsten and Motwani, Rajeev and Sudan, Madhu + and Szegedy, Mario}, + title = {Proof Verification and the Hardness of Approximation Problems}, + journal = {Journal of the ACM}, + volume = {45}, + number = {3}, + pages = {501--555}, + year = {1998}, + doi = {10.1145/278298.278306} +} + +@article{AroraSafra1998, + author = {Arora, Sanjeev and Safra, Shmuel}, + title = {Probabilistic Checking of Proofs: A New Characterization of {NP}}, + journal = {Journal of the ACM}, + volume = {45}, + number = {1}, + pages = {70--122}, + year = {1998}, + doi = {10.1145/273865.273901} +} + @article{AroraSafra1998, author = {Arora, Sanjeev and Safra, Shmuel}, title = {Probabilistic Checking of Proofs: A New Characterization of {NP}}, @@ -297,6 +373,25 @@ @inproceedings{Babai1985 doi = {10.1145/22145.22192} } +@inproceedings{Babai1985, + author = {Babai, L{\'a}szl{\'o}}, + title = {Trading Group Theory for Randomness}, + booktitle = {Proceedings of the 17th Annual ACM Symposium on Theory of Computing}, + pages = {421--429}, + year = {1985}, + doi = {10.1145/22145.22192} +} + +@inproceedings{BabaiFortnowLevinSzegedy1991, + author = {Babai, L{\'a}szl{\'o} and Fortnow, Lance and Levin, Leonid A. and Szegedy, Mario}, + title = {Checking Computations in Polylogarithmic Time}, + booktitle = {Proceedings of the 23rd Annual ACM Symposium on Theory of Computing}, + series = {STOC 1991}, + pages = {21--32}, + year = {1991}, + doi = {10.1145/103418.103428} +} + @inproceedings{BabaiFortnowLevinSzegedy1991, author = {Babai, L{\'a}szl{\'o} and Fortnow, Lance and Levin, Leonid A. and Szegedy, Mario}, title = {Checking Computations in Polylogarithmic Time}, @@ -330,6 +425,29 @@ @article{BabaiFortnowNisanWigderson1993 doi = {10.1007/BF01275486} } +@article{BabaiFortnowNisanWigderson1993, + author = {Babai, L{\'a}szl{\'o} and Fortnow, Lance and Nisan, Noam and Wigderson, Avi}, + title = {{BPP} Has Subexponential Time Simulations Unless {EXPTIME} Has Publishable Proofs}, + journal = {Computational Complexity}, + volume = {3}, + number = {4}, + pages = {307--318}, + year = {1993}, + doi = {10.1007/BF01275486} +} + +@article{BabaiMoran1988, + author = {Babai, L{\'a}szl{\'o} and Moran, Shlomo}, + title = {{Arthur}--{Merlin} Games: A Randomized Proof System, and a Hierarchy of Complexity + Classes}, + journal = {Journal of Computer and System Sciences}, + volume = {36}, + number = {2}, + pages = {254--276}, + year = {1988}, + doi = {10.1016/0022-0000(88)90028-1} +} + @article{BabaiMoran1988, author = {Babai, L{\'a}szl{\'o} and Moran, Shlomo}, title = {{Arthur}--{Merlin} Games: A Randomized Proof System, and a Hierarchy of Complexity @@ -353,6 +471,17 @@ @article{BakerGillSolovay1975 doi = {10.1137/0204037} } +@article{BakerGillSolovay1975, + author = {Baker, Theodore and Gill, John and Solovay, Robert}, + title = {Relativizations of the {$\mathcal{P} =? \mathcal{NP}$} Question}, + journal = {SIAM Journal on Computing}, + volume = {4}, + number = {4}, + pages = {431--442}, + year = {1975}, + doi = {10.1137/0204037} +} + @misc{Barak2016PRG, author = {Barak, Boaz}, title = {{CS 127: Pseudorandomness}}, @@ -388,6 +517,28 @@ @article{BarringtonImmermanStraubing1990 doi = {10.1016/0022-0000(90)90022-D} } +@article{BarringtonImmermanStraubing1990, + author = {Barrington, David A. Mix and Immerman, Neil and Straubing, Howard}, + title = {On Uniformity within {NC}$^1$}, + journal = {Journal of Computer and System Sciences}, + volume = {41}, + number = {3}, + pages = {274--306}, + year = {1990}, + doi = {10.1016/0022-0000(90)90022-D} +} + +@article{BarringtonTherien1988, + author = {Barrington, David A. Mix and Th{\'e}rien, Denis}, + title = {Finite Monoids and the Fine Structure of {NC}$^1$}, + journal = {Journal of the ACM}, + volume = {35}, + number = {4}, + pages = {941--952}, + year = {1988}, + doi = {10.1145/48014.63138} +} + @article{BarringtonTherien1988, author = {Barrington, David A. Mix and Th{\'e}rien, Denis}, title = {Finite Monoids and the Fine Structure of {NC}$^1$}, @@ -423,6 +574,18 @@ @article{BealsBuhrmanCleveMoscaDeWolf2001 doi = {10.1145/502090.502097} } +@article{BealsBuhrmanCleveMoscaDeWolf2001, + author = {Beals, Robert and Buhrman, Harry and Cleve, Richard and Mosca, Michele and + de Wolf, Ronald}, + title = {Quantum Lower Bounds by Polynomials}, + journal = {Journal of the ACM}, + volume = {48}, + number = {4}, + pages = {778--797}, + year = {2001}, + doi = {10.1145/502090.502097} +} + @article{BeameEtAl1998, author = {Beame, Paul and Cook, Stephen and Edmonds, Jeff and Impagliazzo, Russell and Pitassi, Toniann}, title = {The Relative Complexity of {NP} Search Problems}, @@ -489,6 +652,16 @@ @inproceedings{BellareRompel1994 doi = {10.1109/SFCS.1994.365687} } +@inproceedings{BellareRompel1994, + author = {Bellare, Mihir and Rompel, John}, + title = {Randomness-Efficient Oblivious Sampling}, + booktitle = {Proceedings of the 35th Annual Symposium on Foundations of Computer Science}, + series = {FOCS 1994}, + pages = {276--287}, + year = {1994}, + doi = {10.1109/SFCS.1994.365687} +} + @inproceedings{BenOrGoldwasserKilianWigderson1988, author = {Ben-Or, Michael and Goldwasser, Shafi and Kilian, Joe and Wigderson, Avi}, title = {Multi-prover Interactive Proofs: How to Remove Intractability Assumptions}, @@ -525,6 +698,18 @@ @article{BenSassonKopparty2012 note = {Theorem numbers follow the ECCC report TR10-044 (2010)} } +@article{BenSassonKopparty2012, + author = {Ben-Sasson, Eli and Kopparty, Swastik}, + title = {Affine Dispersers from Subspace Polynomials}, + journal = {SIAM Journal on Computing}, + volume = {41}, + number = {4}, + pages = {880--914}, + year = {2012}, + doi = {10.1137/110826254}, + note = {Theorem numbers follow the ECCC report TR10-044 (2010)} +} + @article{BenSassonWigderson2001, author = {Ben-Sasson, Eli and Wigderson, Avi}, title = {Short Proofs Are Narrow---Resolution Made Simple}, @@ -549,6 +734,29 @@ @article{Berkowitz1984 doi = {10.1016/0020-0190(84)90018-8} } +@article{Berkowitz1984, + author = {Berkowitz, Stuart J.}, + title = {On Computing the Determinant in Small Parallel Time Using a Small Number + of Processors}, + journal = {Information Processing Letters}, + volume = {18}, + number = {3}, + pages = {147--150}, + year = {1984}, + doi = {10.1016/0020-0190(84)90018-8} +} + +@article{Bird2011, + author = {Bird, Richard S.}, + title = {A Simple Division-Free Algorithm for Computing Determinants}, + journal = {Information Processing Letters}, + volume = {111}, + number = {21--22}, + pages = {1072--1074}, + year = {2011}, + doi = {10.1016/j.ipl.2011.08.006} +} + @article{Bird2011, author = {Bird, Richard S.}, title = {A Simple Division-Free Algorithm for Computing Determinants}, @@ -581,6 +789,17 @@ @article{BlassGurevich1982 doi = {10.1016/S0019-9958(82)90439-9} } +@article{BlassGurevich1982, + author = {Blass, Andreas and Gurevich, Yuri}, + title = {On the Unique Satisfiability Problem}, + journal = {Information and Control}, + volume = {55}, + number = {1--3}, + pages = {80--88}, + year = {1982}, + doi = {10.1016/S0019-9958(82)90439-9} +} + @inproceedings{BlumImpagliazzo1987, author = {Blum, Manuel and Impagliazzo, Russell}, title = {Generic oracles and oracle classes}, @@ -600,6 +819,27 @@ @article{BlumLubyRubinfeld1993 doi = {10.1016/0022-0000(93)90044-W} } +@article{BlumLubyRubinfeld1993, + author = {Blum, Manuel and Luby, Michael and Rubinfeld, Ronitt}, + title = {Self-Testing/Correcting with Applications to Numerical Problems}, + journal = {Journal of Computer and System Sciences}, + volume = {47}, + number = {3}, + pages = {549--595}, + year = {1993}, + doi = {10.1016/0022-0000(93)90044-W} +} + +@article{BoehlerGlasserMeister2006, + author = {B{\"o}hler, Elmar and Gla{\ss}er, Christian and Meister, Daniel}, + title = {Error-bounded probabilistic computations between {MA} and {AM}}, + journal = {Journal of Computer and System Sciences}, + volume = {72}, + number = {6}, + pages = {1043--1076}, + year = {2006} +} + @article{BoehlerGlasserMeister2006, author = {B{\"o}hler, Elmar and Gla{\ss}er, Christian and Meister, Daniel}, title = {Error-bounded probabilistic computations between {MA} and {AM}}, @@ -621,6 +861,17 @@ @article{Bonami1970 doi = {10.5802/aif.357} } +@article{Bonami1970, + author = {Bonami, Aline}, + title = {{\'E}tude des coefficients de {Fourier} des fonctions de {$L^p(G)$}}, + journal = {Annales de l'Institut Fourier}, + volume = {20}, + number = {2}, + pages = {335--402}, + year = {1970}, + doi = {10.5802/aif.357} +} + @inproceedings{BonehHaleviHamburgOstrovsky2008, author = {Dan Boneh and Shai Halevi and Mike Hamburg and Rafail Ostrovsky}, title = {Circular-Secure Encryption from Decision Diffie-Hellman}, @@ -652,11 +903,22 @@ @article{Book1974 doi = {10.1016/S0022-0000(74)80008-5} } -@article{BoppanaHastadZachos1987, - author = {Boppana, Ravi B. and H{\aa}stad, Johan and Zachos, Stathis}, - title = {Does co-{NP} Have Short Interactive Proofs?}, - journal = {Information Processing Letters}, - volume = {25}, +@article{Book1974, + author = {Book, Ronald V.}, + title = {Comparing Complexity Classes}, + journal = {Journal of Computer and System Sciences}, + volume = {9}, + number = {2}, + pages = {213--229}, + year = {1974}, + doi = {10.1016/S0022-0000(74)80008-5} +} + +@article{BoppanaHastadZachos1987, + author = {Boppana, Ravi B. and H{\aa}stad, Johan and Zachos, Stathis}, + title = {Does co-{NP} Have Short Interactive Proofs?}, + journal = {Information Processing Letters}, + volume = {25}, number = {2}, pages = {127--132}, year = {1987}, @@ -674,6 +936,27 @@ @article{BorchertHemaspaandraRothe2000 url = {https://arxiv.org/abs/cs/9907041} } +@article{BorchertHemaspaandraRothe2000, + author = {Borchert, Bernd and Hemaspaandra, Lane A. and Rothe, J{\"o}rg}, + title = {Restrictive Acceptance Suffices for Equivalence Problems}, + journal = {LMS Journal of Computation and Mathematics}, + volume = {3}, + pages = {86--95}, + year = {2000}, + doi = {10.1112/S146115700000022X}, + url = {https://arxiv.org/abs/cs/9907041} +} + +@book{BorweinErdelyi1995, + author = {Borwein, Peter and Erd{\'e}lyi, Tam{\'a}s}, + title = {Polynomials and Polynomial Inequalities}, + series = {Graduate Texts in Mathematics}, + volume = {161}, + publisher = {Springer}, + year = {1995}, + doi = {10.1007/978-1-4612-0793-1} +} + @book{BorweinErdelyi1995, author = {Borwein, Peter and Erd{\'e}lyi, Tam{\'a}s}, title = {Polynomials and Polynomial Inequalities}, @@ -695,6 +978,26 @@ @article{Bourgain2007 doi = {10.1007/s00039-007-0593-z} } +@article{Bourgain2007, + author = {Bourgain, Jean}, + title = {On the Construction of Affine Extractors}, + journal = {Geometric and Functional Analysis}, + volume = {17}, + number = {1}, + pages = {33--57}, + year = {2007}, + doi = {10.1007/s00039-007-0593-z} +} + +@inproceedings{BovaCapelliMengelSlivovsky2016, + author = {Bova, Simone and Capelli, Florent and Mengel, Stefan and Slivovsky, Friedrich}, + title = {Knowledge Compilation Meets Communication Complexity}, + booktitle = {Proceedings of the Twenty-Fifth International Joint Conference on Artificial + Intelligence (IJCAI 2016)}, + pages = {1008--1014}, + year = {2016} +} + @inproceedings{BovaCapelliMengelSlivovsky2016, author = {Bova, Simone and Capelli, Florent and Mengel, Stefan and Slivovsky, Friedrich}, title = {Knowledge Compilation Meets Communication Complexity}, @@ -744,6 +1047,25 @@ @book{Burgisser2000 doi = {10.1007/978-3-662-04179-6} } +@book{Burgisser2000, + author = {B{\"u}rgisser, Peter}, + title = {Completeness and Reduction in Algebraic Complexity Theory}, + series = {Algorithms and Computation in Mathematics}, + volume = {7}, + publisher = {Springer}, + year = {2000}, + doi = {10.1007/978-3-662-04179-6} +} + +@book{BurgisserClausenShokrollahi1997, + author = {B{\"u}rgisser, Peter and Clausen, Michael and Shokrollahi, M. Amin}, + title = {Algebraic Complexity Theory}, + series = {Grundlehren der mathematischen Wissenschaften}, + publisher = {Springer}, + year = {1997}, + doi = {10.1007/978-3-662-03338-8} +} + @book{BurgisserClausenShokrollahi1997, author = {B{\"u}rgisser, Peter and Clausen, Michael and Shokrollahi, M. Amin}, title = {Algebraic Complexity Theory}, @@ -775,6 +1097,29 @@ @article{CaiChenDowneyFellows1997 doi = {10.1016/S0168-0072(96)00020-6} } +@article{CaiChenDowneyFellows1997, + author = {Cai, Liming and Chen, Jianer and Downey, Rodney G. and Fellows, Michael R.}, + title = {Advice classes of parameterized tractability}, + journal = {Annals of Pure and Applied Logic}, + volume = {84}, + number = {1}, + pages = {119--138}, + year = {1997}, + doi = {10.1016/S0168-0072(96)00020-6} +} + +@article{CaiEtAl1988, + author = {Cai, Jin-Yi and Gundermann, Thomas and Hartmanis, Juris and + Hemachandra, Lane A. and Sewelson, Vivian and Wagner, Klaus and Wechsung, Gerd}, + title = {The Boolean Hierarchy {I}: Structural Properties}, + journal = {SIAM Journal on Computing}, + volume = {17}, + number = {6}, + pages = {1232--1252}, + year = {1988}, + doi = {10.1137/0217078} +} + @article{CaiEtAl1988, author = {Cai, Jin-Yi and Gundermann, Thomas and Hartmanis, Juris and Hemachandra, Lane A. and Sewelson, Vivian and Wagner, Klaus and Wechsung, Gerd}, @@ -805,6 +1150,22 @@ @phdthesis{Calabro2009 year = {2009} } +@phdthesis{Calabro2009, + author = {Calabro, Chris}, + title = {The Exponential Complexity of Satisfiability Problems}, + school = {University of California, San Diego}, + year = {2009} +} + +@inproceedings{CalabroImpagliazzoPaturi2006, + author = {Calabro, Chris and Impagliazzo, Russell and Paturi, Ramamohan}, + title = {A Duality between Clause Width and Clause Density for {SAT}}, + booktitle = {21st Annual IEEE Conference on Computational Complexity (CCC 2006)}, + pages = {252--260}, + year = {2006}, + doi = {10.1109/CCC.2006.6} +} + @inproceedings{CalabroImpagliazzoPaturi2006, author = {Calabro, Chris and Impagliazzo, Russell and Paturi, Ramamohan}, title = {A Duality between Clause Width and Clause Density for {SAT}}, @@ -844,6 +1205,17 @@ @article{CarterWegman1979 doi = {10.1016/0022-0000(79)90044-8} } +@article{CarterWegman1979, + author = {Carter, J. Lawrence and Wegman, Mark N.}, + title = {Universal Classes of Hash Functions}, + journal = {Journal of Computer and System Sciences}, + volume = {18}, + number = {2}, + pages = {143--154}, + year = {1979}, + doi = {10.1016/0022-0000(79)90044-8} +} + @techreport{CavalarOliveira2025, author = {Bruno P. Cavalar and Igor C. Oliveira}, title = {Boolean Circuit Complexity and Two-Dimensional Cover Problems}, @@ -864,6 +1236,17 @@ @article{ChandraKozenStockmeyer1981 doi = {10.1145/322234.322243} } +@article{ChandraKozenStockmeyer1981, + author = {Chandra, Ashok K. and Kozen, Dexter C. and Stockmeyer, Larry J.}, + title = {Alternation}, + journal = {Journal of the ACM}, + volume = {28}, + number = {1}, + pages = {114--133}, + year = {1981}, + doi = {10.1145/322234.322243} +} + @article{ChandraStockmeyerVishkin1984, author = {Chandra, Ashok K. and Stockmeyer, Larry and Vishkin, Uzi}, title = {Constant Depth Reducibility}, @@ -907,6 +1290,29 @@ @article{Chashkin2024 note = {In Russian} } +@article{Chashkin2024, + author = {Chashkin, A. V.}, + title = {On Computing Partial {B}oolean Functions}, + journal = {Matematicheskie Voprosy Kibernetiki}, + volume = {22}, + pages = {152--222}, + year = {2024}, + publisher = {Fizmatlit}, + doi = {10.20948/mvk-2024-152}, + url = {https://library.keldysh.ru/mvk.asp?id=2024-152}, + note = {In Russian} +} + +@article{Chebyshev1852, + author = {Tchebichef, Pafnuty L.}, + title = {M{\'e}moire sur les nombres premiers}, + journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, + series = {1}, + volume = {17}, + pages = {366--390}, + year = {1852} +} + @article{Chebyshev1852, author = {Tchebichef, Pafnuty L.}, title = {M{\'e}moire sur les nombres premiers}, @@ -928,6 +1334,28 @@ @inproceedings{ChenKabanets2015 doi = {10.1007/978-3-319-21398-9_17} } +@inproceedings{ChenKabanets2015, + author = {Chen, Ruiwen and Kabanets, Valentine}, + title = {Correlation Bounds and {\#SAT} Algorithms for Small Linear-Size Circuits}, + booktitle = {Computing and Combinatorics (COCOON 2015)}, + series = {Lecture Notes in Computer Science}, + pages = {211--222}, + publisher = {Springer}, + year = {2015}, + doi = {10.1007/978-3-319-21398-9_17} +} + +@misc{CHLLTV2026, + author = {Chattopadhyay, Eshan and Hatami, Pooya and Lee, Chin Ho and Lovett, Shachar and + Tal, Avishay and Viola, Emanuele}, + title = {Exponential Correlation Bounds for Polynomials}, + year = {2026}, + eprint = {2609.28839}, + archivePrefix = {arXiv}, + primaryClass = {cs.CC}, + url = {https://arxiv.org/abs/2609.28839} +} + @misc{CHLLTV2026, author = {Chattopadhyay, Eshan and Hatami, Pooya and Lee, Chin Ho and Lovett, Shachar and Tal, Avishay and Viola, Emanuele}, @@ -951,6 +1379,29 @@ @article{ChorGoldreich1988 doi = {10.1137/0217015} } +@article{ChorGoldreich1988, + author = {Chor, Benny and Goldreich, Oded}, + title = {Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication + Complexity}, + journal = {SIAM Journal on Computing}, + volume = {17}, + number = {2}, + pages = {230--261}, + year = {1988}, + doi = {10.1137/0217015} +} + +@article{ChungGrahamFranklShearer1986, + author = {Chung, Fan R. K. and Graham, Ronald L. and Frankl, Peter and Shearer, James B.}, + title = {Some Intersection Theorems for Ordered Sets and Graphs}, + journal = {Journal of Combinatorial Theory, Series A}, + volume = {43}, + number = {1}, + pages = {23--37}, + year = {1986}, + doi = {10.1016/0097-3165(86)90019-1} +} + @article{ChungGrahamFranklShearer1986, author = {Chung, Fan R. K. and Graham, Ronald L. and Frankl, Peter and Shearer, James B.}, title = {Some Intersection Theorems for Ordered Sets and Graphs}, @@ -982,6 +1433,26 @@ @article{CMWBS2012 year = {2012} } +@article{CMWBS2012, + author = {Cook, Stephen and McKenzie, Pierre and Wehr, Dustin and Braverman, Mark and + Santhanam, Rahul}, + title = {Pebbles and Branching Programs for Tree Evaluation}, + journal = {ACM Transactions on Computation Theory}, + volume = {3}, + number = {2}, + pages = {4:1--4:43}, + year = {2012} +} + +@techreport{CookMertz2024, + author = {Cook, James and Mertz, Ian}, + title = {Tree Evaluation is in Space {$O(\log n \cdot \log\log n)$}}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR23-174}, + note = {Conference version in STOC 2024}, + year = {2023} +} + @techreport{CookMertz2024, author = {Cook, James and Mertz, Ian}, title = {Tree Evaluation is in Space {$O(\log n \cdot \log\log n)$}}, @@ -1002,6 +1473,28 @@ @article{CookReckhow1973 doi = {10.1016/S0022-0000(73)80029-7} } +@article{CookReckhow1973, + author = {Cook, Stephen A. and Reckhow, Robert A.}, + title = {Time Bounded Random Access Machines}, + journal = {Journal of Computer and System Sciences}, + volume = {7}, + number = {4}, + year = {1973}, + pages = {354--375}, + doi = {10.1016/S0022-0000(73)80029-7} +} + +@article{CookReckhow1979, + author = {Cook, Stephen A. and Reckhow, Robert A.}, + title = {The Relative Efficiency of Propositional Proof Systems}, + journal = {The Journal of Symbolic Logic}, + volume = {44}, + number = {1}, + pages = {36--50}, + year = {1979}, + doi = {10.2307/2273702} +} + @article{CookReckhow1979, author = {Cook, Stephen A. and Reckhow, Robert A.}, title = {The Relative Efficiency of Propositional Proof Systems}, @@ -1032,15 +1525,35 @@ @article{CoppersmithRivlin1992 doi = {10.1137/0523054} } -@book{CoverThomas2006, - author = {Cover, Thomas M. and Thomas, Joy A.}, - title = {Elements of Information Theory}, - edition = {2}, - publisher = {Wiley-Interscience}, - year = {2006}, - doi = {10.1002/047174882X} -} - +@article{CoppersmithRivlin1992, + author = {Coppersmith, Don and Rivlin, Theodore J.}, + title = {The Growth of Polynomials Bounded at Equally Spaced Points}, + journal = {SIAM Journal on Mathematical Analysis}, + volume = {23}, + number = {4}, + pages = {970--983}, + year = {1992}, + doi = {10.1137/0523054} +} + +@book{CoverThomas2006, + author = {Cover, Thomas M. and Thomas, Joy A.}, + title = {Elements of Information Theory}, + edition = {2}, + publisher = {Wiley-Interscience}, + year = {2006}, + doi = {10.1002/047174882X} +} + +@book{CoverThomas2006, + author = {Cover, Thomas M. and Thomas, Joy A.}, + title = {Elements of Information Theory}, + edition = {2}, + publisher = {Wiley-Interscience}, + year = {2006}, + doi = {10.1002/047174882X} +} + @inproceedings{CramerDamgardSchoenmakers1994, author = {Ronald Cramer and Ivan Damg{\aa}rd and Berry Schoenmakers}, title = {Proofs of Partial Knowledge and Simplified Design of Witness Hiding Protocols}, @@ -1072,6 +1585,16 @@ @book{CyganEtAl2015 doi = {10.1007/978-3-319-21275-3} } +@book{CyganEtAl2015, + author = {Cygan, Marek and Fomin, Fedor V. and Kowalik, {\L}ukasz and Lokshtanov, Daniel and + Marx, D{\'a}niel and Pilipczuk, Marcin and Pilipczuk, Micha{\l} and Saurabh, Saket}, + title = {Parameterized Algorithms}, + publisher = {Springer}, + address = {Cham}, + year = {2015}, + doi = {10.1007/978-3-319-21275-3} +} + @inproceedings{Danielsson2008, author = {Danielsson, Nils Anders}, title = {Lightweight semiformal time complexity analysis for purely functional data structures}, @@ -1101,6 +1624,17 @@ @article{Darwiche2001 doi = {10.1145/502090.502091} } +@article{Darwiche2001, + author = {Darwiche, Adnan}, + title = {Decomposable Negation Normal Form}, + journal = {Journal of the ACM}, + volume = {48}, + number = {4}, + pages = {608--647}, + year = {2001}, + doi = {10.1145/502090.502091} +} + @article{DavisLogemannLoveland1962, author = {Davis, Martin and Logemann, George and Loveland, Donald}, title = {A Machine Program for Theorem-Proving}, @@ -1124,6 +1658,17 @@ @article{DavisPutnam1960 doi = {10.1145/321033.321034} } +@article{DavisPutnam1960, + author = {Davis, Martin and Putnam, Hilary}, + title = {A Computing Procedure for Quantification Theory}, + journal = {Journal of the ACM}, + volume = {7}, + number = {3}, + pages = {201--215}, + year = {1960}, + doi = {10.1145/321033.321034} +} + @misc{Demaine2015VanEmdeBoas, author = {Demaine, Erik}, title = {Lecture 4: Divide and Conquer: van {Emde Boas} Trees}, @@ -1165,6 +1710,26 @@ @techreport{Dinur2005 year = {2005} } +@techreport{Dinur2005, + author = {Dinur, Irit}, + title = {The {PCP} Theorem by Gap Amplification}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR05-046}, + note = {Revision 1, 26 September 2005}, + year = {2005} +} + +@article{Dinur2007, + author = {Dinur, Irit}, + title = {The {PCP} Theorem by Gap Amplification}, + journal = {Journal of the ACM}, + volume = {54}, + number = {3}, + pages = {12}, + year = {2007}, + doi = {10.1145/1236457.1236459} +} + @article{Dinur2007, author = {Dinur, Irit}, title = {The {PCP} Theorem by Gap Amplification}, @@ -1186,6 +1751,16 @@ @book{DowneyFellows1999 doi = {10.1007/978-1-4612-0515-9} } +@book{DowneyFellows1999, + author = {Downey, Rodney G. and Fellows, Michael R.}, + title = {Parameterized Complexity}, + series = {Monographs in Computer Science}, + publisher = {Springer}, + address = {New York}, + year = {1999}, + doi = {10.1007/978-1-4612-0515-9} +} + @article{DucasEtAl2018, author = {Ducas, L{\'e}o and Kiltz, Eike and Lepoint, Tancr{\`e}de and Lyubashevsky, Vadim and Schwabe, Peter and Seiler, Gregor and Stehl{\'e}, Damien}, @@ -1235,6 +1810,28 @@ @article{EhlichZeller1964 doi = {10.1007/BF01111276} } +@article{EhlichZeller1964, + author = {Ehlich, Hartmut and Zeller, Karl}, + title = {Schwankung von {Polynomen} zwischen {Gitterpunkten}}, + journal = {Mathematische Zeitschrift}, + volume = {86}, + number = {1}, + pages = {41--44}, + year = {1964}, + doi = {10.1007/BF01111276} +} + +@incollection{ErdosLovasz1975, + author = {Erd{\H{o}}s, Paul and Lov{\'a}sz, L{\'a}szl{\'o}}, + title = {Problems and Results on 3-Chromatic Hypergraphs and Some Related Questions}, + booktitle = {Infinite and Finite Sets}, + series = {Colloquia Mathematica Societatis J{\'a}nos Bolyai}, + volume = {10}, + publisher = {North-Holland}, + pages = {609--627}, + year = {1975} +} + @incollection{ErdosLovasz1975, author = {Erd{\H{o}}s, Paul and Lov{\'a}sz, L{\'a}szl{\'o}}, title = {Problems and Results on 3-Chromatic Hypergraphs and Some Related Questions}, @@ -1268,6 +1865,17 @@ @article{ErdosSpencer1991 doi = {10.1016/0166-218X(91)90040-4} } +@article{ErdosSpencer1991, + author = {Erd{\H{o}}s, Paul and Spencer, Joel}, + title = {Lopsided {Lov{\'a}sz} Local Lemma and {Latin} Transversals}, + journal = {Discrete Applied Mathematics}, + volume = {30}, + number = {2--3}, + pages = {151--154}, + year = {1991}, + doi = {10.1016/0166-218X(91)90040-4} +} + @article{EricksonVanDerHoogMiltzow2024, author = {Erickson, Jeff and van der Hoog, Ivor and Miltzow, Tillmann}, title = {Smoothing the Gap Between {NP} and {$\exists\mathbb{R}$}}, @@ -1303,6 +1911,28 @@ @article{EvenSelmanYacobi1984 doi = {10.1016/S0019-9958(84)80056-X} } +@article{EvenSelmanYacobi1984, + author = {Even, Shimon and Selman, Alan L. and Yacobi, Yacov}, + title = {The Complexity of Promise Problems with Applications to Public-Key Cryptography}, + journal = {Information and Control}, + volume = {61}, + number = {2}, + pages = {159--173}, + year = {1984}, + doi = {10.1016/S0019-9958(84)80056-X} +} + +@incollection{Fagin1974, + author = {Fagin, Ronald}, + title = {Generalized First-Order Spectra and Polynomial-Time Recognizable Sets}, + booktitle = {Complexity of Computation}, + editor = {Karp, Richard M.}, + series = {SIAM-AMS Proceedings}, + volume = {7}, + pages = {43--73}, + year = {1974} +} + @incollection{Fagin1974, author = {Fagin, Ronald}, title = {Generalized First-Order Spectra and Polynomial-Time Recognizable Sets}, @@ -1338,6 +1968,30 @@ @article{Fenner2003 url = {https://eccc.weizmann.ac.il/report/2002/036/} } +@article{Fenner2003, + author = {Fenner, Stephen A.}, + title = {{PP}-Lowness and a Simple Definition of {AWPP}}, + journal = {Theory of Computing Systems}, + volume = {36}, + number = {2}, + pages = {199--212}, + year = {2003}, + doi = {10.1007/s00224-002-1089-8}, + url = {https://eccc.weizmann.ac.il/report/2002/036/} +} + +@article{FennerEtAl2003, + author = {Fenner, Stephen A. and Fortnow, Lance J. and Kurtz, Stuart A. and Li, Lide}, + title = {An Oracle Builder's Toolkit}, + journal = {Information and Computation}, + volume = {182}, + number = {2}, + pages = {95--136}, + year = {2003}, + doi = {10.1016/S0890-5401(03)00018-X}, + url = {https://lance.fortnow.com/papers/files/obt.pdf} +} + @article{FennerEtAl2003, author = {Fenner, Stephen A. and Fortnow, Lance J. and Kurtz, Stuart A. and Li, Lide}, title = {An Oracle Builder's Toolkit}, @@ -1361,6 +2015,17 @@ @article{FennerFortnowKurtz1994 doi = {10.1016/S0022-0000(05)80024-8} } +@article{FennerFortnowKurtz1994, + author = {Fenner, Stephen A. and Fortnow, Lance J. and Kurtz, Stuart A.}, + title = {Gap-Definable Counting Classes}, + journal = {Journal of Computer and System Sciences}, + volume = {48}, + number = {1}, + pages = {116--148}, + year = {1994}, + doi = {10.1016/S0022-0000(05)80024-8} +} + @inproceedings{FiatShamir1986, author = {Amos Fiat and Adi Shamir}, title = {How to Prove Yourself: Practical Solutions to Identification and Signature @@ -1386,6 +2051,18 @@ @inproceedings{FindGolovnevHirschKulikov2016 doi = {10.1109/FOCS.2016.19} } +@inproceedings{FindGolovnevHirschKulikov2016, + author = {Find, Magnus Gausdal and Golovnev, Alexander and Hirsch, Edward A. and + Kulikov, Alexander S.}, + title = {A Better-than-{$3n$} Lower Bound for the Circuit Complexity of an Explicit + Function}, + booktitle = {IEEE 57th Annual Symposium on Foundations of Computer Science}, + series = {FOCS 2016}, + pages = {89--98}, + year = {2016}, + doi = {10.1109/FOCS.2016.19} +} + @article{ FLP1985, author = {Fischer, Michael J. and Lynch, Nancy A. and Paterson, Michael S.}, title = {Impossibility of Distributed Consensus with One Faulty Process}, @@ -1414,6 +2091,23 @@ @book{FlumGrohe2006 doi = {10.1007/3-540-29953-X} } +@book{FlumGrohe2006, + author = {Flum, J{\"o}rg and Grohe, Martin}, + title = {Parameterized Complexity Theory}, + series = {Texts in Theoretical Computer Science. An EATCS Series}, + publisher = {Springer}, + address = {Berlin, Heidelberg}, + year = {2006}, + doi = {10.1007/3-540-29953-X} +} + +@book{Forney1966, + author = {Forney, Jr., G. David}, + title = {Concatenated Codes}, + publisher = {MIT Press}, + year = {1966} +} + @book{Forney1966, author = {Forney, Jr., G. David}, title = {Concatenated Codes}, @@ -1433,6 +2127,18 @@ @incollection{Fortnow1997 url = {https://lance.fortnow.com/papers/files/counting.pdf} } +@incollection{Fortnow1997, + author = {Fortnow, Lance}, + title = {Counting Complexity}, + booktitle = {Complexity Theory Retrospective II}, + editor = {Hemaspaandra, Lane A. and Selman, Alan L.}, + publisher = {Springer}, + pages = {81--107}, + year = {1997}, + doi = {10.1007/978-1-4612-1872-2_4}, + url = {https://lance.fortnow.com/papers/files/counting.pdf} +} + @article{FortnowReingold1996, author = {Fortnow, Lance and Reingold, Nick}, title = {{PP} Is Closed under Truth-Table Reductions}, @@ -1466,6 +2172,17 @@ @article{Friedgut1998 doi = {10.1007/PL00009809} } +@article{Friedgut1998, + author = {Friedgut, Ehud}, + title = {{Boolean} Functions With Low Average Sensitivity Depend On Few Coordinates}, + journal = {Combinatorica}, + volume = {18}, + number = {1}, + pages = {27--35}, + year = {1998}, + doi = {10.1007/PL00009809} +} + @article{FurstSaxeSipser1984, author = {Furst, Merrick and Saxe, James B. and Sipser, Michael}, title = {Parity, Circuits, and the Polynomial-Time Hierarchy}, @@ -1505,6 +2222,14 @@ @book{GareyJohnson1979 year = {1979} } +@book{GareyJohnson1979, + author = {Garey, Michael R. and Johnson, David S.}, + title = {Computers and Intractability: A Guide to the Theory of {NP}-Completeness}, + publisher = {W. H. Freeman}, + address = {San Francisco}, + year = {1979} +} + @inproceedings{Gentry2009, author = {Gentry, Craig}, title = {Fully Homomorphic Encryption Using Ideal Lattices}, @@ -1542,6 +2267,17 @@ @article{GGM1986 url = {https://www.wisdom.weizmann.ac.il/~oded/X/ggm-jacm.pdf} } +@article{GGM1986, + author = {Goldreich, Oded and Goldwasser, Shafi and Micali, Silvio}, + title = {How to Construct Random Functions}, + journal = {Journal of the ACM}, + volume = {33}, + number = {4}, + pages = {792--807}, + year = {1986}, + url = {https://www.wisdom.weizmann.ac.il/~oded/X/ggm-jacm.pdf} +} + @article{Gilbert1952, author = {Gilbert, E. N.}, title = {A Comparison of Signalling Alphabets}, @@ -1553,10 +2289,32 @@ @article{Gilbert1952 doi = {10.1002/j.1538-7305.1952.tb01393.x} } -@article{Gill1977, - author = {Gill, John}, - title = {Computational Complexity of Probabilistic {T}uring Machines}, - journal = {SIAM Journal on Computing}, +@article{Gilbert1952, + author = {Gilbert, E. N.}, + title = {A Comparison of Signalling Alphabets}, + journal = {Bell System Technical Journal}, + volume = {31}, + number = {3}, + pages = {504--522}, + year = {1952}, + doi = {10.1002/j.1538-7305.1952.tb01393.x} +} + +@article{Gill1977, + author = {Gill, John}, + title = {Computational Complexity of Probabilistic {T}uring Machines}, + journal = {SIAM Journal on Computing}, + volume = {6}, + number = {4}, + pages = {675--695}, + year = {1977}, + doi = {10.1137/0206049} +} + +@article{Gill1977, + author = {Gill, John}, + title = {Computational Complexity of Probabilistic {T}uring Machines}, + journal = {SIAM Journal on Computing}, volume = {6}, number = {4}, pages = {675--695}, @@ -1605,6 +2363,20 @@ @inproceedings{GKST2016 doi = {10.4230/LIPIcs.MFCS.2016.45} } +@inproceedings{GKST2016, + author = {Golovnev, Alexander and Kulikov, Alexander S. and Smal, Alexander V. and + Tamaki, Suguru}, + title = {Circuit Size Lower Bounds and {\#SAT} Upper Bounds Through a General Framework}, + booktitle = {41st International Symposium on Mathematical Foundations of Computer Science + (MFCS 2016)}, + series = {Leibniz International Proceedings in Informatics (LIPIcs)}, + volume = {58}, + pages = {45:1--45:16}, + publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, + year = {2016}, + doi = {10.4230/LIPIcs.MFCS.2016.45} +} + @book{Goldreich2001, author = {Goldreich, Oded}, title = {Foundations of Cryptography, Volume 1: Basic Tools}, @@ -1625,6 +2397,27 @@ @incollection{Goldreich2006 doi = {10.1007/11685654_12} } +@incollection{Goldreich2006, + author = {Goldreich, Oded}, + title = {On Promise Problems: A Survey}, + booktitle = {Theoretical Computer Science: Essays in Memory of {S}himon {E}ven}, + series = {Lecture Notes in Computer Science}, + volume = {3895}, + pages = {254--290}, + publisher = {Springer}, + year = {2006}, + doi = {10.1007/11685654_12} +} + +@techreport{Goldreich2024TreeEval, + author = {Goldreich, Oded}, + title = {On the {Cook}--{Mertz} Tree Evaluation procedure}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR24-109}, + note = {Revision 2, 27 December 2024}, + year = {2024} +} + @techreport{Goldreich2024TreeEval, author = {Goldreich, Oded}, title = {On the {Cook}--{Mertz} Tree Evaluation procedure}, @@ -1656,6 +2449,18 @@ @article{GoldreichMicaliWigderson1991 doi = {10.1145/116825.116852} } +@article{GoldreichMicaliWigderson1991, + author = {Goldreich, Oded and Micali, Silvio and Wigderson, Avi}, + title = {Proofs that Yield Nothing but Their Validity or All Languages in {NP} Have + Zero-Knowledge Proof Systems}, + journal = {Journal of the ACM}, + volume = {38}, + number = {3}, + pages = {690--728}, + year = {1991}, + doi = {10.1145/116825.116852} +} + @article{Goldschlager1977, author = {Goldschlager, Leslie M.}, title = {The Monotone and Planar Circuit Value Problems Are Log Space Complete for {$P$}}, @@ -1667,6 +2472,28 @@ @article{Goldschlager1977 doi = {10.1145/1008354.1008356} } +@article{GoldwasserKalaiRothblum2015, + author = {Goldwasser, Shafi and Kalai, Yael Tauman and Rothblum, Guy N.}, + title = {Delegating Computation: Interactive Proofs for Muggles}, + journal = {Journal of the ACM}, + volume = {62}, + number = {4}, + pages = {27:1--27:64}, + year = {2015}, + doi = {10.1145/2699436} +} + +@article{GoldwasserMicaliRackoff1989, + author = {Goldwasser, Shafi and Micali, Silvio and Rackoff, Charles}, + title = {The Knowledge Complexity of Interactive Proof Systems}, + journal = {SIAM Journal on Computing}, + volume = {18}, + number = {1}, + pages = {186--208}, + year = {1989}, + doi = {10.1137/0218012} +} + @article{GoldwasserMicaliRackoff1989, author = {Goldwasser, Shafi and Micali, Silvio and Rackoff, Charles}, title = {The Knowledge Complexity of Interactive Proof Systems}, @@ -1699,6 +2526,26 @@ @inproceedings{GoldwasserSipser1986 doi = {10.1145/12130.12137} } +@inproceedings{GoldwasserSipser1986, + author = {Goldwasser, Shafi and Sipser, Michael}, + title = {Private Coins versus Public Coins in Interactive Proof Systems}, + booktitle = {Proceedings of the 18th Annual ACM Symposium on Theory of Computing}, + pages = {59--68}, + year = {1986}, + doi = {10.1145/12130.12137} +} + +@article{GotsmanLinial1992, + author = {Gotsman, Craig and Linial, Nathan}, + title = {The Equivalence of Two Problems on the Cube}, + journal = {Journal of Combinatorial Theory, Series A}, + volume = {61}, + number = {1}, + pages = {142--146}, + year = {1992}, + doi = {10.1016/0097-3165(92)90060-8} +} + @article{GotsmanLinial1992, author = {Gotsman, Craig and Linial, Nathan}, title = {The Equivalence of Two Problems on the Cube}, @@ -1728,6 +2575,14 @@ @misc{GuruswamiRudraSudan2019 url = {https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/} } +@misc{GuruswamiRudraSudan2019, + author = {Guruswami, Venkatesan and Rudra, Atri and Sudan, Madhu}, + title = {Essential Coding Theory}, + note = {Draft book, 2019}, + year = {2019}, + url = {https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/} +} + @inproceedings{Hagerup1998, author = {Hagerup, Torben}, title = {Sorting and Searching on the Word {RAM}}, @@ -1749,6 +2604,15 @@ @techreport{HaitnerVadhan2017 url = {https://eccc.weizmann.ac.il/report/2017/084/} } +@techreport{HaitnerVadhan2017, + author = {Haitner, Iftach and Vadhan, Salil}, + title = {The Many Entropies in One-Way Functions}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR17-084}, + year = {2017}, + url = {https://eccc.weizmann.ac.il/report/2017/084/} +} + @article{Haken1985, author = {Haken, Armin}, title = {The Intractability of Resolution}, @@ -1771,6 +2635,27 @@ @article{Hamming1950 doi = {10.1002/j.1538-7305.1950.tb00463.x} } +@article{Hamming1950, + author = {Hamming, R. W.}, + title = {Error Detecting and Error Correcting Codes}, + journal = {The Bell System Technical Journal}, + volume = {29}, + number = {2}, + pages = {147--160}, + year = {1950}, + doi = {10.1002/j.1538-7305.1950.tb00463.x} +} + +@inproceedings{HartmanisSimon1974, + author = {Hartmanis, Juris and Simon, Janos}, + title = {On the Power of Multiplication in Random Access Machines}, + booktitle = {15th Annual Symposium on Switching and Automata Theory (SWAT 1974)}, + year = {1974}, + pages = {13--23}, + publisher = {IEEE}, + doi = {10.1109/SWAT.1974.20} +} + @inproceedings{HartmanisSimon1974, author = {Hartmanis, Juris and Simon, Janos}, title = {On the Power of Multiplication in Random Access Machines}, @@ -1826,6 +2711,19 @@ @inproceedings{HemaspaandraEtAl2018 doi = {10.4230/LIPIcs.MFCS.2018.51} } +@inproceedings{HemaspaandraEtAl2018, + author = {Hemaspaandra, Edith and Hemaspaandra, Lane A. and Spakowski, Holger and + Watanabe, Osamu}, + title = {The Robustness of {LWPP} and {WPP}, with an Application to Graph Reconstruction}, + booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science}, + series = {Leibniz International Proceedings in Informatics}, + volume = {117}, + pages = {51:1--51:14}, + year = {2018}, + publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, + doi = {10.4230/LIPIcs.MFCS.2018.51} +} + @article{ Hennessy1985, author = {Matthew Hennessy and Robin Milner}, @@ -1854,6 +2752,28 @@ @article{HILL1999 url = {https://doi.org/10.1137/S0097539793244708} } +@article{HILL1999, + author = {Håstad, Johan and Impagliazzo, Russell and Levin, Leonid A. and Luby, Michael}, + title = {A Pseudorandom Generator from Any One-Way Function}, + journal = {SIAM Journal on Computing}, + volume = {28}, + number = {4}, + pages = {1364--1396}, + year = {1999}, + doi = {10.1137/S0097539793244708}, + url = {https://doi.org/10.1137/S0097539793244708} +} + +@inproceedings{Hirahara2018, + author = {Hirahara, Shuichi}, + title = {Non-Black-Box Worst-Case to Average-Case Reductions within {NP}}, + booktitle = {2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)}, + publisher = {IEEE Computer Society}, + year = {2018}, + pages = {247--258}, + doi = {10.1109/FOCS.2018.00032} +} + @inproceedings{Hirahara2018, author = {Hirahara, Shuichi}, title = {Non-Black-Box Worst-Case to Average-Case Reductions within {NP}}, @@ -1888,6 +2808,29 @@ @article{HMPST1993 doi = {10.1016/0022-0000(93)90001-D} } +@article{HMPST1993, + author = {Hajnal, Andr{\'a}s and Maass, Wolfgang and Pudl{\'a}k, Pavel and + Szegedy, M{\'a}ri{\'o} and Tur{\'a}n, Gy{\"o}rgy}, + title = {Threshold Circuits of Bounded Depth}, + journal = {Journal of Computer and System Sciences}, + volume = {46}, + number = {2}, + pages = {129--154}, + year = {1993}, + doi = {10.1016/0022-0000(93)90001-D} +} + +@article{Hoeffding1963, + author = {Hoeffding, Wassily}, + title = {Probability Inequalities for Sums of Bounded Random Variables}, + journal = {Journal of the American Statistical Association}, + volume = {58}, + number = {301}, + pages = {13--30}, + year = {1963}, + doi = {10.1080/01621459.1963.10500830} +} + @article{Hoeffding1963, author = {Hoeffding, Wassily}, title = {Probability Inequalities for Sums of Bounded Random Variables}, @@ -1917,6 +2860,23 @@ @inproceedings{Holenstein2005 url = {https://crypto.ethz.ch/publications/files/Holens05.pdf} } +@inproceedings{Holenstein2005, + author = {Holenstein, Thomas}, + title = {Key Agreement from Weak Bit Agreement}, + booktitle = {Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing}, + year = {2005}, + url = {https://crypto.ethz.ch/publications/files/Holens05.pdf} +} + +@inproceedings{Holenstein2006, + author = {Holenstein, Thomas}, + title = {Pseudorandom Generators from One-Way Functions: A Simple Construction for + Any Hardness}, + booktitle = {Theory of Cryptography (TCC 2006)}, + year = {2006}, + url = {https://crypto.ethz.ch/publications/files/Holens06.pdf} +} + @inproceedings{Holenstein2006, author = {Holenstein, Thomas}, title = {Pseudorandom Generators from One-Way Functions: A Simple Construction for @@ -1937,6 +2897,17 @@ @article{HooryLinialWigderson2006 doi = {10.1090/S0273-0979-06-01126-8} } +@article{HooryLinialWigderson2006, + author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi}, + title = {Expander Graphs and Their Applications}, + journal = {Bulletin of the American Mathematical Society}, + volume = {43}, + number = {4}, + pages = {439--561}, + year = {2006}, + doi = {10.1090/S0273-0979-06-01126-8} +} + @book{ Hopcroft2006, author = {Hopcroft, John E. and Motwani, Rajeev and Ullman, Jeffrey D.}, title = {Introduction to Automata Theory, Languages, and Computation (3rd Edition)}, @@ -1957,6 +2928,17 @@ @article{Huang2019 doi = {10.4007/annals.2019.190.3.6} } +@article{Huang2019, + author = {Huang, Hao}, + title = {Induced Subgraphs of Hypercubes and a Proof of the Sensitivity Conjecture}, + journal = {Annals of Mathematics}, + volume = {190}, + number = {3}, + pages = {949--955}, + year = {2019}, + doi = {10.4007/annals.2019.190.3.6} +} + @inproceedings{Ilango2020, author = {Ilango, Rahul}, title = {Approaching {MCSP} from Above and Below: Hardness for a Conditional Variant and {$\mathsf{AC}^0[p]$}}, @@ -2035,17 +3017,49 @@ @inproceedings{Impagliazzo1995 doi = {10.1109/SFCS.1995.492584} } -@article{ImpagliazzoPaturi2001, - author = {Impagliazzo, Russell and Paturi, Ramamohan}, - title = {On the Complexity of {$k$-SAT}}, - journal = {Journal of Computer and System Sciences}, - volume = {62}, - number = {2}, +@inproceedings{Impagliazzo1995, + author = {Impagliazzo, Russell}, + title = {Hard-Core Distributions for Somewhat Hard Problems}, + booktitle = {Proceedings of the 36th Annual Symposium on Foundations of Computer Science}, + series = {FOCS 1995}, + pages = {538--545}, + year = {1995}, + doi = {10.1109/SFCS.1995.492584} +} + +@article{ImpagliazzoPaturi2001, + author = {Impagliazzo, Russell and Paturi, Ramamohan}, + title = {On the Complexity of {$k$-SAT}}, + journal = {Journal of Computer and System Sciences}, + volume = {62}, + number = {2}, pages = {367--375}, year = {2001}, doi = {10.1006/jcss.2000.1727} } +@article{ImpagliazzoPaturi2001, + author = {Impagliazzo, Russell and Paturi, Ramamohan}, + title = {On the Complexity of {$k$-SAT}}, + journal = {Journal of Computer and System Sciences}, + volume = {62}, + number = {2}, + pages = {367--375}, + year = {2001}, + doi = {10.1006/jcss.2000.1727} +} + +@article{ImpagliazzoPaturiZane2001, + author = {Impagliazzo, Russell and Paturi, Ramamohan and Zane, Francis}, + title = {Which Problems Have Strongly Exponential Complexity?}, + journal = {Journal of Computer and System Sciences}, + volume = {63}, + number = {4}, + pages = {512--530}, + year = {2001}, + doi = {10.1006/jcss.2001.1774} +} + @article{ImpagliazzoPaturiZane2001, author = {Impagliazzo, Russell and Paturi, Ramamohan and Zane, Francis}, title = {Which Problems Have Strongly Exponential Complexity?}, @@ -2067,6 +3081,27 @@ @inproceedings{ImpagliazzoWigderson1997 doi = {10.1145/258533.258590} } +@inproceedings{ImpagliazzoWigderson1997, + author = {Impagliazzo, Russell and Wigderson, Avi}, + title = {{P} = {BPP} if {E} Requires Exponential Circuits: Derandomizing the {XOR} Lemma}, + booktitle = {Proceedings of the 29th Annual ACM Symposium on Theory of Computing}, + series = {STOC 1997}, + pages = {220--229}, + year = {1997}, + doi = {10.1145/258533.258590} +} + +@article{Johnson1962, + author = {Johnson, Selmer M.}, + title = {A New Upper Bound for Error-Correcting Codes}, + journal = {IRE Transactions on Information Theory}, + volume = {8}, + number = {3}, + pages = {203--207}, + year = {1962}, + doi = {10.1109/TIT.1962.1057714} +} + @article{Johnson1962, author = {Johnson, Selmer M.}, title = {A New Upper Bound for Error-Correcting Codes}, @@ -2100,6 +3135,16 @@ @book{Jukna2011 doi = {10.1007/978-3-642-17364-6} } +@book{Jukna2011, + author = {Jukna, Stasys}, + title = {Extremal Combinatorics: With Applications in Computer Science}, + edition = {2}, + series = {Texts in Theoretical Computer Science. An EATCS Series}, + publisher = {Springer}, + year = {2011}, + doi = {10.1007/978-3-642-17364-6} +} + @book{Jukna2012, author = {Stasys Jukna}, title = {Boolean Function Complexity: Advances and Frontiers}, @@ -2122,6 +3167,17 @@ @article{Justesen1972 doi = {10.1109/TIT.1972.1054893} } +@article{Justesen1972, + author = {Justesen, J{\o}rn}, + title = {Class of Constructive Asymptotically Good Algebraic Codes}, + journal = {IEEE Transactions on Information Theory}, + volume = {18}, + number = {5}, + pages = {652--656}, + year = {1972}, + doi = {10.1109/TIT.1972.1054893} +} + @inproceedings{KabanetsCai2000, author = {Kabanets, Valentine and Cai, Jin-Yi}, title = {Circuit Minimization Problem}, @@ -2131,6 +3187,26 @@ @inproceedings{KabanetsCai2000 doi = {10.1145/335305.335314} } +@inproceedings{KabanetsCai2000, + author = {Kabanets, Valentine and Cai, Jin-Yi}, + title = {Circuit Minimization Problem}, + booktitle = {Proceedings of the Thirty-Second Annual {ACM} Symposium on Theory of Computing (STOC 2000)}, + pages = {73--79}, + year = {2000}, + doi = {10.1145/335305.335314} +} + +@article{Kadin1988, + author = {Kadin, Jim}, + title = {The Polynomial Time Hierarchy Collapses if the {Boolean} Hierarchy Collapses}, + journal = {SIAM Journal on Computing}, + volume = {17}, + number = {6}, + pages = {1263--1282}, + year = {1988}, + doi = {10.1137/0217080} +} + @article{Kadin1988, author = {Kadin, Jim}, title = {The Polynomial Time Hierarchy Collapses if the {Boolean} Hierarchy Collapses}, @@ -2164,6 +3240,16 @@ @inproceedings{KahnKalaiLinial1988 doi = {10.1109/SFCS.1988.21923} } +@inproceedings{KahnKalaiLinial1988, + author = {Kahn, Jeff and Kalai, Gil and Linial, Nathan}, + title = {The Influence of Variables on {Boolean} Functions}, + booktitle = {29th Annual Symposium on Foundations of Computer Science}, + publisher = {IEEE Computer Society}, + year = {1988}, + pages = {68--80}, + doi = {10.1109/SFCS.1988.21923} +} + @article{KalyanasundaramSchnitger1992, author = {Kalyanasundaram, Bala and Schnitger, Georg}, title = {The Probabilistic Communication Complexity of Set Intersection}, @@ -2186,6 +3272,17 @@ @article{Kannan1982 doi = {10.1016/S0019-9958(82)90382-5} } +@article{Kannan1982, + author = {Kannan, Ravi}, + title = {Circuit-Size Lower Bounds and Non-Reducibility to Sparse Sets}, + journal = {Information and Control}, + volume = {55}, + number = {1--3}, + pages = {40--56}, + year = {1982}, + doi = {10.1016/S0019-9958(82)90382-5} +} + @article{KarchmerWigderson1990, author = {Karchmer, Mauricio and Wigderson, Avi}, title = {Monotone circuits for connectivity require super-logarithmic depth}, @@ -2217,6 +3314,16 @@ @inproceedings{KarpLipton1980 doi = {10.1145/800141.804678} } +@inproceedings{KarpLipton1980, + author = {Karp, Richard M. and Lipton, Richard J.}, + title = {Some Connections between Nonuniform and Uniform Complexity Classes}, + booktitle = {Proceedings of the 12th Annual ACM Symposium on Theory of Computing}, + series = {STOC 1980}, + pages = {302--309}, + year = {1980}, + doi = {10.1145/800141.804678} +} + @book{ KatzLindell2020, author = {Jonathan Katz and Yehuda Lindell}, @@ -2247,6 +3354,17 @@ @inproceedings{Khot2002 doi = {10.1145/509907.510017} } +@inproceedings{Khot2002, + author = {Khot, Subhash}, + title = {On the Power of Unique 2-Prover 1-Round Games}, + booktitle = {Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing}, + series = {STOC '02}, + pages = {767--775}, + publisher = {ACM}, + year = {2002}, + doi = {10.1145/509907.510017} +} + @article{Khrapchenko1971, author = {Khrapchenko, V. M.}, title = {Complexity of the realization of a linear function in the class of $\Pi$-circuits}, @@ -2323,6 +3441,25 @@ @techreport{Korten2026 year = {2026} } +@techreport{Korten2026, + author = {Korten, Oliver}, + title = {Top-Down Lower Bounds for All Depths}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR26-221}, + year = {2026} +} + +@article{KushilevitzMansour1993, + author = {Kushilevitz, Eyal and Mansour, Yishay}, + title = {Learning Decision Trees Using the {F}ourier Spectrum}, + journal = {SIAM Journal on Computing}, + volume = {22}, + number = {6}, + pages = {1331--1348}, + year = {1993}, + doi = {10.1137/0222080} +} + @article{KushilevitzMansour1993, author = {Kushilevitz, Eyal and Mansour, Yishay}, title = {Learning Decision Trees Using the {F}ourier Spectrum}, @@ -2352,6 +3489,17 @@ @article{Ladner1975 doi = {10.1145/990518.990519} } +@article{Ladner1975, + author = {Ladner, Richard E.}, + title = {The Circuit Value Problem Is Log Space Complete for {$P$}}, + journal = {ACM SIGACT News}, + volume = {7}, + number = {1}, + pages = {18--20}, + year = {1975}, + doi = {10.1145/990518.990519} +} + @article{LadnerFischer1980, author = {Ladner, Richard E. and Fischer, Michael J.}, title = {Parallel Prefix Computation}, @@ -2396,6 +3544,17 @@ @article{Lautemann1983 doi = {10.1016/0020-0190(83)90044-3} } +@article{Lautemann1983, + author = {Lautemann, Clemens}, + title = {{BPP} and the Polynomial Hierarchy}, + journal = {Information Processing Letters}, + volume = {17}, + number = {4}, + pages = {215--217}, + year = {1983}, + doi = {10.1016/0020-0190(83)90044-3} +} + @misc{LeanCanny2026, author = {{lean-canny contributors}}, title = {lean-canny}, @@ -2412,6 +3571,14 @@ @misc{LecomteRamakrishnan2026 archivePrefix = {arXiv} } +@misc{LecomteRamakrishnan2026, + author = {Lecomte, Victor and Ramakrishnan, Prasanna}, + title = {Optimal Shallow Circuits for Majority}, + year = {2026}, + eprint = {2609.34029}, + archivePrefix = {arXiv} +} + @article{Levin1973, author = {Levin, Leonid A.}, title = {Universal Sequential Search Problems}, @@ -2444,6 +3611,27 @@ @article{LewisPapadimitriou1982 doi = {10.1016/0304-3975(82)90058-5} } +@article{LewisPapadimitriou1982, + author = {Lewis, Harry R. and Papadimitriou, Christos H.}, + title = {Symmetric Space-Bounded Computation}, + journal = {Theoretical Computer Science}, + volume = {19}, + number = {2}, + pages = {161--187}, + year = {1982}, + doi = {10.1016/0304-3975(82)90058-5} +} + +@inproceedings{Li2023, + author = {Li, Xin}, + title = {Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More}, + booktitle = {Proceedings of the 64th Annual IEEE Symposium on Foundations of Computer Science}, + series = {FOCS 2023}, + pages = {1271--1281}, + year = {2023}, + doi = {10.1109/FOCS57990.2023.00075} +} + @inproceedings{Li2023, author = {Li, Xin}, title = {Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More}, @@ -2476,6 +3664,28 @@ @book{LidlNiederreiter1997 doi = {10.1017/CBO9780511525926} } +@book{LidlNiederreiter1997, + author = {Lidl, Rudolf and Niederreiter, Harald}, + title = {Finite Fields}, + edition = {2}, + series = {Encyclopedia of Mathematics and its Applications}, + volume = {20}, + publisher = {Cambridge University Press}, + year = {1997}, + doi = {10.1017/CBO9780511525926} +} + +@article{LinialMansourNisan1993, + author = {Linial, Nathan and Mansour, Yishay and Nisan, Noam}, + title = {Constant Depth Circuits, {F}ourier Transform, and Learnability}, + journal = {Journal of the ACM}, + volume = {40}, + number = {3}, + pages = {607--620}, + year = {1993}, + doi = {10.1145/174130.174138} +} + @article{LinialMansourNisan1993, author = {Linial, Nathan and Mansour, Yishay and Nisan, Noam}, title = {Constant Depth Circuits, {F}ourier Transform, and Learnability}, @@ -2496,6 +3706,15 @@ @inproceedings{LiptonYoung1994 doi = {10.1145/195058.195447} } +@inproceedings{LiptonYoung1994, + author = {Lipton, Richard J. and Young, Neal E.}, + title = {Simple Strategies for Large Zero-Sum Games with Applications to Complexity Theory}, + booktitle = {Proceedings of the Twenty-Sixth Annual {ACM} Symposium on Theory of Computing (STOC 1994)}, + pages = {734--740}, + year = {1994}, + doi = {10.1145/195058.195447} +} + @inproceedings{LiuPass2020, author = {Liu, Yanyi and Pass, Rafael}, title = {On One-way Functions and {Kolmogorov} Complexity}, @@ -2529,6 +3748,17 @@ @book{LiVitanyi2019 doi = {10.1007/978-3-030-11298-1} } +@book{LiVitanyi2019, + author = {Li, Ming and Vit{\'a}nyi, Paul}, + title = {An Introduction to {K}olmogorov Complexity and Its Applications}, + edition = {4th}, + series = {Texts in Computer Science}, + publisher = {Springer}, + address = {Cham}, + year = {2019}, + doi = {10.1007/978-3-030-11298-1} +} + @inproceedings{LiYang2022, author = {Li, Jiatu and Yang, Tianqi}, title = {{$3.1n - o(n)$} Circuit Lower Bounds for Explicit Functions}, @@ -2539,8 +3769,27 @@ @inproceedings{LiYang2022 doi = {10.1145/3519935.3519976} } -@article{LokshtanovMarxSaurabh2011, - author = {Lokshtanov, Daniel and Marx, D{\'a}niel and Saurabh, Saket}, +@inproceedings{LiYang2022, + author = {Li, Jiatu and Yang, Tianqi}, + title = {{$3.1n - o(n)$} Circuit Lower Bounds for Explicit Functions}, + booktitle = {Proceedings of the 54th Annual ACM Symposium on Theory of Computing}, + series = {STOC 2022}, + pages = {1180--1193}, + year = {2022}, + doi = {10.1145/3519935.3519976} +} + +@article{LokshtanovMarxSaurabh2011, + author = {Lokshtanov, Daniel and Marx, D{\'a}niel and Saurabh, Saket}, + title = {Lower bounds based on the {Exponential Time Hypothesis}}, + journal = {Bulletin of the EATCS}, + volume = {105}, + pages = {41--72}, + year = {2011} +} + +@article{LokshtanovMarxSaurabh2011, + author = {Lokshtanov, Daniel and Marx, D{\'a}niel and Saurabh, Saket}, title = {Lower bounds based on the {Exponential Time Hypothesis}}, journal = {Bulletin of the EATCS}, volume = {105}, @@ -2570,6 +3819,27 @@ @article{LoomisWhitney1949 doi = {10.1090/S0002-9904-1949-09320-5} } +@article{LoomisWhitney1949, + author = {Loomis, Lynn H. and Whitney, Hassler}, + title = {An Inequality Related to the Isoperimetric Inequality}, + journal = {Bulletin of the American Mathematical Society}, + volume = {55}, + number = {10}, + pages = {961--962}, + year = {1949}, + doi = {10.1090/S0002-9904-1949-09320-5} +} + +@inproceedings{LovaszSaks1988, + author = {Lov{\'a}sz, L{\'a}szl{\'o} and Saks, Michael}, + title = {Lattices, {M{\"o}bius} Functions and Communication Complexity}, + booktitle = {Proceedings of the 29th Annual Symposium on Foundations of Computer Science}, + series = {FOCS 1988}, + pages = {81--90}, + year = {1988}, + doi = {10.1109/SFCS.1988.21924} +} + @inproceedings{LovaszSaks1988, author = {Lov{\'a}sz, L{\'a}szl{\'o} and Saks, Michael}, title = {Lattices, {M{\"o}bius} Functions and Communication Complexity}, @@ -2602,6 +3872,17 @@ @article{LundFortnowKarloffNisan1992 doi = {10.1145/146585.146605} } +@article{LundFortnowKarloffNisan1992, + author = {Lund, Carsten and Fortnow, Lance and Karloff, Howard and Nisan, Noam}, + title = {Algebraic Methods for Interactive Proof Systems}, + journal = {Journal of the ACM}, + volume = {39}, + number = {4}, + pages = {859--868}, + year = {1992}, + doi = {10.1145/146585.146605} +} + @article{Lupanov1958, author = {Oleg B. Lupanov}, title = {On a Method of Circuit Synthesis}, @@ -2624,6 +3905,16 @@ @article{Lupanov1965 note = {In Russian} } +@article{Lupanov1965, + author = {Lupanov, O. B.}, + title = {On an Approach to the Synthesis of Control Systems: the Principle of Local Coding}, + journal = {Problemy Kibernetiki}, + volume = {14}, + pages = {31--110}, + year = {1965}, + note = {In Russian} +} + @inproceedings{Lyubashevsky2009, author = {Lyubashevsky, Vadim}, title = {Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based @@ -2658,6 +3949,15 @@ @book{MacWilliamsSloane1977 year = {1977} } +@book{MacWilliamsSloane1977, + author = {MacWilliams, F. J. and Sloane, N. J. A.}, + title = {The Theory of Error-Correcting Codes}, + series = {North-Holland Mathematical Library}, + volume = {16}, + publisher = {North-Holland}, + year = {1977} +} + @misc{ Malkin2024, author = {Tal Malkin}, title = {{COMS W3261: Computer Science Theory, Handout 3: The Myhill-Nerode Theorem and Implications}}, @@ -2680,6 +3980,19 @@ @incollection{McDiarmid1989 doi = {10.1017/CBO9781107359949.008} } +@incollection{McDiarmid1989, + author = {McDiarmid, Colin}, + title = {On the Method of Bounded Differences}, + booktitle = {Surveys in Combinatorics, 1989}, + editor = {Siemons, J.}, + series = {London Mathematical Society Lecture Note Series}, + volume = {141}, + pages = {148--188}, + publisher = {Cambridge University Press}, + year = {1989}, + doi = {10.1017/CBO9781107359949.008} +} + @article{MegiddoPapadimitriou1991, author = {Megiddo, Nimrod and Papadimitriou, Christos H.}, title = {On Total Functions, Existence Theorems and Computational Complexity}, @@ -2702,6 +4015,16 @@ @inproceedings{MehlhornSchmidt1982 doi = {10.1145/800070.802208} } +@inproceedings{MehlhornSchmidt1982, + author = {Mehlhorn, Kurt and Schmidt, Erik M.}, + title = {{Las Vegas} Is Better than Determinism in {VLSI} and Distributed Computing}, + booktitle = {Proceedings of the 14th Annual ACM Symposium on Theory of Computing}, + series = {STOC 1982}, + pages = {330--337}, + year = {1982}, + doi = {10.1145/800070.802208} +} + @book{ MilewskiDao, author = {Milewski, Bartosz}, title = {The Dao of Functional Programming}, @@ -2732,6 +4055,22 @@ @book{MinskyPapert1969 year = {1969} } +@book{MinskyPapert1969, + author = {Minsky, Marvin and Papert, Seymour}, + title = {Perceptrons: An Introduction to Computational Geometry}, + publisher = {MIT Press}, + year = {1969} +} + +@techreport{MishraSanthanam2026, + author = {Mishra, Om Swostik and Santhanam, Rahul}, + title = {An Explicit {$3.5n - o(n)$} Size Lower Bound against General {Boolean} Circuits}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR26-260}, + year = {2026}, + url = {https://eccc.weizmann.ac.il/report/2026/260/} +} + @techreport{MishraSanthanam2026, author = {Mishra, Om Swostik and Santhanam, Rahul}, title = {An Explicit {$3.5n - o(n)$} Size Lower Bound against General {Boolean} Circuits}, @@ -2778,6 +4117,15 @@ @book{MitzenmacherUpfal2017 year = {2017} } +@book{MitzenmacherUpfal2017, + author = {Mitzenmacher, Michael and Upfal, Eli}, + title = {Probability and Computing: Randomization and Probabilistic Techniques in + Algorithms and Data Analysis}, + edition = {2}, + publisher = {Cambridge University Press}, + year = {2017} +} + @Book{ Montesi2023, title = {Introduction to {Choreographies}}, author = {Montesi, Fabrizio}, @@ -2833,6 +4181,28 @@ @article{MoserTardos2010 doi = {10.1145/1667053.1667060} } +@article{MoserTardos2010, + author = {Moser, Robin A. and Tardos, G{\'a}bor}, + title = {A Constructive Proof of the General {Lov{\'a}sz} Local Lemma}, + journal = {Journal of the ACM}, + volume = {57}, + number = {2}, + pages = {11:1--11:15}, + year = {2010}, + doi = {10.1145/1667053.1667060} +} + +@article{Muller1954, + author = {Muller, David E.}, + title = {Application of {Boolean} Algebra to Switching Circuit Design and to Error Detection}, + journal = {Transactions of the I.R.E. Professional Group on Electronic Computers}, + volume = {EC-3}, + number = {3}, + pages = {6--12}, + year = {1954}, + doi = {10.1109/IREPGELC.1954.6499441} +} + @article{Muller1954, author = {Muller, David E.}, title = {Application of {Boolean} Algebra to Switching Circuit Design and to Error Detection}, @@ -2855,6 +4225,17 @@ @article{MulmuleyVaziraniVazirani1987 doi = {10.1007/BF02579206} } +@article{MulmuleyVaziraniVazirani1987, + author = {Mulmuley, Ketan and Vazirani, Umesh V. and Vazirani, Vijay V.}, + title = {Matching Is as Easy as Matrix Inversion}, + journal = {Combinatorica}, + volume = {7}, + number = {1}, + pages = {105--113}, + year = {1987}, + doi = {10.1007/BF02579206} +} + @article{Naor1991, author = {Naor, Moni}, title = {Bit Commitment Using Pseudorandomness}, @@ -2878,6 +4259,29 @@ @article{NaorNaor1993 doi = {10.1137/0222053} } +@article{NaorNaor1993, + author = {Naor, Joseph and Naor, Moni}, + title = {Small-Bias Probability Spaces: Efficient Constructions and Applications}, + journal = {SIAM Journal on Computing}, + volume = {22}, + number = {4}, + pages = {838--856}, + year = {1993}, + doi = {10.1137/0222053} +} + +@article{Nechiporuk1965, + author = {Nechiporuk, E. I.}, + title = {On the Complexity of Gating Circuits Realizing {B}oolean Matrices with Undefined + Elements}, + journal = {Doklady Akademii Nauk SSSR}, + volume = {163}, + number = {1}, + pages = {40--42}, + year = {1965}, + note = {In Russian} +} + @article{Nechiporuk1965, author = {Nechiporuk, E. I.}, title = {On the Complexity of Gating Circuits Realizing {B}oolean Matrices with Undefined @@ -2922,6 +4326,17 @@ @article{Newman1991 doi = {10.1016/0020-0190(91)90157-D} } +@article{Newman1991, + author = {Newman, Ilan}, + title = {Private vs. Common Random Bits in Communication Complexity}, + journal = {Information Processing Letters}, + volume = {39}, + number = {2}, + pages = {67--71}, + year = {1991}, + doi = {10.1016/0020-0190(91)90157-D} +} + @article{ Nipkow2001, title = {More {Church-Rosser} Proofs (in {Isabelle/HOL})}, author = {Nipkow, Tobias}, @@ -2953,6 +4368,28 @@ @article{NisanSzegedy1994 doi = {10.1007/BF01263419} } +@article{NisanSzegedy1994, + author = {Nisan, Noam and Szegedy, Mario}, + title = {On the Degree of {Boolean} Functions as Real Polynomials}, + journal = {Computational Complexity}, + volume = {4}, + number = {4}, + pages = {301--313}, + year = {1994}, + doi = {10.1007/BF01263419} +} + +@article{NisanWigderson1994, + author = {Nisan, Noam and Wigderson, Avi}, + title = {Hardness vs Randomness}, + journal = {Journal of Computer and System Sciences}, + volume = {49}, + number = {2}, + pages = {149--167}, + year = {1994}, + doi = {10.1016/S0022-0000(05)80043-1} +} + @article{NisanWigderson1994, author = {Nisan, Noam and Wigderson, Avi}, title = {Hardness vs Randomness}, @@ -3015,6 +4452,18 @@ @inproceedings{OliveiraPichSanthanam2019 doi = {10.4230/LIPIcs.CCC.2019.27} } +@inproceedings{OliveiraPichSanthanam2019, + author = {Oliveira, Igor Carboni and Pich, J{\'a}n and Santhanam, Rahul}, + title = {Hardness Magnification near State-Of-The-Art Lower Bounds}, + booktitle = {34th Computational Complexity Conference (CCC 2019)}, + series = {Leibniz International Proceedings in Informatics (LIPIcs)}, + volume = {137}, + pages = {27:1--27:29}, + publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, + year = {2019}, + doi = {10.4230/LIPIcs.CCC.2019.27} +} + @misc{OpenAI2026RationalHitting, author = {{OpenAI}}, title = {{Rational} matrix hitting lists for noncommutative rational formulas}, @@ -3031,6 +4480,25 @@ @misc{OpenAI2026Sensitivity url = {https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Sensitivity} } +@misc{OpenAI2026Sensitivity, + author = {{OpenAI}}, + title = {{Sensitivity} versus block sensitivity: a superquadratic separation}, + year = {2026}, + note = {Lean formalization, revision adc7f1241b42e322a6451854ab7e4b4c146bf78a}, + url = {https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Sensitivity} +} + +@article{Ore1933, + author = {Ore, Oystein}, + title = {On a Special Class of Polynomials}, + journal = {Transactions of the American Mathematical Society}, + volume = {35}, + number = {3}, + pages = {559--584}, + year = {1933}, + doi = {10.1090/S0002-9947-1933-1501703-0} +} + @article{Ore1933, author = {Ore, Oystein}, title = {On a Special Class of Polynomials}, @@ -3053,6 +4521,17 @@ @article{Oum2005 doi = {10.1016/j.jctb.2005.03.003} } +@article{Oum2005, + author = {Oum, Sang-il}, + title = {Rank-width and Vertex-minors}, + journal = {Journal of Combinatorial Theory, Series B}, + volume = {95}, + number = {1}, + pages = {79--100}, + year = {2005}, + doi = {10.1016/j.jctb.2005.03.003} +} + @article{Papadimitriou1994PPAD, author = {Papadimitriou, Christos H.}, title = {On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence}, @@ -3097,11 +4576,20 @@ @article{PaturiPudlakZane1999 year = {1999} } -@book{ Pierce2002, - author = {Benjamin C. Pierce}, - title = {Types and programming languages}, - publisher = {{MIT} Press}, - year = {2002}, +@article{PaturiPudlakZane1999, + author = {Paturi, Ramamohan and Pudl{\'a}k, Pavel and Zane, Francis}, + title = {Satisfiability Coding Lemma}, + journal = {Chicago Journal of Theoretical Computer Science}, + volume = {1999}, + number = {11}, + year = {1999} +} + +@book{ Pierce2002, + author = {Benjamin C. Pierce}, + title = {Types and programming languages}, + publisher = {{MIT} Press}, + year = {2002}, isbn = {978-0-262-16209-8} } @@ -3115,6 +4603,16 @@ @inproceedings{Radhakrishnan2006 doi = {10.1007/11786986_10} } +@inproceedings{Radhakrishnan2006, + author = {Radhakrishnan, Jaikumar}, + title = {Gap Amplification in {PCP}s Using Lazy Random Walks}, + booktitle = {Automata, Languages and Programming (ICALP 2006)}, + series = {Lecture Notes in Computer Science}, + pages = {96--107}, + year = {2006}, + doi = {10.1007/11786986_10} +} + @article{Razborov1985, author = {Razborov, Alexander A.}, title = {Lower Bounds on the Monotone Complexity of Some {B}oolean Functions}, @@ -3156,6 +4654,17 @@ @article{Razborov1992 doi = {10.1016/0304-3975(92)90260-M} } +@article{Razborov1992, + author = {Razborov, Alexander A.}, + title = {On the Distributional Complexity of Disjointness}, + journal = {Theoretical Computer Science}, + volume = {106}, + number = {2}, + pages = {385--390}, + year = {1992}, + doi = {10.1016/0304-3975(92)90260-M} +} + @incollection{Razborov1995, author = {Razborov, Alexander A.}, title = {Bounded Arithmetic and Lower Bounds in {B}oolean Complexity}, @@ -3179,6 +4688,17 @@ @article{RazborovRudich1997 doi = {10.1006/jcss.1997.1494} } +@article{RazborovRudich1997, + author = {Razborov, Alexander A. and Rudich, Steven}, + title = {Natural Proofs}, + journal = {Journal of Computer and System Sciences}, + volume = {55}, + number = {1}, + pages = {24--35}, + year = {1997}, + doi = {10.1006/jcss.1997.1494} +} + @article{RazWigderson1992, author = {Raz, Ran and Wigderson, Avi}, title = {Monotone Circuits for Matching Require Linear Depth}, @@ -3201,6 +4721,28 @@ @article{Reed1954 doi = {10.1109/TIT.1954.1057465} } +@article{Reed1954, + author = {Reed, Irving S.}, + title = {A Class of Multiple-Error-Correcting Codes and the Decoding Scheme}, + journal = {Transactions of the IRE Professional Group on Information Theory}, + volume = {4}, + number = {4}, + pages = {38--49}, + year = {1954}, + doi = {10.1109/TIT.1954.1057465} +} + +@article{ReedSolomon1960, + author = {Reed, Irving S. and Solomon, Gustave}, + title = {Polynomial Codes Over Certain Finite Fields}, + journal = {Journal of the Society for Industrial and Applied Mathematics}, + volume = {8}, + number = {2}, + pages = {300--304}, + year = {1960}, + doi = {10.1137/0108018} +} + @article{ReedSolomon1960, author = {Reed, Irving S. and Solomon, Gustave}, title = {Polynomial Codes Over Certain Finite Fields}, @@ -3223,6 +4765,28 @@ @article{Reingold2008 doi = {10.1145/1391289.1391291} } +@article{Reingold2008, + author = {Reingold, Omer}, + title = {Undirected Connectivity in Log-Space}, + journal = {Journal of the ACM}, + volume = {55}, + number = {4}, + pages = {17:1--17:24}, + year = {2008}, + doi = {10.1145/1391289.1391291} +} + +@article{ReingoldVadhanWigderson2002, + author = {Reingold, Omer and Vadhan, Salil and Wigderson, Avi}, + title = {Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders}, + journal = {Annals of Mathematics}, + volume = {155}, + number = {1}, + pages = {157--187}, + year = {2002}, + doi = {10.2307/3062153} +} + @article{ReingoldVadhanWigderson2002, author = {Reingold, Omer and Vadhan, Salil and Wigderson, Avi}, title = {Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders}, @@ -3254,6 +4818,25 @@ @article{Rice1953 doi = {10.1090/S0002-9947-1953-0053041-6} } +@article{Rice1953, + author = {Rice, Henry Gordon}, + title = {Classes of Recursively Enumerable Sets and Their Decision Problems}, + journal = {Transactions of the American Mathematical Society}, + volume = {74}, + number = {2}, + pages = {358--366}, + year = {1953}, + doi = {10.1090/S0002-9947-1953-0053041-6} +} + +@book{Rivlin1990, + author = {Rivlin, Theodore J.}, + title = {Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory}, + edition = {2}, + publisher = {Wiley}, + year = {1990} +} + @book{Rivlin1990, author = {Rivlin, Theodore J.}, title = {Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory}, @@ -3296,6 +4879,17 @@ @article{Rubinstein1995 doi = {10.1007/BF01200762} } +@article{Rubinstein1995, + author = {Rubinstein, David}, + title = {Sensitivity vs. Block Sensitivity of {Boolean} Functions}, + journal = {Combinatorica}, + volume = {15}, + number = {2}, + pages = {297--299}, + year = {1995}, + doi = {10.1007/BF01200762} +} + @Book{ Sangiorgi2011, location = {Cambridge}, title = {Introduction to Bisimulation and Coinduction}, @@ -3386,6 +4980,14 @@ @misc{Schlesinger2026Sensitivity url = {https://github.com/SamuelSchlesinger/complexitylib-v1/tree/add0aca8ee0d71e68969f2cb45cf91554a50f787/Complexitylib/BooleanAnalysis/Sensitivity} } +@misc{Schlesinger2026Sensitivity, + author = {Schlesinger, Samuel}, + title = {{Sensitivity} separation in {Complexitylib}}, + year = {2026}, + note = {Source of ported Lean proofs, revision add0aca8ee0d71e68969f2cb45cf91554a50f787}, + url = {https://github.com/SamuelSchlesinger/complexitylib-v1/tree/add0aca8ee0d71e68969f2cb45cf91554a50f787/Complexitylib/BooleanAnalysis/Sensitivity} +} + @article{Schnorr1974, author = {Schnorr, Claus-Peter}, title = {Zwei lineare untere {S}chranken f{\"u}r die {K}omplexit{\"a}t {B}oolescher {F}unktionen}, @@ -3452,1989 +5054,84 @@ @article{ Shamir1979 doi = {10.1145/359168.359176} } -@article{BarringtonImmermanStraubing1990, - author = {Barrington, David A. Mix and Immerman, Neil and Straubing, Howard}, - title = {On Uniformity within {NC}$^1$}, - journal = {Journal of Computer and System Sciences}, - volume = {41}, - number = {3}, - pages = {274--306}, - year = {1990}, - doi = {10.1016/0022-0000(90)90022-D} +@article{Shamir1992, + author = {Shamir, Adi}, + title = {{IP} = {PSPACE}}, + journal = {Journal of the ACM}, + volume = {39}, + number = {4}, + pages = {869--877}, + year = {1992}, + doi = {10.1145/146585.146609} } -@article{Hamming1950, - author = {Hamming, R. W.}, - title = {Error Detecting and Error Correcting Codes}, - journal = {The Bell System Technical Journal}, - volume = {29}, - number = {2}, - pages = {147--160}, - year = {1950}, - doi = {10.1002/j.1538-7305.1950.tb00463.x} +@article{Shannon1949, + author = {Claude E. Shannon}, + title = {The Synthesis of Two-Terminal Switching Circuits}, + journal = {Bell System Technical Journal}, + volume = {28}, + number = {1}, + pages = {59--98}, + year = {1949}, + doi = {10.1002/j.1538-7305.1949.tb03624.x}, + url = {https://deeplearning.cs.cmu.edu/F24/document/readings/Shannon49.pdf} } -@book{MacWilliamsSloane1977, - author = {MacWilliams, F. J. and Sloane, N. J. A.}, - title = {The Theory of Error-Correcting Codes}, - series = {North-Holland Mathematical Library}, - volume = {16}, - publisher = {North-Holland}, - year = {1977} +@article{Shen1992, + author = {Shen, A.}, + title = {{IP} = {SPACE}: Simplified Proof}, + journal = {Journal of the ACM}, + volume = {39}, + number = {4}, + pages = {878--880}, + year = {1992}, + doi = {10.1145/146585.146613} } -@article{DavisPutnam1960, - author = {Davis, Martin and Putnam, Hilary}, - title = {A Computing Procedure for Quantification Theory}, - journal = {Journal of the ACM}, - volume = {7}, - number = {3}, - pages = {201--215}, - year = {1960}, - doi = {10.1145/321033.321034} +@article{ ShepherdsonSturgis1963, + author = {Shepherdson, J. C. and Sturgis, H. E.}, + title = {Computability of Recursive Functions}, + journal = {Journal of the ACM}, + volume = {10}, + number = {2}, + year = {1963}, + pages = {217--255}, + doi = {10.1145/321160.321170}, + publisher = {Association for Computing Machinery}, + address = {New York, NY, USA} } -@article{LinialMansourNisan1993, - author = {Linial, Nathan and Mansour, Yishay and Nisan, Noam}, - title = {Constant Depth Circuits, {F}ourier Transform, and Learnability}, - journal = {Journal of the ACM}, - volume = {40}, - number = {3}, - pages = {607--620}, - year = {1993}, - doi = {10.1145/174130.174138} +@article{Sheppard1899, + author = {Sheppard, W. F.}, + title = {On the Application of the Theory of Error to Cases of Normal Distribution and Normal + Correlation}, + journal = {Philosophical Transactions of the Royal Society of London. Series A}, + volume = {192}, + year = {1899}, + pages = {101--167}, + doi = {10.1098/rsta.1899.0003} } -@article{PaturiPudlakZane1999, - author = {Paturi, Ramamohan and Pudl{\'a}k, Pavel and Zane, Francis}, - title = {Satisfiability Coding Lemma}, - journal = {Chicago Journal of Theoretical Computer Science}, - volume = {1999}, - number = {11}, - year = {1999} +@article{Sheppard1899, + author = {Sheppard, W. F.}, + title = {On the Application of the Theory of Error to Cases of Normal Distribution and Normal + Correlation}, + journal = {Philosophical Transactions of the Royal Society of London. Series A}, + volume = {192}, + year = {1899}, + pages = {101--167}, + doi = {10.1098/rsta.1899.0003} } -@article{ImpagliazzoPaturiZane2001, - author = {Impagliazzo, Russell and Paturi, Ramamohan and Zane, Francis}, - title = {Which Problems Have Strongly Exponential Complexity?}, - journal = {Journal of Computer and System Sciences}, - volume = {63}, - number = {4}, - pages = {512--530}, - year = {2001}, - doi = {10.1006/jcss.2001.1774} -} - -@inproceedings{CalabroImpagliazzoPaturi2006, - author = {Calabro, Chris and Impagliazzo, Russell and Paturi, Ramamohan}, - title = {A Duality between Clause Width and Clause Density for {SAT}}, - booktitle = {21st Annual IEEE Conference on Computational Complexity (CCC 2006)}, - pages = {252--260}, - year = {2006}, - doi = {10.1109/CCC.2006.6} -} - -@phdthesis{Calabro2009, - author = {Calabro, Chris}, - title = {The Exponential Complexity of Satisfiability Problems}, - school = {University of California, San Diego}, - year = {2009} -} - -@article{Gilbert1952, - author = {Gilbert, E. 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An EATCS Series}, - publisher = {Springer}, - address = {Berlin, Heidelberg}, - year = {2006}, - doi = {10.1007/3-540-29953-X} -} - -@article{CaiChenDowneyFellows1997, - author = {Cai, Liming and Chen, Jianer and Downey, Rodney G. and Fellows, Michael R.}, - title = {Advice classes of parameterized tractability}, - journal = {Annals of Pure and Applied Logic}, - volume = {84}, - number = {1}, - pages = {119--138}, - year = {1997}, - doi = {10.1016/S0168-0072(96)00020-6} -} - -@article{LokshtanovMarxSaurabh2011, - author = {Lokshtanov, Daniel and Marx, D{\'a}niel and Saurabh, Saket}, - title = {Lower bounds based on the {Exponential Time Hypothesis}}, - journal = {Bulletin of the EATCS}, - volume = {105}, - pages = {41--72}, - year = {2011} -} - -@article{Reingold2008, - author = {Reingold, Omer}, - title = {Undirected Connectivity in Log-Space}, - journal = {Journal of the ACM}, - volume = {55}, - number = {4}, - pages = {17:1--17:24}, - year = {2008}, - doi = {10.1145/1391289.1391291} -} - -@article{LewisPapadimitriou1982, - author = {Lewis, Harry R. and Papadimitriou, Christos H.}, - title = {Symmetric Space-Bounded Computation}, - journal = {Theoretical Computer Science}, - volume = {19}, - number = {2}, - pages = {161--187}, - year = {1982}, - doi = {10.1016/0304-3975(82)90058-5} -} - -@inproceedings{KahnKalaiLinial1988, - author = {Kahn, Jeff and Kalai, Gil and Linial, Nathan}, - title = {The Influence of Variables on {Boolean} Functions}, - booktitle = {29th Annual Symposium on Foundations of Computer Science}, - publisher = {IEEE Computer Society}, - year = {1988}, - pages = {68--80}, - doi = {10.1109/SFCS.1988.21923} -} - -@article{Talagrand1994, - author = {Talagrand, Michel}, - title = {On {Russo}'s Approximate Zero-One Law}, - journal = {The Annals of Probability}, - volume = {22}, - number = {3}, - pages = {1576--1587}, - year = {1994}, - doi = {10.1214/aop/1176988612} -} - -@article{Friedgut1998, - author = {Friedgut, Ehud}, - title = {{Boolean} Functions With Low Average Sensitivity Depend On Few Coordinates}, - journal = {Combinatorica}, - volume = {18}, - number = {1}, - pages = {27--35}, - year = {1998}, - doi = {10.1007/PL00009809} -} - -@article{LundFortnowKarloffNisan1992, - author = {Lund, Carsten and Fortnow, Lance and Karloff, Howard and Nisan, Noam}, - title = {Algebraic Methods for Interactive Proof Systems}, - journal = {Journal of the ACM}, - volume = {39}, - number = {4}, - pages = {859--868}, - year = {1992}, - doi = {10.1145/146585.146605} -} - -@article{Shamir1992, - author = {Shamir, Adi}, - title = {{IP} = {PSPACE}}, - journal = {Journal of the ACM}, - volume = {39}, - number = {4}, - pages = {869--877}, - year = {1992}, - doi = {10.1145/146585.146609} -} - -@article{Shannon1949, - author = {Claude E. Shannon}, - title = {The Synthesis of Two-Terminal Switching Circuits}, - journal = {Bell System Technical Journal}, - volume = {28}, - number = {1}, - pages = {59--98}, - year = {1949}, - doi = {10.1002/j.1538-7305.1949.tb03624.x}, - url = {https://deeplearning.cs.cmu.edu/F24/document/readings/Shannon49.pdf} -} - -@article{Shen1992, - author = {Shen, A.}, - title = {{IP} = {SPACE}: Simplified Proof}, - journal = {Journal of the ACM}, - volume = {39}, - number = {4}, - pages = {878--880}, - year = {1992}, - doi = {10.1145/146585.146613} -} - -@article{ ShepherdsonSturgis1963, - author = {Shepherdson, J. C. and Sturgis, H. E.}, - title = {Computability of Recursive Functions}, - journal = {Journal of the ACM}, - volume = {10}, - number = {2}, - year = {1963}, - pages = {217--255}, - doi = {10.1145/321160.321170}, - publisher = {Association for Computing Machinery}, - address = {New York, NY, USA} -} - -@article{Sheppard1899, - author = {Sheppard, W. F.}, - title = {On the Application of the Theory of Error to Cases of Normal Distribution and Normal - Correlation}, - journal = {Philosophical Transactions of the Royal Society of London. Series A}, - volume = {192}, - year = {1899}, - pages = {101--167}, - doi = {10.1098/rsta.1899.0003} +@article{Sholomov1969, + author = {Sholomov, L. A.}, + title = {On the Realization of Incompletely Defined {B}oolean Functions by Circuits of + Functional Elements}, + journal = {Problemy Kibernetiki}, + volume = {21}, + pages = {215--226}, + year = {1969}, + note = {In Russian} } @article{Sholomov1969, @@ -5466,6 +5163,28 @@ @article{Singleton1964 doi = {10.1109/TIT.1964.1053661} } +@article{Singleton1964, + author = {Singleton, Richard C.}, + title = {Maximum Distance $q$-Nary Codes}, + journal = {IEEE Transactions on Information Theory}, + volume = {10}, + number = {2}, + pages = {116--118}, + year = {1964}, + doi = {10.1109/TIT.1964.1053661} +} + +@article{Sion1958, + author = {Sion, Maurice}, + title = {On General Minimax Theorems}, + journal = {Pacific Journal of Mathematics}, + volume = {8}, + number = {1}, + pages = {171--176}, + year = {1958}, + doi = {10.2140/pjm.1958.8.171} +} + @article{Sion1958, author = {Sion, Maurice}, title = {On General Minimax Theorems}, @@ -5487,6 +5206,16 @@ @inproceedings{Sipser1983 url = {https://doi.org/10.1145/800061.808762} } +@inproceedings{Sipser1983, + author = {Sipser, Michael}, + title = {A Complexity Theoretic Approach to Randomness}, + booktitle = {Proceedings of the 15th Annual ACM Symposium on Theory of Computing}, + pages = {330--335}, + year = {1983}, + doi = {10.1145/800061.808762}, + url = {https://doi.org/10.1145/800061.808762} +} + @book{Sipser2013, author = {Sipser, Michael}, title = {Introduction to the Theory of Computation}, @@ -5506,6 +5235,17 @@ @article{SkyumValiant1985 doi = {10.1145/3149.3158} } +@article{SkyumValiant1985, + author = {Skyum, Sven and Valiant, Leslie G.}, + title = {A Complexity Theory Based on {B}oolean Algebra}, + journal = {Journal of the ACM}, + volume = {32}, + number = {2}, + pages = {484--502}, + year = {1985}, + doi = {10.1145/3149.3158} +} + @inproceedings{Smolensky1987, author = {Smolensky, Roman}, title = {Algebraic Methods in the Theory of Lower Bounds for {B}oolean Circuit Complexity}, @@ -5547,6 +5287,28 @@ @inproceedings{StearnsHartmanisLewis1965 doi = {10.1109/FOCS.1965.11} } +@inproceedings{StearnsHartmanisLewis1965, + author = {Stearns, Richard E. and Hartmanis, Juris and Lewis, II, Philip M.}, + title = {Hierarchies of Memory Limited Computations}, + booktitle = {Proceedings of the 6th Annual Symposium on Switching Circuit Theory and Logical + Design}, + series = {SWCT 1965}, + pages = {179--190}, + year = {1965}, + doi = {10.1109/FOCS.1965.11} +} + +@article{Stockmeyer1976, + author = {Stockmeyer, Larry J.}, + title = {The Polynomial-Time Hierarchy}, + journal = {Theoretical Computer Science}, + volume = {3}, + number = {1}, + pages = {1--22}, + year = {1976}, + doi = {10.1016/0304-3975(76)90061-X} +} + @article{Stockmeyer1976, author = {Stockmeyer, Larry J.}, title = {The Polynomial-Time Hierarchy}, @@ -5566,6 +5328,23 @@ @inproceedings{Stockmeyer1983 year = {1983} } +@inproceedings{Stockmeyer1983, + author = {Stockmeyer, Larry J.}, + title = {The Complexity of Approximate Counting}, + booktitle = {Proceedings of the 15th Annual ACM Symposium on Theory of Computing}, + pages = {118--126}, + year = {1983} +} + +@inproceedings{StockmeyerMeyer1973, + author = {Stockmeyer, Larry J. and Meyer, Albert R.}, + title = {Word Problems Requiring Exponential Time (Preliminary Report)}, + booktitle = {Proceedings of the Fifth Annual {ACM} Symposium on Theory of Computing (STOC 1973)}, + pages = {1--9}, + year = {1973}, + doi = {10.1145/800125.804029} +} + @inproceedings{StockmeyerMeyer1973, author = {Stockmeyer, Larry J. and Meyer, Albert R.}, title = {Word Problems Requiring Exponential Time (Preliminary Report)}, @@ -5596,6 +5375,28 @@ @article{Sudan1997 doi = {10.1006/jcom.1997.0439} } +@article{Sudan1997, + author = {Sudan, Madhu}, + title = {Decoding of {Reed Solomon} Codes beyond the Error-Correction Bound}, + journal = {Journal of Complexity}, + volume = {13}, + number = {1}, + pages = {180--193}, + year = {1997}, + doi = {10.1006/jcom.1997.0439} +} + +@article{SudanTrevisanVadhan2001, + author = {Sudan, Madhu and Trevisan, Luca and Vadhan, Salil}, + title = {Pseudorandom Generators without the {XOR} Lemma}, + journal = {Journal of Computer and System Sciences}, + volume = {62}, + number = {2}, + pages = {236--266}, + year = {2001}, + doi = {10.1006/jcss.2000.1730} +} + @article{SudanTrevisanVadhan2001, author = {Sudan, Madhu and Trevisan, Luca and Vadhan, Salil}, title = {Pseudorandom Generators without the {XOR} Lemma}, @@ -5627,6 +5428,17 @@ @article{Talagrand1994 doi = {10.1214/aop/1176988612} } +@article{Talagrand1994, + author = {Talagrand, Michel}, + title = {On {Russo}'s Approximate Zero-One Law}, + journal = {The Annals of Probability}, + volume = {22}, + number = {3}, + pages = {1576--1587}, + year = {1994}, + doi = {10.1214/aop/1176988612} +} + @article{Tardos1989, author = {Tardos, G{\'a}bor}, title = {Query complexity, or why is it difficult to separate {NP}$^A \cap$ co{NP}$^A$ from @@ -5638,6 +5450,17 @@ @article{Tardos1989 year = {1989} } +@article{Thaler2022, + author = {Thaler, Justin}, + title = {Proofs, Arguments, and Zero-Knowledge}, + journal = {Foundations and Trends in Privacy and Security}, + volume = {4}, + number = {2--4}, + pages = {117--660}, + year = {2022}, + doi = {10.1561/3300000030} +} + @incollection{ Thomas1990, author = {Wolfgang Thomas}, editor = {Jan van Leeuwen}, @@ -5657,6 +5480,15 @@ @misc{ThresholdCircuitRepresentations2026 url = {https://github.com/SamuelSchlesinger/threshold-circuits/blob/791fa2749317fbcc7b5c5b030a1e6fadd28cc44c/threshold_circuit_representations.tex} } +@misc{ThresholdCircuitRepresentations2026, + title = {Threshold Circuits as Decision Trees and Finite-State Programs}, + year = {2026}, + howpublished = {Research draft, second pass}, + note = {October 18, 2026; repository revision + 791fa2749317fbcc7b5c5b030a1e6fadd28cc44c}, + url = {https://github.com/SamuelSchlesinger/threshold-circuits/blob/791fa2749317fbcc7b5c5b030a1e6fadd28cc44c/threshold_circuit_representations.tex} +} + @article{Toda1991, author = {Toda, Seinosuke}, title = {{PP} Is as Hard as the Polynomial-Time Hierarchy}, @@ -5721,6 +5553,17 @@ @article{Turing1936 doi = {10.1112/plms/s2-42.1.230} } +@article{Turing1936, + author = {Turing, Alan M.}, + title = {On Computable Numbers, with an Application to the {Entscheidungsproblem}}, + journal = {Proceedings of the London Mathematical Society}, + volume = {s2-42}, + number = {1}, + pages = {230--265}, + year = {1936}, + doi = {10.1112/plms/s2-42.1.230} +} + @incollection{Uhlig1992, author = {Uhlig, Dietmar}, title = {Networks Computing {Boolean} Functions for Multiple Input Values}, @@ -5758,6 +5601,17 @@ @article{Vadhan2012 doi = {10.1561/0400000010} } +@article{Vadhan2012, + author = {Vadhan, Salil P.}, + title = {Pseudorandomness}, + journal = {Foundations and Trends in Theoretical Computer Science}, + volume = {7}, + number = {1--3}, + pages = {1--336}, + year = {2012}, + doi = {10.1561/0400000010} +} + @inproceedings{Valiant1977, author = {Valiant, Leslie G.}, title = {Graph-Theoretic Arguments in Low-Level Complexity}, @@ -5781,6 +5635,28 @@ @article{Valiant1979 doi = {10.1016/0304-3975(79)90044-6} } +@article{Valiant1979, + author = {Valiant, Leslie G.}, + title = {The Complexity of Computing the Permanent}, + journal = {Theoretical Computer Science}, + volume = {8}, + number = {2}, + pages = {189--201}, + year = {1979}, + doi = {10.1016/0304-3975(79)90044-6} +} + +@inproceedings{Valiant1979Algebra, + author = {Valiant, Leslie G.}, + title = {Completeness Classes in Algebra}, + booktitle = {Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing}, + series = {STOC '79}, + pages = {249--261}, + publisher = {ACM}, + year = {1979}, + doi = {10.1145/800135.804419} +} + @inproceedings{Valiant1979Algebra, author = {Valiant, Leslie G.}, title = {Completeness Classes in Algebra}, @@ -5816,6 +5692,16 @@ @article{ValiantVazirani1986 doi = {10.1016/0304-3975(86)90135-0} } +@article{ValiantVazirani1986, + author = {Valiant, Leslie G. and Vazirani, Vijay V.}, + title = {{NP} is as Easy as Detecting Unique Solutions}, + journal = {Theoretical Computer Science}, + volume = {47}, + pages = {85--93}, + year = {1986}, + doi = {10.1016/0304-3975(86)90135-0} +} + @article{VanEmdeBoasKaasZijlstra1977, author = {van Emde Boas, Peter and Kaas, R. and Zijlstra, E.}, title = {Design and Implementation of an Efficient Priority Queue}, @@ -5846,6 +5732,22 @@ @article{Varshamov1957 year = {1957} } +@article{Varshamov1957, + author = {Varshamov, R. R.}, + title = {Estimate of the Number of Signals in Error Correcting Codes}, + journal = {Doklady Akademii Nauk SSSR}, + volume = {117}, + pages = {739--741}, + year = {1957} +} + +@phdthesis{Vazirani1986, + author = {Vazirani, Umesh V.}, + title = {Randomness, Adversaries and Computation}, + school = {University of California, Berkeley}, + year = {1986} +} + @phdthesis{Vazirani1986, author = {Vazirani, Umesh V.}, title = {Randomness, Adversaries and Computation}, @@ -5886,6 +5788,17 @@ @article{vonNeumann1928 doi = {10.1007/BF01448847} } +@article{vonNeumann1928, + author = {von Neumann, John}, + title = {Zur {Theorie} der {Gesellschaftsspiele}}, + journal = {Mathematische Annalen}, + volume = {100}, + number = {1}, + pages = {295--320}, + year = {1928}, + doi = {10.1007/BF01448847} +} + @article{Wagner1986, author = {Wagner, Klaus W.}, title = {The Complexity of Combinatorial Problems with Succinct Input Representation}, @@ -5906,6 +5819,15 @@ @techreport{Watson2012 url = {https://eccc.weizmann.ac.il/report/2012/070/} } +@techreport{Watson2012, + author = {Watson, Thomas}, + title = {The Complexity of Estimating Min-Entropy}, + institution = {Electronic Colloquium on Computational Complexity}, + number = {TR12-070}, + year = {2012}, + url = {https://eccc.weizmann.ac.il/report/2012/070/} +} + @book{Wegener1987, author = {Wegener, Ingo}, title = {The Complexity of {Boolean} Functions}, @@ -5925,6 +5847,24 @@ @article{WegmanCarter1981 doi = {10.1016/0022-0000(81)90033-7} } +@article{WegmanCarter1981, + author = {Wegman, Mark N. and Carter, J. Lawrence}, + title = {New Hash Functions and Their Use in Authentication and Set Equality}, + journal = {Journal of Computer and System Sciences}, + volume = {22}, + number = {3}, + pages = {265--279}, + year = {1981}, + doi = {10.1016/0022-0000(81)90033-7} +} + +@misc{WelchBerlekamp1986, + author = {Welch, Lloyd R. and Berlekamp, Elwyn R.}, + title = {Error Correction for Algebraic Block Codes}, + note = {U.S. Patent 4,633,470}, + year = {1986} +} + @misc{WelchBerlekamp1986, author = {Welch, Lloyd R. and Berlekamp, Elwyn R.}, title = {Error Correction for Algebraic Block Codes}, @@ -5952,6 +5892,17 @@ @article{Williams2005 doi = {10.1016/j.tcs.2005.09.023} } +@article{Williams2005, + author = {Williams, Ryan}, + title = {A New Algorithm for Optimal 2-Constraint Satisfaction and Its Implications}, + journal = {Theoretical Computer Science}, + volume = {348}, + number = {2--3}, + pages = {357--365}, + year = {2005}, + doi = {10.1016/j.tcs.2005.09.023} +} + @article{Williams2014, author = {Williams, Ryan}, title = {Nonuniform {ACC} Circuit Lower Bounds}, @@ -5999,6 +5950,28 @@ @article{Wrathall1976 doi = {10.1016/0304-3975(76)90062-1} } +@article{Wrathall1976, + author = {Wrathall, Celia}, + title = {Complete Sets and the Polynomial-Time Hierarchy}, + journal = {Theoretical Computer Science}, + volume = {3}, + number = {1}, + pages = {23--33}, + year = {1976}, + doi = {10.1016/0304-3975(76)90062-1} +} + +@article{Yablonskii1954, + author = {Yablonskii, S. V.}, + title = {Realization of a linear function in the class of $\pi$-schemes}, + journal = {Doklady Akademii Nauk SSSR}, + volume = {94}, + number = {5}, + pages = {805--806}, + year = {1954}, + note = {In Russian} +} + @article{Yablonskii1954, author = {Yablonskii, S. V.}, title = {Realization of a linear function in the class of $\pi$-schemes}, @@ -6020,6 +5993,26 @@ @inproceedings{Yao1977 doi = {10.1109/SFCS.1977.24} } +@inproceedings{Yao1977, + author = {Yao, Andrew Chi-Chih}, + title = {Probabilistic Computations: Toward a Unified Measure of Complexity}, + booktitle = {Proceedings of the 18th Annual Symposium on Foundations of Computer Science}, + series = {FOCS 1977}, + pages = {222--227}, + year = {1977}, + doi = {10.1109/SFCS.1977.24} +} + +@inproceedings{Yao1982, + author = {Yao, Andrew C.}, + title = {Theory and Application of Trapdoor Functions}, + booktitle = {Proceedings of the 23rd Annual Symposium on Foundations of Computer Science}, + series = {FOCS 1982}, + pages = {80--91}, + year = {1982}, + doi = {10.1109/SFCS.1982.45} +} + @inproceedings{Yao1982, author = {Yao, Andrew C.}, title = {Theory and Application of Trapdoor Functions}, @@ -6036,3 +6029,10 @@ @misc{Zhao2023 howpublished = {Lecture notes, MIT 18.226}, year = {2023} } + +@misc{Zhao2023, + author = {Zhao, Yufei}, + title = {Probabilistic Methods in Combinatorics}, + howpublished = {Lecture notes, MIT 18.226}, + year = {2023} +} From c5638e73726d0ec4a2104b3bf74d88b3c3d41105 Mon Sep 17 00:00:00 2001 From: Samuel Giuseppe Schlesinger Date: Sat, 24 Oct 2026 23:07:15 -0400 Subject: [PATCH 4/5] fix(bib): remove entries duplicated by a faulty restack; keep only this PR's additions, sorted Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01Dvi6MU29zU9t8zYfeU51sB --- references.bib | 1959 +----------------------------------------------- 1 file changed, 27 insertions(+), 1932 deletions(-) diff --git a/references.bib b/references.bib index ff7ea22e21..38e90da6b7 100644 --- a/references.bib +++ b/references.bib @@ -49,16 +49,6 @@ @article{AFvMV2006 year = {2006} } -@article{AFvMV2006, - author = {Antunes, Lu{\'i}s and Fortnow, Lance and van Melkebeek, Dieter and - Vinodchandran, N. V.}, - title = {Computational Depth: Concept and Applications}, - journal = {Theoretical Computer Science}, - volume = {354}, - pages = {391--404}, - year = {2006} -} - @article{Ajtai1983, author = {Ajtai, Mikl{\'o}s}, title = {{$\Sigma^1_1$}-Formulae on Finite Structures}, @@ -135,17 +125,6 @@ @article{AllenderRubinstein1988 doi = {10.1137/0217075} } -@article{AllenderRubinstein1988, - author = {Allender, Eric W. and Rubinstein, Roy S.}, - title = {{P}-Printable Sets}, - journal = {SIAM Journal on Computing}, - volume = {17}, - number = {6}, - pages = {1193--1202}, - year = {1988}, - doi = {10.1137/0217075} -} - @article{AlonBoppana1987, author = {Alon, Noga and Boppana, Ravi B.}, title = {The Monotone Circuit Complexity of {B}oolean Functions}, @@ -168,26 +147,6 @@ @article{AlonGoldreichHastadPeralta1992 doi = {10.1002/rsa.3240030308} } -@article{AlonGoldreichHastadPeralta1992, - author = {Alon, Noga and Goldreich, Oded and H{\aa}stad, Johan and Peralta, Ren{\'e}}, - title = {Simple Constructions of Almost $k$-wise Independent Random Variables}, - journal = {Random Structures \& Algorithms}, - volume = {3}, - number = {3}, - pages = {289--304}, - year = {1992}, - doi = {10.1002/rsa.3240030308} -} - -@book{AlonSpencer2016, - author = {Alon, Noga and Spencer, Joel H.}, - title = {The Probabilistic Method}, - edition = {4}, - publisher = {Wiley}, - year = {2016}, - isbn = {978-1-119-06195-3} -} - @book{AlonSpencer2016, author = {Alon, Noga and Spencer, Joel H.}, title = {The Probabilistic Method}, @@ -245,18 +204,6 @@ @article{Andreev1989 note = {In Russian} } -@article{Andreev1989, - author = {Andreev, A. E.}, - title = {On the Complexity of the Realization of Partial {B}oolean Functions by Circuits of - Functional Elements}, - journal = {Diskretnaya Matematika}, - volume = {1}, - number = {4}, - pages = {36--45}, - year = {1989}, - note = {In Russian} -} - @article{Angluin1980, author = {Angluin, Dana}, title = {Inductive Inference of Formal Languages from Positive Data}, @@ -310,29 +257,6 @@ @article{AroraLundMotwaniSudanSzegedy1998 doi = {10.1145/278298.278306} } -@article{AroraLundMotwaniSudanSzegedy1998, - author = {Arora, Sanjeev and Lund, Carsten and Motwani, Rajeev and Sudan, Madhu - and Szegedy, Mario}, - title = {Proof Verification and the Hardness of Approximation Problems}, - journal = {Journal of the ACM}, - volume = {45}, - number = {3}, - pages = {501--555}, - year = {1998}, - doi = {10.1145/278298.278306} -} - -@article{AroraSafra1998, - author = {Arora, Sanjeev and Safra, Shmuel}, - title = {Probabilistic Checking of Proofs: A New Characterization of {NP}}, - journal = {Journal of the ACM}, - volume = {45}, - number = {1}, - pages = {70--122}, - year = {1998}, - doi = {10.1145/273865.273901} -} - @article{AroraSafra1998, author = {Arora, Sanjeev and Safra, Shmuel}, title = {Probabilistic Checking of Proofs: A New Characterization of {NP}}, @@ -373,25 +297,6 @@ @inproceedings{Babai1985 doi = {10.1145/22145.22192} } -@inproceedings{Babai1985, - author = {Babai, L{\'a}szl{\'o}}, - title = {Trading Group Theory for Randomness}, - booktitle = {Proceedings of the 17th Annual ACM Symposium on Theory of Computing}, - pages = {421--429}, - year = {1985}, - doi = {10.1145/22145.22192} -} - -@inproceedings{BabaiFortnowLevinSzegedy1991, - author = {Babai, L{\'a}szl{\'o} and Fortnow, Lance and Levin, Leonid A. and Szegedy, Mario}, - title = {Checking Computations in Polylogarithmic Time}, - booktitle = {Proceedings of the 23rd Annual ACM Symposium on Theory of Computing}, - series = {STOC 1991}, - pages = {21--32}, - year = {1991}, - doi = {10.1145/103418.103428} -} - @inproceedings{BabaiFortnowLevinSzegedy1991, author = {Babai, L{\'a}szl{\'o} and Fortnow, Lance and Levin, Leonid A. and Szegedy, Mario}, title = {Checking Computations in Polylogarithmic Time}, @@ -425,29 +330,6 @@ @article{BabaiFortnowNisanWigderson1993 doi = {10.1007/BF01275486} } -@article{BabaiFortnowNisanWigderson1993, - author = {Babai, L{\'a}szl{\'o} and Fortnow, Lance and Nisan, Noam and Wigderson, Avi}, - title = {{BPP} Has Subexponential Time Simulations Unless {EXPTIME} Has Publishable Proofs}, - journal = {Computational Complexity}, - volume = {3}, - number = {4}, - pages = {307--318}, - year = {1993}, - doi = {10.1007/BF01275486} -} - -@article{BabaiMoran1988, - author = {Babai, L{\'a}szl{\'o} and Moran, Shlomo}, - title = {{Arthur}--{Merlin} Games: A Randomized Proof System, and a Hierarchy of Complexity - Classes}, - journal = {Journal of Computer and System Sciences}, - volume = {36}, - number = {2}, - pages = {254--276}, - year = {1988}, - doi = {10.1016/0022-0000(88)90028-1} -} - @article{BabaiMoran1988, author = {Babai, L{\'a}szl{\'o} and Moran, Shlomo}, title = {{Arthur}--{Merlin} Games: A Randomized Proof System, and a Hierarchy of Complexity @@ -471,17 +353,6 @@ @article{BakerGillSolovay1975 doi = {10.1137/0204037} } -@article{BakerGillSolovay1975, - author = {Baker, Theodore and Gill, John and Solovay, Robert}, - title = {Relativizations of the {$\mathcal{P} =? \mathcal{NP}$} Question}, - journal = {SIAM Journal on Computing}, - volume = {4}, - number = {4}, - pages = {431--442}, - year = {1975}, - doi = {10.1137/0204037} -} - @misc{Barak2016PRG, author = {Barak, Boaz}, title = {{CS 127: Pseudorandomness}}, @@ -517,28 +388,6 @@ @article{BarringtonImmermanStraubing1990 doi = {10.1016/0022-0000(90)90022-D} } -@article{BarringtonImmermanStraubing1990, - author = {Barrington, David A. Mix and Immerman, Neil and Straubing, Howard}, - title = {On Uniformity within {NC}$^1$}, - journal = {Journal of Computer and System Sciences}, - volume = {41}, - number = {3}, - pages = {274--306}, - year = {1990}, - doi = {10.1016/0022-0000(90)90022-D} -} - -@article{BarringtonTherien1988, - author = {Barrington, David A. Mix and Th{\'e}rien, Denis}, - title = {Finite Monoids and the Fine Structure of {NC}$^1$}, - journal = {Journal of the ACM}, - volume = {35}, - number = {4}, - pages = {941--952}, - year = {1988}, - doi = {10.1145/48014.63138} -} - @article{BarringtonTherien1988, author = {Barrington, David A. Mix and Th{\'e}rien, Denis}, title = {Finite Monoids and the Fine Structure of {NC}$^1$}, @@ -574,18 +423,6 @@ @article{BealsBuhrmanCleveMoscaDeWolf2001 doi = {10.1145/502090.502097} } -@article{BealsBuhrmanCleveMoscaDeWolf2001, - author = {Beals, Robert and Buhrman, Harry and Cleve, Richard and Mosca, Michele and - de Wolf, Ronald}, - title = {Quantum Lower Bounds by Polynomials}, - journal = {Journal of the ACM}, - volume = {48}, - number = {4}, - pages = {778--797}, - year = {2001}, - doi = {10.1145/502090.502097} -} - @article{BeameEtAl1998, author = {Beame, Paul and Cook, Stephen and Edmonds, Jeff and Impagliazzo, Russell and Pitassi, Toniann}, title = {The Relative Complexity of {NP} Search Problems}, @@ -652,16 +489,6 @@ @inproceedings{BellareRompel1994 doi = {10.1109/SFCS.1994.365687} } -@inproceedings{BellareRompel1994, - author = {Bellare, Mihir and Rompel, John}, - title = {Randomness-Efficient Oblivious Sampling}, - booktitle = {Proceedings of the 35th Annual Symposium on Foundations of Computer Science}, - series = {FOCS 1994}, - pages = {276--287}, - year = {1994}, - doi = {10.1109/SFCS.1994.365687} -} - @inproceedings{BenOrGoldwasserKilianWigderson1988, author = {Ben-Or, Michael and Goldwasser, Shafi and Kilian, Joe and Wigderson, Avi}, title = {Multi-prover Interactive Proofs: How to Remove Intractability Assumptions}, @@ -698,18 +525,6 @@ @article{BenSassonKopparty2012 note = {Theorem numbers follow the ECCC report TR10-044 (2010)} } -@article{BenSassonKopparty2012, - author = {Ben-Sasson, Eli and Kopparty, Swastik}, - title = {Affine Dispersers from Subspace Polynomials}, - journal = {SIAM Journal on Computing}, - volume = {41}, - number = {4}, - pages = {880--914}, - year = {2012}, - doi = {10.1137/110826254}, - note = {Theorem numbers follow the ECCC report TR10-044 (2010)} -} - @article{BenSassonWigderson2001, author = {Ben-Sasson, Eli and Wigderson, Avi}, title = {Short Proofs Are Narrow---Resolution Made Simple}, @@ -734,29 +549,6 @@ @article{Berkowitz1984 doi = {10.1016/0020-0190(84)90018-8} } -@article{Berkowitz1984, - author = {Berkowitz, Stuart J.}, - title = {On Computing the Determinant in Small Parallel Time Using a Small Number - of Processors}, - journal = {Information Processing Letters}, - volume = {18}, - number = {3}, - pages = {147--150}, - year = {1984}, - doi = {10.1016/0020-0190(84)90018-8} -} - -@article{Bird2011, - author = {Bird, Richard S.}, - title = {A Simple Division-Free Algorithm for Computing Determinants}, - journal = {Information Processing Letters}, - volume = {111}, - number = {21--22}, - pages = {1072--1074}, - year = {2011}, - doi = {10.1016/j.ipl.2011.08.006} -} - @article{Bird2011, author = {Bird, Richard S.}, title = {A Simple Division-Free Algorithm for Computing Determinants}, @@ -789,17 +581,6 @@ @article{BlassGurevich1982 doi = {10.1016/S0019-9958(82)90439-9} } -@article{BlassGurevich1982, - author = {Blass, Andreas and Gurevich, Yuri}, - title = {On the Unique Satisfiability Problem}, - journal = {Information and Control}, - volume = {55}, - number = {1--3}, - pages = {80--88}, - year = {1982}, - doi = {10.1016/S0019-9958(82)90439-9} -} - @inproceedings{BlumImpagliazzo1987, author = {Blum, Manuel and Impagliazzo, Russell}, title = {Generic oracles and oracle classes}, @@ -819,27 +600,6 @@ @article{BlumLubyRubinfeld1993 doi = {10.1016/0022-0000(93)90044-W} } -@article{BlumLubyRubinfeld1993, - author = {Blum, Manuel and Luby, Michael and Rubinfeld, Ronitt}, - title = {Self-Testing/Correcting with Applications to Numerical Problems}, - journal = {Journal of Computer and System Sciences}, - volume = {47}, - number = {3}, - pages = {549--595}, - year = {1993}, - doi = {10.1016/0022-0000(93)90044-W} -} - -@article{BoehlerGlasserMeister2006, - author = {B{\"o}hler, Elmar and Gla{\ss}er, Christian and Meister, Daniel}, - title = {Error-bounded probabilistic computations between {MA} and {AM}}, - journal = {Journal of Computer and System Sciences}, - volume = {72}, - number = {6}, - pages = {1043--1076}, - year = {2006} -} - @article{BoehlerGlasserMeister2006, author = {B{\"o}hler, Elmar and Gla{\ss}er, Christian and Meister, Daniel}, title = {Error-bounded probabilistic computations between {MA} and {AM}}, @@ -861,17 +621,6 @@ @article{Bonami1970 doi = {10.5802/aif.357} } -@article{Bonami1970, - author = {Bonami, Aline}, - title = {{\'E}tude des coefficients de {Fourier} des fonctions de {$L^p(G)$}}, - journal = {Annales de l'Institut Fourier}, - volume = {20}, - number = {2}, - pages = {335--402}, - year = {1970}, - doi = {10.5802/aif.357} -} - @inproceedings{BonehHaleviHamburgOstrovsky2008, author = {Dan Boneh and Shai Halevi and Mike Hamburg and Rafail Ostrovsky}, title = {Circular-Secure Encryption from Decision Diffie-Hellman}, @@ -903,17 +652,6 @@ @article{Book1974 doi = {10.1016/S0022-0000(74)80008-5} } -@article{Book1974, - author = {Book, Ronald V.}, - title = {Comparing Complexity Classes}, - journal = {Journal of Computer and System Sciences}, - volume = {9}, - number = {2}, - pages = {213--229}, - year = {1974}, - doi = {10.1016/S0022-0000(74)80008-5} -} - @article{BoppanaHastadZachos1987, author = {Boppana, Ravi B. and H{\aa}stad, Johan and Zachos, Stathis}, title = {Does co-{NP} Have Short Interactive Proofs?}, @@ -936,27 +674,6 @@ @article{BorchertHemaspaandraRothe2000 url = {https://arxiv.org/abs/cs/9907041} } -@article{BorchertHemaspaandraRothe2000, - author = {Borchert, Bernd and Hemaspaandra, Lane A. and Rothe, J{\"o}rg}, - title = {Restrictive Acceptance Suffices for Equivalence Problems}, - journal = {LMS Journal of Computation and Mathematics}, - volume = {3}, - pages = {86--95}, - year = {2000}, - doi = {10.1112/S146115700000022X}, - url = {https://arxiv.org/abs/cs/9907041} -} - -@book{BorweinErdelyi1995, - author = {Borwein, Peter and Erd{\'e}lyi, Tam{\'a}s}, - title = {Polynomials and Polynomial Inequalities}, - series = {Graduate Texts in Mathematics}, - volume = {161}, - publisher = {Springer}, - year = {1995}, - doi = {10.1007/978-1-4612-0793-1} -} - @book{BorweinErdelyi1995, author = {Borwein, Peter and Erd{\'e}lyi, Tam{\'a}s}, title = {Polynomials and Polynomial Inequalities}, @@ -978,26 +695,6 @@ @article{Bourgain2007 doi = {10.1007/s00039-007-0593-z} } -@article{Bourgain2007, - author = {Bourgain, Jean}, - title = {On the Construction of Affine Extractors}, - journal = {Geometric and Functional Analysis}, - volume = {17}, - number = {1}, - pages = {33--57}, - year = {2007}, - doi = {10.1007/s00039-007-0593-z} -} - -@inproceedings{BovaCapelliMengelSlivovsky2016, - author = {Bova, Simone and Capelli, Florent and Mengel, Stefan and Slivovsky, Friedrich}, - title = {Knowledge Compilation Meets Communication Complexity}, - booktitle = {Proceedings of the Twenty-Fifth International Joint Conference on Artificial - Intelligence (IJCAI 2016)}, - pages = {1008--1014}, - year = {2016} -} - @inproceedings{BovaCapelliMengelSlivovsky2016, author = {Bova, Simone and Capelli, Florent and Mengel, Stefan and Slivovsky, Friedrich}, title = {Knowledge Compilation Meets Communication Complexity}, @@ -1047,25 +744,6 @@ @book{Burgisser2000 doi = {10.1007/978-3-662-04179-6} } -@book{Burgisser2000, - author = {B{\"u}rgisser, Peter}, - title = {Completeness and Reduction in Algebraic Complexity Theory}, - series = {Algorithms and Computation in Mathematics}, - volume = {7}, - publisher = {Springer}, - year = {2000}, - doi = {10.1007/978-3-662-04179-6} -} - -@book{BurgisserClausenShokrollahi1997, - author = {B{\"u}rgisser, Peter and Clausen, Michael and Shokrollahi, M. Amin}, - title = {Algebraic Complexity Theory}, - series = {Grundlehren der mathematischen Wissenschaften}, - publisher = {Springer}, - year = {1997}, - doi = {10.1007/978-3-662-03338-8} -} - @book{BurgisserClausenShokrollahi1997, author = {B{\"u}rgisser, Peter and Clausen, Michael and Shokrollahi, M. Amin}, title = {Algebraic Complexity Theory}, @@ -1097,29 +775,6 @@ @article{CaiChenDowneyFellows1997 doi = {10.1016/S0168-0072(96)00020-6} } -@article{CaiChenDowneyFellows1997, - author = {Cai, Liming and Chen, Jianer and Downey, Rodney G. and Fellows, Michael R.}, - title = {Advice classes of parameterized tractability}, - journal = {Annals of Pure and Applied Logic}, - volume = {84}, - number = {1}, - pages = {119--138}, - year = {1997}, - doi = {10.1016/S0168-0072(96)00020-6} -} - -@article{CaiEtAl1988, - author = {Cai, Jin-Yi and Gundermann, Thomas and Hartmanis, Juris and - Hemachandra, Lane A. and Sewelson, Vivian and Wagner, Klaus and Wechsung, Gerd}, - title = {The Boolean Hierarchy {I}: Structural Properties}, - journal = {SIAM Journal on Computing}, - volume = {17}, - number = {6}, - pages = {1232--1252}, - year = {1988}, - doi = {10.1137/0217078} -} - @article{CaiEtAl1988, author = {Cai, Jin-Yi and Gundermann, Thomas and Hartmanis, Juris and Hemachandra, Lane A. and Sewelson, Vivian and Wagner, Klaus and Wechsung, Gerd}, @@ -1150,22 +805,6 @@ @phdthesis{Calabro2009 year = {2009} } -@phdthesis{Calabro2009, - author = {Calabro, Chris}, - title = {The Exponential Complexity of Satisfiability Problems}, - school = {University of California, San Diego}, - year = {2009} -} - -@inproceedings{CalabroImpagliazzoPaturi2006, - author = {Calabro, Chris and Impagliazzo, Russell and Paturi, Ramamohan}, - title = {A Duality between Clause Width and Clause Density for {SAT}}, - booktitle = {21st Annual IEEE Conference on Computational Complexity (CCC 2006)}, - pages = {252--260}, - year = {2006}, - doi = {10.1109/CCC.2006.6} -} - @inproceedings{CalabroImpagliazzoPaturi2006, author = {Calabro, Chris and Impagliazzo, Russell and Paturi, Ramamohan}, title = {A Duality between Clause Width and Clause Density for {SAT}}, @@ -1205,17 +844,6 @@ @article{CarterWegman1979 doi = {10.1016/0022-0000(79)90044-8} } -@article{CarterWegman1979, - author = {Carter, J. Lawrence and Wegman, Mark N.}, - title = {Universal Classes of Hash Functions}, - journal = {Journal of Computer and System Sciences}, - volume = {18}, - number = {2}, - pages = {143--154}, - year = {1979}, - doi = {10.1016/0022-0000(79)90044-8} -} - @techreport{CavalarOliveira2025, author = {Bruno P. Cavalar and Igor C. Oliveira}, title = {Boolean Circuit Complexity and Two-Dimensional Cover Problems}, @@ -1236,17 +864,6 @@ @article{ChandraKozenStockmeyer1981 doi = {10.1145/322234.322243} } -@article{ChandraKozenStockmeyer1981, - author = {Chandra, Ashok K. and Kozen, Dexter C. and Stockmeyer, Larry J.}, - title = {Alternation}, - journal = {Journal of the ACM}, - volume = {28}, - number = {1}, - pages = {114--133}, - year = {1981}, - doi = {10.1145/322234.322243} -} - @article{ChandraStockmeyerVishkin1984, author = {Chandra, Ashok K. and Stockmeyer, Larry and Vishkin, Uzi}, title = {Constant Depth Reducibility}, @@ -1290,29 +907,6 @@ @article{Chashkin2024 note = {In Russian} } -@article{Chashkin2024, - author = {Chashkin, A. V.}, - title = {On Computing Partial {B}oolean Functions}, - journal = {Matematicheskie Voprosy Kibernetiki}, - volume = {22}, - pages = {152--222}, - year = {2024}, - publisher = {Fizmatlit}, - doi = {10.20948/mvk-2024-152}, - url = {https://library.keldysh.ru/mvk.asp?id=2024-152}, - note = {In Russian} -} - -@article{Chebyshev1852, - author = {Tchebichef, Pafnuty L.}, - title = {M{\'e}moire sur les nombres premiers}, - journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, - series = {1}, - volume = {17}, - pages = {366--390}, - year = {1852} -} - @article{Chebyshev1852, author = {Tchebichef, Pafnuty L.}, title = {M{\'e}moire sur les nombres premiers}, @@ -1334,28 +928,6 @@ @inproceedings{ChenKabanets2015 doi = {10.1007/978-3-319-21398-9_17} } -@inproceedings{ChenKabanets2015, - author = {Chen, Ruiwen and Kabanets, Valentine}, - title = {Correlation Bounds and {\#SAT} Algorithms for Small Linear-Size Circuits}, - booktitle = {Computing and Combinatorics (COCOON 2015)}, - series = {Lecture Notes in Computer Science}, - pages = {211--222}, - publisher = {Springer}, - year = {2015}, - doi = {10.1007/978-3-319-21398-9_17} -} - -@misc{CHLLTV2026, - author = {Chattopadhyay, Eshan and Hatami, Pooya and Lee, Chin Ho and Lovett, Shachar and - Tal, Avishay and Viola, Emanuele}, - title = {Exponential Correlation Bounds for Polynomials}, - year = {2026}, - eprint = {2609.28839}, - archivePrefix = {arXiv}, - primaryClass = {cs.CC}, - url = {https://arxiv.org/abs/2609.28839} -} - @misc{CHLLTV2026, author = {Chattopadhyay, Eshan and Hatami, Pooya and Lee, Chin Ho and Lovett, Shachar and Tal, Avishay and Viola, Emanuele}, @@ -1379,29 +951,6 @@ @article{ChorGoldreich1988 doi = {10.1137/0217015} } -@article{ChorGoldreich1988, - author = {Chor, Benny and Goldreich, Oded}, - title = {Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication - Complexity}, - journal = {SIAM Journal on Computing}, - volume = {17}, - number = {2}, - pages = {230--261}, - year = {1988}, - doi = {10.1137/0217015} -} - -@article{ChungGrahamFranklShearer1986, - author = {Chung, Fan R. K. and Graham, Ronald L. and Frankl, Peter and Shearer, James B.}, - title = {Some Intersection Theorems for Ordered Sets and Graphs}, - journal = {Journal of Combinatorial Theory, Series A}, - volume = {43}, - number = {1}, - pages = {23--37}, - year = {1986}, - doi = {10.1016/0097-3165(86)90019-1} -} - @article{ChungGrahamFranklShearer1986, author = {Chung, Fan R. K. and Graham, Ronald L. and Frankl, Peter and Shearer, James B.}, title = {Some Intersection Theorems for Ordered Sets and Graphs}, @@ -1433,17 +982,6 @@ @article{CMWBS2012 year = {2012} } -@article{CMWBS2012, - author = {Cook, Stephen and McKenzie, Pierre and Wehr, Dustin and Braverman, Mark and - Santhanam, Rahul}, - title = {Pebbles and Branching Programs for Tree Evaluation}, - journal = {ACM Transactions on Computation Theory}, - volume = {3}, - number = {2}, - pages = {4:1--4:43}, - year = {2012} -} - @techreport{CookMertz2024, author = {Cook, James and Mertz, Ian}, title = {Tree Evaluation is in Space {$O(\log n \cdot \log\log n)$}}, @@ -1453,26 +991,6 @@ @techreport{CookMertz2024 year = {2023} } -@techreport{CookMertz2024, - author = {Cook, James and Mertz, Ian}, - title = {Tree Evaluation is in Space {$O(\log n \cdot \log\log n)$}}, - institution = {Electronic Colloquium on Computational Complexity}, - number = {TR23-174}, - note = {Conference version in STOC 2024}, - year = {2023} -} - -@article{CookReckhow1973, - author = {Cook, Stephen A. and Reckhow, Robert A.}, - title = {Time Bounded Random Access Machines}, - journal = {Journal of Computer and System Sciences}, - volume = {7}, - number = {4}, - year = {1973}, - pages = {354--375}, - doi = {10.1016/S0022-0000(73)80029-7} -} - @article{CookReckhow1973, author = {Cook, Stephen A. and Reckhow, Robert A.}, title = {Time Bounded Random Access Machines}, @@ -1495,17 +1013,6 @@ @article{CookReckhow1979 doi = {10.2307/2273702} } -@article{CookReckhow1979, - author = {Cook, Stephen A. and Reckhow, Robert A.}, - title = {The Relative Efficiency of Propositional Proof Systems}, - journal = {The Journal of Symbolic Logic}, - volume = {44}, - number = {1}, - pages = {36--50}, - year = {1979}, - doi = {10.2307/2273702} -} - @mastersthesis{Copes2018, author = {Copes, Martín}, title = {A machine-checked proof of the Standardization Theorem @@ -1525,26 +1032,6 @@ @article{CoppersmithRivlin1992 doi = {10.1137/0523054} } -@article{CoppersmithRivlin1992, - author = {Coppersmith, Don and Rivlin, Theodore J.}, - title = {The Growth of Polynomials Bounded at Equally Spaced Points}, - journal = {SIAM Journal on Mathematical Analysis}, - volume = {23}, - number = {4}, - pages = {970--983}, - year = {1992}, - doi = {10.1137/0523054} -} - -@book{CoverThomas2006, - author = {Cover, Thomas M. and Thomas, Joy A.}, - title = {Elements of Information Theory}, - edition = {2}, - publisher = {Wiley-Interscience}, - year = {2006}, - doi = {10.1002/047174882X} -} - @book{CoverThomas2006, author = {Cover, Thomas M. and Thomas, Joy A.}, title = {Elements of Information Theory}, @@ -1585,16 +1072,6 @@ @book{CyganEtAl2015 doi = {10.1007/978-3-319-21275-3} } -@book{CyganEtAl2015, - author = {Cygan, Marek and Fomin, Fedor V. and Kowalik, {\L}ukasz and Lokshtanov, Daniel and - Marx, D{\'a}niel and Pilipczuk, Marcin and Pilipczuk, Micha{\l} and Saurabh, Saket}, - title = {Parameterized Algorithms}, - publisher = {Springer}, - address = {Cham}, - year = {2015}, - doi = {10.1007/978-3-319-21275-3} -} - @inproceedings{Danielsson2008, author = {Danielsson, Nils Anders}, title = {Lightweight semiformal time complexity analysis for purely functional data structures}, @@ -1624,17 +1101,6 @@ @article{Darwiche2001 doi = {10.1145/502090.502091} } -@article{Darwiche2001, - author = {Darwiche, Adnan}, - title = {Decomposable Negation Normal Form}, - journal = {Journal of the ACM}, - volume = {48}, - number = {4}, - pages = {608--647}, - year = {2001}, - doi = {10.1145/502090.502091} -} - @article{DavisLogemannLoveland1962, author = {Davis, Martin and Logemann, George and Loveland, Donald}, title = {A Machine Program for Theorem-Proving}, @@ -1658,17 +1124,6 @@ @article{DavisPutnam1960 doi = {10.1145/321033.321034} } -@article{DavisPutnam1960, - author = {Davis, Martin and Putnam, Hilary}, - title = {A Computing Procedure for Quantification Theory}, - journal = {Journal of the ACM}, - volume = {7}, - number = {3}, - pages = {201--215}, - year = {1960}, - doi = {10.1145/321033.321034} -} - @misc{Demaine2015VanEmdeBoas, author = {Demaine, Erik}, title = {Lecture 4: Divide and Conquer: van {Emde Boas} Trees}, @@ -1710,26 +1165,6 @@ @techreport{Dinur2005 year = {2005} } -@techreport{Dinur2005, - author = {Dinur, Irit}, - title = {The {PCP} Theorem by Gap Amplification}, - institution = {Electronic Colloquium on Computational Complexity}, - number = {TR05-046}, - note = {Revision 1, 26 September 2005}, - year = {2005} -} - -@article{Dinur2007, - author = {Dinur, Irit}, - title = {The {PCP} Theorem by Gap Amplification}, - journal = {Journal of the ACM}, - volume = {54}, - number = {3}, - pages = {12}, - year = {2007}, - doi = {10.1145/1236457.1236459} -} - @article{Dinur2007, author = {Dinur, Irit}, title = {The {PCP} Theorem by Gap Amplification}, @@ -1751,16 +1186,6 @@ @book{DowneyFellows1999 doi = {10.1007/978-1-4612-0515-9} } -@book{DowneyFellows1999, - author = {Downey, Rodney G. and Fellows, Michael R.}, - title = {Parameterized Complexity}, - series = {Monographs in Computer Science}, - publisher = {Springer}, - address = {New York}, - year = {1999}, - doi = {10.1007/978-1-4612-0515-9} -} - @article{DucasEtAl2018, author = {Ducas, L{\'e}o and Kiltz, Eike and Lepoint, Tancr{\`e}de and Lyubashevsky, Vadim and Schwabe, Peter and Seiler, Gregor and Stehl{\'e}, Damien}, @@ -1810,28 +1235,6 @@ @article{EhlichZeller1964 doi = {10.1007/BF01111276} } -@article{EhlichZeller1964, - author = {Ehlich, Hartmut and Zeller, Karl}, - title = {Schwankung von {Polynomen} zwischen {Gitterpunkten}}, - journal = {Mathematische Zeitschrift}, - volume = {86}, - number = {1}, - pages = {41--44}, - year = {1964}, - doi = {10.1007/BF01111276} -} - -@incollection{ErdosLovasz1975, - author = {Erd{\H{o}}s, Paul and Lov{\'a}sz, L{\'a}szl{\'o}}, - title = {Problems and Results on 3-Chromatic Hypergraphs and Some Related Questions}, - booktitle = {Infinite and Finite Sets}, - series = {Colloquia Mathematica Societatis J{\'a}nos Bolyai}, - volume = {10}, - publisher = {North-Holland}, - pages = {609--627}, - year = {1975} -} - @incollection{ErdosLovasz1975, author = {Erd{\H{o}}s, Paul and Lov{\'a}sz, L{\'a}szl{\'o}}, title = {Problems and Results on 3-Chromatic Hypergraphs and Some Related Questions}, @@ -1865,17 +1268,6 @@ @article{ErdosSpencer1991 doi = {10.1016/0166-218X(91)90040-4} } -@article{ErdosSpencer1991, - author = {Erd{\H{o}}s, Paul and Spencer, Joel}, - title = {Lopsided {Lov{\'a}sz} Local Lemma and {Latin} Transversals}, - journal = {Discrete Applied Mathematics}, - volume = {30}, - number = {2--3}, - pages = {151--154}, - year = {1991}, - doi = {10.1016/0166-218X(91)90040-4} -} - @article{EricksonVanDerHoogMiltzow2024, author = {Erickson, Jeff and van der Hoog, Ivor and Miltzow, Tillmann}, title = {Smoothing the Gap Between {NP} and {$\exists\mathbb{R}$}}, @@ -1911,28 +1303,6 @@ @article{EvenSelmanYacobi1984 doi = {10.1016/S0019-9958(84)80056-X} } -@article{EvenSelmanYacobi1984, - author = {Even, Shimon and Selman, Alan L. and Yacobi, Yacov}, - title = {The Complexity of Promise Problems with Applications to Public-Key Cryptography}, - journal = {Information and Control}, - volume = {61}, - number = {2}, - pages = {159--173}, - year = {1984}, - doi = {10.1016/S0019-9958(84)80056-X} -} - -@incollection{Fagin1974, - author = {Fagin, Ronald}, - title = {Generalized First-Order Spectra and Polynomial-Time Recognizable Sets}, - booktitle = {Complexity of Computation}, - editor = {Karp, Richard M.}, - series = {SIAM-AMS Proceedings}, - volume = {7}, - pages = {43--73}, - year = {1974} -} - @incollection{Fagin1974, author = {Fagin, Ronald}, title = {Generalized First-Order Spectra and Polynomial-Time Recognizable Sets}, @@ -1968,30 +1338,6 @@ @article{Fenner2003 url = {https://eccc.weizmann.ac.il/report/2002/036/} } -@article{Fenner2003, - author = {Fenner, Stephen A.}, - title = {{PP}-Lowness and a Simple Definition of {AWPP}}, - journal = {Theory of Computing Systems}, - volume = {36}, - number = {2}, - pages = {199--212}, - year = {2003}, - doi = {10.1007/s00224-002-1089-8}, - url = {https://eccc.weizmann.ac.il/report/2002/036/} -} - -@article{FennerEtAl2003, - author = {Fenner, Stephen A. and Fortnow, Lance J. and Kurtz, Stuart A. and Li, Lide}, - title = {An Oracle Builder's Toolkit}, - journal = {Information and Computation}, - volume = {182}, - number = {2}, - pages = {95--136}, - year = {2003}, - doi = {10.1016/S0890-5401(03)00018-X}, - url = {https://lance.fortnow.com/papers/files/obt.pdf} -} - @article{FennerEtAl2003, author = {Fenner, Stephen A. and Fortnow, Lance J. and Kurtz, Stuart A. and Li, Lide}, title = {An Oracle Builder's Toolkit}, @@ -2015,17 +1361,6 @@ @article{FennerFortnowKurtz1994 doi = {10.1016/S0022-0000(05)80024-8} } -@article{FennerFortnowKurtz1994, - author = {Fenner, Stephen A. and Fortnow, Lance J. and Kurtz, Stuart A.}, - title = {Gap-Definable Counting Classes}, - journal = {Journal of Computer and System Sciences}, - volume = {48}, - number = {1}, - pages = {116--148}, - year = {1994}, - doi = {10.1016/S0022-0000(05)80024-8} -} - @inproceedings{FiatShamir1986, author = {Amos Fiat and Adi Shamir}, title = {How to Prove Yourself: Practical Solutions to Identification and Signature @@ -2035,20 +1370,8 @@ @inproceedings{FiatShamir1986 volume = {263}, pages = {186--194}, publisher = {Springer}, - year = {1987}, - doi = {10.1007/3-540-47721-7_12} -} - -@inproceedings{FindGolovnevHirschKulikov2016, - author = {Find, Magnus Gausdal and Golovnev, Alexander and Hirsch, Edward A. and - Kulikov, Alexander S.}, - title = {A Better-than-{$3n$} Lower Bound for the Circuit Complexity of an Explicit - Function}, - booktitle = {IEEE 57th Annual Symposium on Foundations of Computer Science}, - series = {FOCS 2016}, - pages = {89--98}, - year = {2016}, - doi = {10.1109/FOCS.2016.19} + year = {1987}, + doi = {10.1007/3-540-47721-7_12} } @inproceedings{FindGolovnevHirschKulikov2016, @@ -2091,23 +1414,6 @@ @book{FlumGrohe2006 doi = {10.1007/3-540-29953-X} } -@book{FlumGrohe2006, - author = {Flum, J{\"o}rg and Grohe, Martin}, - title = {Parameterized Complexity Theory}, - series = {Texts in Theoretical Computer Science. An EATCS Series}, - publisher = {Springer}, - address = {Berlin, Heidelberg}, - year = {2006}, - doi = {10.1007/3-540-29953-X} -} - -@book{Forney1966, - author = {Forney, Jr., G. David}, - title = {Concatenated Codes}, - publisher = {MIT Press}, - year = {1966} -} - @book{Forney1966, author = {Forney, Jr., G. David}, title = {Concatenated Codes}, @@ -2127,18 +1433,6 @@ @incollection{Fortnow1997 url = {https://lance.fortnow.com/papers/files/counting.pdf} } -@incollection{Fortnow1997, - author = {Fortnow, Lance}, - title = {Counting Complexity}, - booktitle = {Complexity Theory Retrospective II}, - editor = {Hemaspaandra, Lane A. and Selman, Alan L.}, - publisher = {Springer}, - pages = {81--107}, - year = {1997}, - doi = {10.1007/978-1-4612-1872-2_4}, - url = {https://lance.fortnow.com/papers/files/counting.pdf} -} - @article{FortnowReingold1996, author = {Fortnow, Lance and Reingold, Nick}, title = {{PP} Is Closed under Truth-Table Reductions}, @@ -2172,17 +1466,6 @@ @article{Friedgut1998 doi = {10.1007/PL00009809} } -@article{Friedgut1998, - author = {Friedgut, Ehud}, - title = {{Boolean} Functions With Low Average Sensitivity Depend On Few Coordinates}, - journal = {Combinatorica}, - volume = {18}, - number = {1}, - pages = {27--35}, - year = {1998}, - doi = {10.1007/PL00009809} -} - @article{FurstSaxeSipser1984, author = {Furst, Merrick and Saxe, James B. and Sipser, Michael}, title = {Parity, Circuits, and the Polynomial-Time Hierarchy}, @@ -2222,14 +1505,6 @@ @book{GareyJohnson1979 year = {1979} } -@book{GareyJohnson1979, - author = {Garey, Michael R. and Johnson, David S.}, - title = {Computers and Intractability: A Guide to the Theory of {NP}-Completeness}, - publisher = {W. H. Freeman}, - address = {San Francisco}, - year = {1979} -} - @inproceedings{Gentry2009, author = {Gentry, Craig}, title = {Fully Homomorphic Encryption Using Ideal Lattices}, @@ -2267,28 +1542,6 @@ @article{GGM1986 url = {https://www.wisdom.weizmann.ac.il/~oded/X/ggm-jacm.pdf} } -@article{GGM1986, - author = {Goldreich, Oded and Goldwasser, Shafi and Micali, Silvio}, - title = {How to Construct Random Functions}, - journal = {Journal of the ACM}, - volume = {33}, - number = {4}, - pages = {792--807}, - year = {1986}, - url = {https://www.wisdom.weizmann.ac.il/~oded/X/ggm-jacm.pdf} -} - -@article{Gilbert1952, - author = {Gilbert, E. N.}, - title = {A Comparison of Signalling Alphabets}, - journal = {Bell System Technical Journal}, - volume = {31}, - number = {3}, - pages = {504--522}, - year = {1952}, - doi = {10.1002/j.1538-7305.1952.tb01393.x} -} - @article{Gilbert1952, author = {Gilbert, E. N.}, title = {A Comparison of Signalling Alphabets}, @@ -2311,17 +1564,6 @@ @article{Gill1977 doi = {10.1137/0206049} } -@article{Gill1977, - author = {Gill, John}, - title = {Computational Complexity of Probabilistic {T}uring Machines}, - journal = {SIAM Journal on Computing}, - volume = {6}, - number = {4}, - pages = {675--695}, - year = {1977}, - doi = {10.1137/0206049} -} - @article{ Girard1987, title={Linear logic}, author={Girard, Jean-Yves}, @@ -2363,20 +1605,6 @@ @inproceedings{GKST2016 doi = {10.4230/LIPIcs.MFCS.2016.45} } -@inproceedings{GKST2016, - author = {Golovnev, Alexander and Kulikov, Alexander S. and Smal, Alexander V. and - Tamaki, Suguru}, - title = {Circuit Size Lower Bounds and {\#SAT} Upper Bounds Through a General Framework}, - booktitle = {41st International Symposium on Mathematical Foundations of Computer Science - (MFCS 2016)}, - series = {Leibniz International Proceedings in Informatics (LIPIcs)}, - volume = {58}, - pages = {45:1--45:16}, - publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, - year = {2016}, - doi = {10.4230/LIPIcs.MFCS.2016.45} -} - @book{Goldreich2001, author = {Goldreich, Oded}, title = {Foundations of Cryptography, Volume 1: Basic Tools}, @@ -2397,27 +1625,6 @@ @incollection{Goldreich2006 doi = {10.1007/11685654_12} } -@incollection{Goldreich2006, - author = {Goldreich, Oded}, - title = {On Promise Problems: A Survey}, - booktitle = {Theoretical Computer Science: Essays in Memory of {S}himon {E}ven}, - series = {Lecture Notes in Computer Science}, - volume = {3895}, - pages = {254--290}, - publisher = {Springer}, - year = {2006}, - doi = {10.1007/11685654_12} -} - -@techreport{Goldreich2024TreeEval, - author = {Goldreich, Oded}, - title = {On the {Cook}--{Mertz} Tree Evaluation procedure}, - institution = {Electronic Colloquium on Computational Complexity}, - number = {TR24-109}, - note = {Revision 2, 27 December 2024}, - year = {2024} -} - @techreport{Goldreich2024TreeEval, author = {Goldreich, Oded}, title = {On the {Cook}--{Mertz} Tree Evaluation procedure}, @@ -2449,18 +1656,6 @@ @article{GoldreichMicaliWigderson1991 doi = {10.1145/116825.116852} } -@article{GoldreichMicaliWigderson1991, - author = {Goldreich, Oded and Micali, Silvio and Wigderson, Avi}, - title = {Proofs that Yield Nothing but Their Validity or All Languages in {NP} Have - Zero-Knowledge Proof Systems}, - journal = {Journal of the ACM}, - volume = {38}, - number = {3}, - pages = {690--728}, - year = {1991}, - doi = {10.1145/116825.116852} -} - @article{Goldschlager1977, author = {Goldschlager, Leslie M.}, title = {The Monotone and Planar Circuit Value Problems Are Log Space Complete for {$P$}}, @@ -2494,17 +1689,6 @@ @article{GoldwasserMicaliRackoff1989 doi = {10.1137/0218012} } -@article{GoldwasserMicaliRackoff1989, - author = {Goldwasser, Shafi and Micali, Silvio and Rackoff, Charles}, - title = {The Knowledge Complexity of Interactive Proof Systems}, - journal = {SIAM Journal on Computing}, - volume = {18}, - number = {1}, - pages = {186--208}, - year = {1989}, - doi = {10.1137/0218012} -} - @article{GoldwasserMicaliRivest1988, author = {Goldwasser, Shafi and Micali, Silvio and Rivest, Ronald L.}, title = {A Digital Signature Scheme Secure Against Adaptive Chosen-Message Attacks}, @@ -2526,26 +1710,6 @@ @inproceedings{GoldwasserSipser1986 doi = {10.1145/12130.12137} } -@inproceedings{GoldwasserSipser1986, - author = {Goldwasser, Shafi and Sipser, Michael}, - title = {Private Coins versus Public Coins in Interactive Proof Systems}, - booktitle = {Proceedings of the 18th Annual ACM Symposium on Theory of Computing}, - pages = {59--68}, - year = {1986}, - doi = {10.1145/12130.12137} -} - -@article{GotsmanLinial1992, - author = {Gotsman, Craig and Linial, Nathan}, - title = {The Equivalence of Two Problems on the Cube}, - journal = {Journal of Combinatorial Theory, Series A}, - volume = {61}, - number = {1}, - pages = {142--146}, - year = {1992}, - doi = {10.1016/0097-3165(92)90060-8} -} - @article{GotsmanLinial1992, author = {Gotsman, Craig and Linial, Nathan}, title = {The Equivalence of Two Problems on the Cube}, @@ -2575,14 +1739,6 @@ @misc{GuruswamiRudraSudan2019 url = {https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/} } -@misc{GuruswamiRudraSudan2019, - author = {Guruswami, Venkatesan and Rudra, Atri and Sudan, Madhu}, - title = {Essential Coding Theory}, - note = {Draft book, 2019}, - year = {2019}, - url = {https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/} -} - @inproceedings{Hagerup1998, author = {Hagerup, Torben}, title = {Sorting and Searching on the Word {RAM}}, @@ -2604,15 +1760,6 @@ @techreport{HaitnerVadhan2017 url = {https://eccc.weizmann.ac.il/report/2017/084/} } -@techreport{HaitnerVadhan2017, - author = {Haitner, Iftach and Vadhan, Salil}, - title = {The Many Entropies in One-Way Functions}, - institution = {Electronic Colloquium on Computational Complexity}, - number = {TR17-084}, - year = {2017}, - url = {https://eccc.weizmann.ac.il/report/2017/084/} -} - @article{Haken1985, author = {Haken, Armin}, title = {The Intractability of Resolution}, @@ -2635,27 +1782,6 @@ @article{Hamming1950 doi = {10.1002/j.1538-7305.1950.tb00463.x} } -@article{Hamming1950, - author = {Hamming, R. W.}, - title = {Error Detecting and Error Correcting Codes}, - journal = {The Bell System Technical Journal}, - volume = {29}, - number = {2}, - pages = {147--160}, - year = {1950}, - doi = {10.1002/j.1538-7305.1950.tb00463.x} -} - -@inproceedings{HartmanisSimon1974, - author = {Hartmanis, Juris and Simon, Janos}, - title = {On the Power of Multiplication in Random Access Machines}, - booktitle = {15th Annual Symposium on Switching and Automata Theory (SWAT 1974)}, - year = {1974}, - pages = {13--23}, - publisher = {IEEE}, - doi = {10.1109/SWAT.1974.20} -} - @inproceedings{HartmanisSimon1974, author = {Hartmanis, Juris and Simon, Janos}, title = {On the Power of Multiplication in Random Access Machines}, @@ -2711,19 +1837,6 @@ @inproceedings{HemaspaandraEtAl2018 doi = {10.4230/LIPIcs.MFCS.2018.51} } -@inproceedings{HemaspaandraEtAl2018, - author = {Hemaspaandra, Edith and Hemaspaandra, Lane A. and Spakowski, Holger and - Watanabe, Osamu}, - title = {The Robustness of {LWPP} and {WPP}, with an Application to Graph Reconstruction}, - booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science}, - series = {Leibniz International Proceedings in Informatics}, - volume = {117}, - pages = {51:1--51:14}, - year = {2018}, - publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, - doi = {10.4230/LIPIcs.MFCS.2018.51} -} - @article{ Hennessy1985, author = {Matthew Hennessy and Robin Milner}, @@ -2752,18 +1865,6 @@ @article{HILL1999 url = {https://doi.org/10.1137/S0097539793244708} } -@article{HILL1999, - author = {Håstad, Johan and Impagliazzo, Russell and Levin, Leonid A. and Luby, Michael}, - title = {A Pseudorandom Generator from Any One-Way Function}, - journal = {SIAM Journal on Computing}, - volume = {28}, - number = {4}, - pages = {1364--1396}, - year = {1999}, - doi = {10.1137/S0097539793244708}, - url = {https://doi.org/10.1137/S0097539793244708} -} - @inproceedings{Hirahara2018, author = {Hirahara, Shuichi}, title = {Non-Black-Box Worst-Case to Average-Case Reductions within {NP}}, @@ -2774,38 +1875,16 @@ @inproceedings{Hirahara2018 doi = {10.1109/FOCS.2018.00032} } -@inproceedings{Hirahara2018, - author = {Hirahara, Shuichi}, - title = {Non-Black-Box Worst-Case to Average-Case Reductions within {NP}}, - booktitle = {2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)}, - publisher = {IEEE Computer Society}, - year = {2018}, - pages = {247--258}, - doi = {10.1109/FOCS.2018.00032} -} - -@inproceedings{Hirahara2022, - author = {Hirahara, Shuichi}, - title = {{NP}-Hardness of Learning Programs and Partial {MCSP}}, - booktitle = {2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)}, - pages = {968--979}, - publisher = {IEEE}, - year = {2022}, - doi = {10.1109/FOCS54457.2022.00095}, - url = {https://eccc.weizmann.ac.il/report/2022/119/download}, - note = {Full version in Electronic Colloquium on Computational Complexity (ECCC), TR22-119} -} - -@article{HMPST1993, - author = {Hajnal, Andr{\'a}s and Maass, Wolfgang and Pudl{\'a}k, Pavel and - Szegedy, M{\'a}ri{\'o} and Tur{\'a}n, Gy{\"o}rgy}, - title = {Threshold Circuits of Bounded Depth}, - journal = {Journal of Computer and System Sciences}, - volume = {46}, - number = {2}, - pages = {129--154}, - year = {1993}, - doi = {10.1016/0022-0000(93)90001-D} +@inproceedings{Hirahara2022, + author = {Hirahara, Shuichi}, + title = {{NP}-Hardness of Learning Programs and Partial {MCSP}}, + booktitle = {2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)}, + pages = {968--979}, + publisher = {IEEE}, + year = {2022}, + doi = {10.1109/FOCS54457.2022.00095}, + url = {https://eccc.weizmann.ac.il/report/2022/119/download}, + note = {Full version in Electronic Colloquium on Computational Complexity (ECCC), TR22-119} } @article{HMPST1993, @@ -2831,17 +1910,6 @@ @article{Hoeffding1963 doi = {10.1080/01621459.1963.10500830} } -@article{Hoeffding1963, - author = {Hoeffding, Wassily}, - title = {Probability Inequalities for Sums of Bounded Random Variables}, - journal = {Journal of the American Statistical Association}, - volume = {58}, - number = {301}, - pages = {13--30}, - year = {1963}, - doi = {10.1080/01621459.1963.10500830} -} - @book{Holcombe1982, title={Algebraic automata theory}, author={Holcombe, W. M. L.}, @@ -2860,23 +1928,6 @@ @inproceedings{Holenstein2005 url = {https://crypto.ethz.ch/publications/files/Holens05.pdf} } -@inproceedings{Holenstein2005, - author = {Holenstein, Thomas}, - title = {Key Agreement from Weak Bit Agreement}, - booktitle = {Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing}, - year = {2005}, - url = {https://crypto.ethz.ch/publications/files/Holens05.pdf} -} - -@inproceedings{Holenstein2006, - author = {Holenstein, Thomas}, - title = {Pseudorandom Generators from One-Way Functions: A Simple Construction for - Any Hardness}, - booktitle = {Theory of Cryptography (TCC 2006)}, - year = {2006}, - url = {https://crypto.ethz.ch/publications/files/Holens06.pdf} -} - @inproceedings{Holenstein2006, author = {Holenstein, Thomas}, title = {Pseudorandom Generators from One-Way Functions: A Simple Construction for @@ -2897,17 +1948,6 @@ @article{HooryLinialWigderson2006 doi = {10.1090/S0273-0979-06-01126-8} } -@article{HooryLinialWigderson2006, - author = {Hoory, Shlomo and Linial, Nathan and Wigderson, Avi}, - title = {Expander Graphs and Their Applications}, - journal = {Bulletin of the American Mathematical Society}, - volume = {43}, - number = {4}, - pages = {439--561}, - year = {2006}, - doi = {10.1090/S0273-0979-06-01126-8} -} - @book{ Hopcroft2006, author = {Hopcroft, John E. and Motwani, Rajeev and Ullman, Jeffrey D.}, title = {Introduction to Automata Theory, Languages, and Computation (3rd Edition)}, @@ -2928,17 +1968,6 @@ @article{Huang2019 doi = {10.4007/annals.2019.190.3.6} } -@article{Huang2019, - author = {Huang, Hao}, - title = {Induced Subgraphs of Hypercubes and a Proof of the Sensitivity Conjecture}, - journal = {Annals of Mathematics}, - volume = {190}, - number = {3}, - pages = {949--955}, - year = {2019}, - doi = {10.4007/annals.2019.190.3.6} -} - @inproceedings{Ilango2020, author = {Ilango, Rahul}, title = {Approaching {MCSP} from Above and Below: Hardness for a Conditional Variant and {$\mathsf{AC}^0[p]$}}, @@ -3017,27 +2046,6 @@ @inproceedings{Impagliazzo1995 doi = {10.1109/SFCS.1995.492584} } -@inproceedings{Impagliazzo1995, - author = {Impagliazzo, Russell}, - title = {Hard-Core Distributions for Somewhat Hard Problems}, - booktitle = {Proceedings of the 36th Annual Symposium on Foundations of Computer Science}, - series = {FOCS 1995}, - pages = {538--545}, - year = {1995}, - doi = {10.1109/SFCS.1995.492584} -} - -@article{ImpagliazzoPaturi2001, - author = {Impagliazzo, Russell and Paturi, Ramamohan}, - title = {On the Complexity of {$k$-SAT}}, - journal = {Journal of Computer and System Sciences}, - volume = {62}, - number = {2}, - pages = {367--375}, - year = {2001}, - doi = {10.1006/jcss.2000.1727} -} - @article{ImpagliazzoPaturi2001, author = {Impagliazzo, Russell and Paturi, Ramamohan}, title = {On the Complexity of {$k$-SAT}}, @@ -3060,27 +2068,6 @@ @article{ImpagliazzoPaturiZane2001 doi = {10.1006/jcss.2001.1774} } -@article{ImpagliazzoPaturiZane2001, - author = {Impagliazzo, Russell and Paturi, Ramamohan and Zane, Francis}, - title = {Which Problems Have Strongly Exponential Complexity?}, - journal = {Journal of Computer and System Sciences}, - volume = {63}, - number = {4}, - pages = {512--530}, - year = {2001}, - doi = {10.1006/jcss.2001.1774} -} - -@inproceedings{ImpagliazzoWigderson1997, - author = {Impagliazzo, Russell and Wigderson, Avi}, - title = {{P} = {BPP} if {E} Requires Exponential Circuits: Derandomizing the {XOR} Lemma}, - booktitle = {Proceedings of the 29th Annual ACM Symposium on Theory of Computing}, - series = {STOC 1997}, - pages = {220--229}, - year = {1997}, - doi = {10.1145/258533.258590} -} - @inproceedings{ImpagliazzoWigderson1997, author = {Impagliazzo, Russell and Wigderson, Avi}, title = {{P} = {BPP} if {E} Requires Exponential Circuits: Derandomizing the {XOR} Lemma}, @@ -3102,17 +2089,6 @@ @article{Johnson1962 doi = {10.1109/TIT.1962.1057714} } -@article{Johnson1962, - author = {Johnson, Selmer M.}, - title = {A New Upper Bound for Error-Correcting Codes}, - journal = {IRE Transactions on Information Theory}, - volume = {8}, - number = {3}, - pages = {203--207}, - year = {1962}, - doi = {10.1109/TIT.1962.1057714} -} - @article{JohnsonPapadimitriouYannakakis1988, author = {Johnson, David S. and Papadimitriou, Christos H. and Yannakakis, Mihalis}, title = {How Easy Is Local Search?}, @@ -3135,16 +2111,6 @@ @book{Jukna2011 doi = {10.1007/978-3-642-17364-6} } -@book{Jukna2011, - author = {Jukna, Stasys}, - title = {Extremal Combinatorics: With Applications in Computer Science}, - edition = {2}, - series = {Texts in Theoretical Computer Science. An EATCS Series}, - publisher = {Springer}, - year = {2011}, - doi = {10.1007/978-3-642-17364-6} -} - @book{Jukna2012, author = {Stasys Jukna}, title = {Boolean Function Complexity: Advances and Frontiers}, @@ -3167,26 +2133,6 @@ @article{Justesen1972 doi = {10.1109/TIT.1972.1054893} } -@article{Justesen1972, - author = {Justesen, J{\o}rn}, - title = {Class of Constructive Asymptotically Good Algebraic Codes}, - journal = {IEEE Transactions on Information Theory}, - volume = {18}, - number = {5}, - pages = {652--656}, - year = {1972}, - doi = {10.1109/TIT.1972.1054893} -} - -@inproceedings{KabanetsCai2000, - author = {Kabanets, Valentine and Cai, Jin-Yi}, - title = {Circuit Minimization Problem}, - booktitle = {Proceedings of the Thirty-Second Annual {ACM} Symposium on Theory of Computing (STOC 2000)}, - pages = {73--79}, - year = {2000}, - doi = {10.1145/335305.335314} -} - @inproceedings{KabanetsCai2000, author = {Kabanets, Valentine and Cai, Jin-Yi}, title = {Circuit Minimization Problem}, @@ -3207,17 +2153,6 @@ @article{Kadin1988 doi = {10.1137/0217080} } -@article{Kadin1988, - author = {Kadin, Jim}, - title = {The Polynomial Time Hierarchy Collapses if the {Boolean} Hierarchy Collapses}, - journal = {SIAM Journal on Computing}, - volume = {17}, - number = {6}, - pages = {1263--1282}, - year = {1988}, - doi = {10.1137/0217080} -} - @article{Kahale1995, author = {Kahale, Nabil}, title = {Eigenvalues and Expansion of Regular Graphs}, @@ -3240,16 +2175,6 @@ @inproceedings{KahnKalaiLinial1988 doi = {10.1109/SFCS.1988.21923} } -@inproceedings{KahnKalaiLinial1988, - author = {Kahn, Jeff and Kalai, Gil and Linial, Nathan}, - title = {The Influence of Variables on {Boolean} Functions}, - booktitle = {29th Annual Symposium on Foundations of Computer Science}, - publisher = {IEEE Computer Society}, - year = {1988}, - pages = {68--80}, - doi = {10.1109/SFCS.1988.21923} -} - @article{KalyanasundaramSchnitger1992, author = {Kalyanasundaram, Bala and Schnitger, Georg}, title = {The Probabilistic Communication Complexity of Set Intersection}, @@ -3272,17 +2197,6 @@ @article{Kannan1982 doi = {10.1016/S0019-9958(82)90382-5} } -@article{Kannan1982, - author = {Kannan, Ravi}, - title = {Circuit-Size Lower Bounds and Non-Reducibility to Sparse Sets}, - journal = {Information and Control}, - volume = {55}, - number = {1--3}, - pages = {40--56}, - year = {1982}, - doi = {10.1016/S0019-9958(82)90382-5} -} - @article{KarchmerWigderson1990, author = {Karchmer, Mauricio and Wigderson, Avi}, title = {Monotone circuits for connectivity require super-logarithmic depth}, @@ -3314,16 +2228,6 @@ @inproceedings{KarpLipton1980 doi = {10.1145/800141.804678} } -@inproceedings{KarpLipton1980, - author = {Karp, Richard M. and Lipton, Richard J.}, - title = {Some Connections between Nonuniform and Uniform Complexity Classes}, - booktitle = {Proceedings of the 12th Annual ACM Symposium on Theory of Computing}, - series = {STOC 1980}, - pages = {302--309}, - year = {1980}, - doi = {10.1145/800141.804678} -} - @book{ KatzLindell2020, author = {Jonathan Katz and Yehuda Lindell}, @@ -3354,17 +2258,6 @@ @inproceedings{Khot2002 doi = {10.1145/509907.510017} } -@inproceedings{Khot2002, - author = {Khot, Subhash}, - title = {On the Power of Unique 2-Prover 1-Round Games}, - booktitle = {Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing}, - series = {STOC '02}, - pages = {767--775}, - publisher = {ACM}, - year = {2002}, - doi = {10.1145/509907.510017} -} - @article{Khrapchenko1971, author = {Khrapchenko, V. M.}, title = {Complexity of the realization of a linear function in the class of $\Pi$-circuits}, @@ -3441,25 +2334,6 @@ @techreport{Korten2026 year = {2026} } -@techreport{Korten2026, - author = {Korten, Oliver}, - title = {Top-Down Lower Bounds for All Depths}, - institution = {Electronic Colloquium on Computational Complexity}, - number = {TR26-221}, - year = {2026} -} - -@article{KushilevitzMansour1993, - author = {Kushilevitz, Eyal and Mansour, Yishay}, - title = {Learning Decision Trees Using the {F}ourier Spectrum}, - journal = {SIAM Journal on Computing}, - volume = {22}, - number = {6}, - pages = {1331--1348}, - year = {1993}, - doi = {10.1137/0222080} -} - @article{KushilevitzMansour1993, author = {Kushilevitz, Eyal and Mansour, Yishay}, title = {Learning Decision Trees Using the {F}ourier Spectrum}, @@ -3489,17 +2363,6 @@ @article{Ladner1975 doi = {10.1145/990518.990519} } -@article{Ladner1975, - author = {Ladner, Richard E.}, - title = {The Circuit Value Problem Is Log Space Complete for {$P$}}, - journal = {ACM SIGACT News}, - volume = {7}, - number = {1}, - pages = {18--20}, - year = {1975}, - doi = {10.1145/990518.990519} -} - @article{LadnerFischer1980, author = {Ladner, Richard E. and Fischer, Michael J.}, title = {Parallel Prefix Computation}, @@ -3544,17 +2407,6 @@ @article{Lautemann1983 doi = {10.1016/0020-0190(83)90044-3} } -@article{Lautemann1983, - author = {Lautemann, Clemens}, - title = {{BPP} and the Polynomial Hierarchy}, - journal = {Information Processing Letters}, - volume = {17}, - number = {4}, - pages = {215--217}, - year = {1983}, - doi = {10.1016/0020-0190(83)90044-3} -} - @misc{LeanCanny2026, author = {{lean-canny contributors}}, title = {lean-canny}, @@ -3571,14 +2423,6 @@ @misc{LecomteRamakrishnan2026 archivePrefix = {arXiv} } -@misc{LecomteRamakrishnan2026, - author = {Lecomte, Victor and Ramakrishnan, Prasanna}, - title = {Optimal Shallow Circuits for Majority}, - year = {2026}, - eprint = {2609.34029}, - archivePrefix = {arXiv} -} - @article{Levin1973, author = {Levin, Leonid A.}, title = {Universal Sequential Search Problems}, @@ -3611,27 +2455,6 @@ @article{LewisPapadimitriou1982 doi = {10.1016/0304-3975(82)90058-5} } -@article{LewisPapadimitriou1982, - author = {Lewis, Harry R. and Papadimitriou, Christos H.}, - title = {Symmetric Space-Bounded Computation}, - journal = {Theoretical Computer Science}, - volume = {19}, - number = {2}, - pages = {161--187}, - year = {1982}, - doi = {10.1016/0304-3975(82)90058-5} -} - -@inproceedings{Li2023, - author = {Li, Xin}, - title = {Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More}, - booktitle = {Proceedings of the 64th Annual IEEE Symposium on Foundations of Computer Science}, - series = {FOCS 2023}, - pages = {1271--1281}, - year = {2023}, - doi = {10.1109/FOCS57990.2023.00075} -} - @inproceedings{Li2023, author = {Li, Xin}, title = {Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More}, @@ -3664,28 +2487,6 @@ @book{LidlNiederreiter1997 doi = {10.1017/CBO9780511525926} } -@book{LidlNiederreiter1997, - author = {Lidl, Rudolf and Niederreiter, Harald}, - title = {Finite Fields}, - edition = {2}, - series = {Encyclopedia of Mathematics and its Applications}, - volume = {20}, - publisher = {Cambridge University Press}, - year = {1997}, - doi = {10.1017/CBO9780511525926} -} - -@article{LinialMansourNisan1993, - author = {Linial, Nathan and Mansour, Yishay and Nisan, Noam}, - title = {Constant Depth Circuits, {F}ourier Transform, and Learnability}, - journal = {Journal of the ACM}, - volume = {40}, - number = {3}, - pages = {607--620}, - year = {1993}, - doi = {10.1145/174130.174138} -} - @article{LinialMansourNisan1993, author = {Linial, Nathan and Mansour, Yishay and Nisan, Noam}, title = {Constant Depth Circuits, {F}ourier Transform, and Learnability}, @@ -3706,15 +2507,6 @@ @inproceedings{LiptonYoung1994 doi = {10.1145/195058.195447} } -@inproceedings{LiptonYoung1994, - author = {Lipton, Richard J. and Young, Neal E.}, - title = {Simple Strategies for Large Zero-Sum Games with Applications to Complexity Theory}, - booktitle = {Proceedings of the Twenty-Sixth Annual {ACM} Symposium on Theory of Computing (STOC 1994)}, - pages = {734--740}, - year = {1994}, - doi = {10.1145/195058.195447} -} - @inproceedings{LiuPass2020, author = {Liu, Yanyi and Pass, Rafael}, title = {On One-way Functions and {Kolmogorov} Complexity}, @@ -3748,27 +2540,6 @@ @book{LiVitanyi2019 doi = {10.1007/978-3-030-11298-1} } -@book{LiVitanyi2019, - author = {Li, Ming and Vit{\'a}nyi, Paul}, - title = {An Introduction to {K}olmogorov Complexity and Its Applications}, - edition = {4th}, - series = {Texts in Computer Science}, - publisher = {Springer}, - address = {Cham}, - year = {2019}, - doi = {10.1007/978-3-030-11298-1} -} - -@inproceedings{LiYang2022, - author = {Li, Jiatu and Yang, Tianqi}, - title = {{$3.1n - o(n)$} Circuit Lower Bounds for Explicit Functions}, - booktitle = {Proceedings of the 54th Annual ACM Symposium on Theory of Computing}, - series = {STOC 2022}, - pages = {1180--1193}, - year = {2022}, - doi = {10.1145/3519935.3519976} -} - @inproceedings{LiYang2022, author = {Li, Jiatu and Yang, Tianqi}, title = {{$3.1n - o(n)$} Circuit Lower Bounds for Explicit Functions}, @@ -3788,15 +2559,6 @@ @article{LokshtanovMarxSaurabh2011 year = {2011} } -@article{LokshtanovMarxSaurabh2011, - author = {Lokshtanov, Daniel and Marx, D{\'a}niel and Saurabh, Saket}, - title = {Lower bounds based on the {Exponential Time Hypothesis}}, - journal = {Bulletin of the EATCS}, - volume = {105}, - pages = {41--72}, - year = {2011} -} - @article{LokshtanovMarxSaurabh2018, author = {Lokshtanov, Daniel and Marx, D{\'a}niel and Saurabh, Saket}, title = {Slightly Superexponential Parameterized Problems}, @@ -3819,27 +2581,6 @@ @article{LoomisWhitney1949 doi = {10.1090/S0002-9904-1949-09320-5} } -@article{LoomisWhitney1949, - author = {Loomis, Lynn H. and Whitney, Hassler}, - title = {An Inequality Related to the Isoperimetric Inequality}, - journal = {Bulletin of the American Mathematical Society}, - volume = {55}, - number = {10}, - pages = {961--962}, - year = {1949}, - doi = {10.1090/S0002-9904-1949-09320-5} -} - -@inproceedings{LovaszSaks1988, - author = {Lov{\'a}sz, L{\'a}szl{\'o} and Saks, Michael}, - title = {Lattices, {M{\"o}bius} Functions and Communication Complexity}, - booktitle = {Proceedings of the 29th Annual Symposium on Foundations of Computer Science}, - series = {FOCS 1988}, - pages = {81--90}, - year = {1988}, - doi = {10.1109/SFCS.1988.21924} -} - @inproceedings{LovaszSaks1988, author = {Lov{\'a}sz, L{\'a}szl{\'o} and Saks, Michael}, title = {Lattices, {M{\"o}bius} Functions and Communication Complexity}, @@ -3872,17 +2613,6 @@ @article{LundFortnowKarloffNisan1992 doi = {10.1145/146585.146605} } -@article{LundFortnowKarloffNisan1992, - author = {Lund, Carsten and Fortnow, Lance and Karloff, Howard and Nisan, Noam}, - title = {Algebraic Methods for Interactive Proof Systems}, - journal = {Journal of the ACM}, - volume = {39}, - number = {4}, - pages = {859--868}, - year = {1992}, - doi = {10.1145/146585.146605} -} - @article{Lupanov1958, author = {Oleg B. Lupanov}, title = {On a Method of Circuit Synthesis}, @@ -3905,16 +2635,6 @@ @article{Lupanov1965 note = {In Russian} } -@article{Lupanov1965, - author = {Lupanov, O. B.}, - title = {On an Approach to the Synthesis of Control Systems: the Principle of Local Coding}, - journal = {Problemy Kibernetiki}, - volume = {14}, - pages = {31--110}, - year = {1965}, - note = {In Russian} -} - @inproceedings{Lyubashevsky2009, author = {Lyubashevsky, Vadim}, title = {Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based @@ -3949,15 +2669,6 @@ @book{MacWilliamsSloane1977 year = {1977} } -@book{MacWilliamsSloane1977, - author = {MacWilliams, F. J. and Sloane, N. J. A.}, - title = {The Theory of Error-Correcting Codes}, - series = {North-Holland Mathematical Library}, - volume = {16}, - publisher = {North-Holland}, - year = {1977} -} - @misc{ Malkin2024, author = {Tal Malkin}, title = {{COMS W3261: Computer Science Theory, Handout 3: The Myhill-Nerode Theorem and Implications}}, @@ -3980,19 +2691,6 @@ @incollection{McDiarmid1989 doi = {10.1017/CBO9781107359949.008} } -@incollection{McDiarmid1989, - author = {McDiarmid, Colin}, - title = {On the Method of Bounded Differences}, - booktitle = {Surveys in Combinatorics, 1989}, - editor = {Siemons, J.}, - series = {London Mathematical Society Lecture Note Series}, - volume = {141}, - pages = {148--188}, - publisher = {Cambridge University Press}, - year = {1989}, - doi = {10.1017/CBO9781107359949.008} -} - @article{MegiddoPapadimitriou1991, author = {Megiddo, Nimrod and Papadimitriou, Christos H.}, title = {On Total Functions, Existence Theorems and Computational Complexity}, @@ -4015,16 +2713,6 @@ @inproceedings{MehlhornSchmidt1982 doi = {10.1145/800070.802208} } -@inproceedings{MehlhornSchmidt1982, - author = {Mehlhorn, Kurt and Schmidt, Erik M.}, - title = {{Las Vegas} Is Better than Determinism in {VLSI} and Distributed Computing}, - booktitle = {Proceedings of the 14th Annual ACM Symposium on Theory of Computing}, - series = {STOC 1982}, - pages = {330--337}, - year = {1982}, - doi = {10.1145/800070.802208} -} - @book{ MilewskiDao, author = {Milewski, Bartosz}, title = {The Dao of Functional Programming}, @@ -4055,22 +2743,6 @@ @book{MinskyPapert1969 year = {1969} } -@book{MinskyPapert1969, - author = {Minsky, Marvin and Papert, Seymour}, - title = {Perceptrons: An Introduction to Computational Geometry}, - publisher = {MIT Press}, - year = {1969} -} - -@techreport{MishraSanthanam2026, - author = {Mishra, Om Swostik and Santhanam, Rahul}, - title = {An Explicit {$3.5n - o(n)$} Size Lower Bound against General {Boolean} Circuits}, - institution = {Electronic Colloquium on Computational Complexity}, - number = {TR26-260}, - year = {2026}, - url = {https://eccc.weizmann.ac.il/report/2026/260/} -} - @techreport{MishraSanthanam2026, author = {Mishra, Om Swostik and Santhanam, Rahul}, title = {An Explicit {$3.5n - o(n)$} Size Lower Bound against General {Boolean} Circuits}, @@ -4117,15 +2789,6 @@ @book{MitzenmacherUpfal2017 year = {2017} } -@book{MitzenmacherUpfal2017, - author = {Mitzenmacher, Michael and Upfal, Eli}, - title = {Probability and Computing: Randomization and Probabilistic Techniques in - Algorithms and Data Analysis}, - edition = {2}, - publisher = {Cambridge University Press}, - year = {2017} -} - @Book{ Montesi2023, title = {Introduction to {Choreographies}}, author = {Montesi, Fabrizio}, @@ -4181,28 +2844,6 @@ @article{MoserTardos2010 doi = {10.1145/1667053.1667060} } -@article{MoserTardos2010, - author = {Moser, Robin A. and Tardos, G{\'a}bor}, - title = {A Constructive Proof of the General {Lov{\'a}sz} Local Lemma}, - journal = {Journal of the ACM}, - volume = {57}, - number = {2}, - pages = {11:1--11:15}, - year = {2010}, - doi = {10.1145/1667053.1667060} -} - -@article{Muller1954, - author = {Muller, David E.}, - title = {Application of {Boolean} Algebra to Switching Circuit Design and to Error Detection}, - journal = {Transactions of the I.R.E. Professional Group on Electronic Computers}, - volume = {EC-3}, - number = {3}, - pages = {6--12}, - year = {1954}, - doi = {10.1109/IREPGELC.1954.6499441} -} - @article{Muller1954, author = {Muller, David E.}, title = {Application of {Boolean} Algebra to Switching Circuit Design and to Error Detection}, @@ -4225,17 +2866,6 @@ @article{MulmuleyVaziraniVazirani1987 doi = {10.1007/BF02579206} } -@article{MulmuleyVaziraniVazirani1987, - author = {Mulmuley, Ketan and Vazirani, Umesh V. and Vazirani, Vijay V.}, - title = {Matching Is as Easy as Matrix Inversion}, - journal = {Combinatorica}, - volume = {7}, - number = {1}, - pages = {105--113}, - year = {1987}, - doi = {10.1007/BF02579206} -} - @article{Naor1991, author = {Naor, Moni}, title = {Bit Commitment Using Pseudorandomness}, @@ -4259,29 +2889,6 @@ @article{NaorNaor1993 doi = {10.1137/0222053} } -@article{NaorNaor1993, - author = {Naor, Joseph and Naor, Moni}, - title = {Small-Bias Probability Spaces: Efficient Constructions and Applications}, - journal = {SIAM Journal on Computing}, - volume = {22}, - number = {4}, - pages = {838--856}, - year = {1993}, - doi = {10.1137/0222053} -} - -@article{Nechiporuk1965, - author = {Nechiporuk, E. I.}, - title = {On the Complexity of Gating Circuits Realizing {B}oolean Matrices with Undefined - Elements}, - journal = {Doklady Akademii Nauk SSSR}, - volume = {163}, - number = {1}, - pages = {40--42}, - year = {1965}, - note = {In Russian} -} - @article{Nechiporuk1965, author = {Nechiporuk, E. I.}, title = {On the Complexity of Gating Circuits Realizing {B}oolean Matrices with Undefined @@ -4326,17 +2933,6 @@ @article{Newman1991 doi = {10.1016/0020-0190(91)90157-D} } -@article{Newman1991, - author = {Newman, Ilan}, - title = {Private vs. Common Random Bits in Communication Complexity}, - journal = {Information Processing Letters}, - volume = {39}, - number = {2}, - pages = {67--71}, - year = {1991}, - doi = {10.1016/0020-0190(91)90157-D} -} - @article{ Nipkow2001, title = {More {Church-Rosser} Proofs (in {Isabelle/HOL})}, author = {Nipkow, Tobias}, @@ -4347,25 +2943,14 @@ @article{ Nipkow2001 publisher = {Kluwer Academic Publishers} } -@article{Nisan1991, - author = {Nisan, Noam}, - title = {{CREW} {PRAM}s and decision trees}, - journal = {SIAM Journal on Computing}, - volume = {20}, - number = {6}, - pages = {999--1007}, - year = {1991} -} - -@article{NisanSzegedy1994, - author = {Nisan, Noam and Szegedy, Mario}, - title = {On the Degree of {Boolean} Functions as Real Polynomials}, - journal = {Computational Complexity}, - volume = {4}, - number = {4}, - pages = {301--313}, - year = {1994}, - doi = {10.1007/BF01263419} +@article{Nisan1991, + author = {Nisan, Noam}, + title = {{CREW} {PRAM}s and decision trees}, + journal = {SIAM Journal on Computing}, + volume = {20}, + number = {6}, + pages = {999--1007}, + year = {1991} } @article{NisanSzegedy1994, @@ -4390,17 +2975,6 @@ @article{NisanWigderson1994 doi = {10.1016/S0022-0000(05)80043-1} } -@article{NisanWigderson1994, - author = {Nisan, Noam and Wigderson, Avi}, - title = {Hardness vs Randomness}, - journal = {Journal of Computer and System Sciences}, - volume = {49}, - number = {2}, - pages = {149--167}, - year = {1994}, - doi = {10.1016/S0022-0000(05)80043-1} -} - @article{NisanZuckerman1996, author = {Nisan, Noam and Zuckerman, David}, title = {Randomness is Linear in Space}, @@ -4452,18 +3026,6 @@ @inproceedings{OliveiraPichSanthanam2019 doi = {10.4230/LIPIcs.CCC.2019.27} } -@inproceedings{OliveiraPichSanthanam2019, - author = {Oliveira, Igor Carboni and Pich, J{\'a}n and Santhanam, Rahul}, - title = {Hardness Magnification near State-Of-The-Art Lower Bounds}, - booktitle = {34th Computational Complexity Conference (CCC 2019)}, - series = {Leibniz International Proceedings in Informatics (LIPIcs)}, - volume = {137}, - pages = {27:1--27:29}, - publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, - year = {2019}, - doi = {10.4230/LIPIcs.CCC.2019.27} -} - @misc{OpenAI2026RationalHitting, author = {{OpenAI}}, title = {{Rational} matrix hitting lists for noncommutative rational formulas}, @@ -4480,25 +3042,6 @@ @misc{OpenAI2026Sensitivity url = {https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Sensitivity} } -@misc{OpenAI2026Sensitivity, - author = {{OpenAI}}, - title = {{Sensitivity} versus block sensitivity: a superquadratic separation}, - year = {2026}, - note = {Lean formalization, revision adc7f1241b42e322a6451854ab7e4b4c146bf78a}, - url = {https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Sensitivity} -} - -@article{Ore1933, - author = {Ore, Oystein}, - title = {On a Special Class of Polynomials}, - journal = {Transactions of the American Mathematical Society}, - volume = {35}, - number = {3}, - pages = {559--584}, - year = {1933}, - doi = {10.1090/S0002-9947-1933-1501703-0} -} - @article{Ore1933, author = {Ore, Oystein}, title = {On a Special Class of Polynomials}, @@ -4521,17 +3064,6 @@ @article{Oum2005 doi = {10.1016/j.jctb.2005.03.003} } -@article{Oum2005, - author = {Oum, Sang-il}, - title = {Rank-width and Vertex-minors}, - journal = {Journal of Combinatorial Theory, Series B}, - volume = {95}, - number = {1}, - pages = {79--100}, - year = {2005}, - doi = {10.1016/j.jctb.2005.03.003} -} - @article{Papadimitriou1994PPAD, author = {Papadimitriou, Christos H.}, title = {On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence}, @@ -4576,15 +3108,6 @@ @article{PaturiPudlakZane1999 year = {1999} } -@article{PaturiPudlakZane1999, - author = {Paturi, Ramamohan and Pudl{\'a}k, Pavel and Zane, Francis}, - title = {Satisfiability Coding Lemma}, - journal = {Chicago Journal of Theoretical Computer Science}, - volume = {1999}, - number = {11}, - year = {1999} -} - @book{ Pierce2002, author = {Benjamin C. Pierce}, title = {Types and programming languages}, @@ -4603,16 +3126,6 @@ @inproceedings{Radhakrishnan2006 doi = {10.1007/11786986_10} } -@inproceedings{Radhakrishnan2006, - author = {Radhakrishnan, Jaikumar}, - title = {Gap Amplification in {PCP}s Using Lazy Random Walks}, - booktitle = {Automata, Languages and Programming (ICALP 2006)}, - series = {Lecture Notes in Computer Science}, - pages = {96--107}, - year = {2006}, - doi = {10.1007/11786986_10} -} - @article{Razborov1985, author = {Razborov, Alexander A.}, title = {Lower Bounds on the Monotone Complexity of Some {B}oolean Functions}, @@ -4654,17 +3167,6 @@ @article{Razborov1992 doi = {10.1016/0304-3975(92)90260-M} } -@article{Razborov1992, - author = {Razborov, Alexander A.}, - title = {On the Distributional Complexity of Disjointness}, - journal = {Theoretical Computer Science}, - volume = {106}, - number = {2}, - pages = {385--390}, - year = {1992}, - doi = {10.1016/0304-3975(92)90260-M} -} - @incollection{Razborov1995, author = {Razborov, Alexander A.}, title = {Bounded Arithmetic and Lower Bounds in {B}oolean Complexity}, @@ -4688,17 +3190,6 @@ @article{RazborovRudich1997 doi = {10.1006/jcss.1997.1494} } -@article{RazborovRudich1997, - author = {Razborov, Alexander A. and Rudich, Steven}, - title = {Natural Proofs}, - journal = {Journal of Computer and System Sciences}, - volume = {55}, - number = {1}, - pages = {24--35}, - year = {1997}, - doi = {10.1006/jcss.1997.1494} -} - @article{RazWigderson1992, author = {Raz, Ran and Wigderson, Avi}, title = {Monotone Circuits for Matching Require Linear Depth}, @@ -4721,28 +3212,6 @@ @article{Reed1954 doi = {10.1109/TIT.1954.1057465} } -@article{Reed1954, - author = {Reed, Irving S.}, - title = {A Class of Multiple-Error-Correcting Codes and the Decoding Scheme}, - journal = {Transactions of the IRE Professional Group on Information Theory}, - volume = {4}, - number = {4}, - pages = {38--49}, - year = {1954}, - doi = {10.1109/TIT.1954.1057465} -} - -@article{ReedSolomon1960, - author = {Reed, Irving S. and Solomon, Gustave}, - title = {Polynomial Codes Over Certain Finite Fields}, - journal = {Journal of the Society for Industrial and Applied Mathematics}, - volume = {8}, - number = {2}, - pages = {300--304}, - year = {1960}, - doi = {10.1137/0108018} -} - @article{ReedSolomon1960, author = {Reed, Irving S. and Solomon, Gustave}, title = {Polynomial Codes Over Certain Finite Fields}, @@ -4765,28 +3234,6 @@ @article{Reingold2008 doi = {10.1145/1391289.1391291} } -@article{Reingold2008, - author = {Reingold, Omer}, - title = {Undirected Connectivity in Log-Space}, - journal = {Journal of the ACM}, - volume = {55}, - number = {4}, - pages = {17:1--17:24}, - year = {2008}, - doi = {10.1145/1391289.1391291} -} - -@article{ReingoldVadhanWigderson2002, - author = {Reingold, Omer and Vadhan, Salil and Wigderson, Avi}, - title = {Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders}, - journal = {Annals of Mathematics}, - volume = {155}, - number = {1}, - pages = {157--187}, - year = {2002}, - doi = {10.2307/3062153} -} - @article{ReingoldVadhanWigderson2002, author = {Reingold, Omer and Vadhan, Salil and Wigderson, Avi}, title = {Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders}, @@ -4818,25 +3265,6 @@ @article{Rice1953 doi = {10.1090/S0002-9947-1953-0053041-6} } -@article{Rice1953, - author = {Rice, Henry Gordon}, - title = {Classes of Recursively Enumerable Sets and Their Decision Problems}, - journal = {Transactions of the American Mathematical Society}, - volume = {74}, - number = {2}, - pages = {358--366}, - year = {1953}, - doi = {10.1090/S0002-9947-1953-0053041-6} -} - -@book{Rivlin1990, - author = {Rivlin, Theodore J.}, - title = {Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory}, - edition = {2}, - publisher = {Wiley}, - year = {1990} -} - @book{Rivlin1990, author = {Rivlin, Theodore J.}, title = {Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory}, @@ -4879,17 +3307,6 @@ @article{Rubinstein1995 doi = {10.1007/BF01200762} } -@article{Rubinstein1995, - author = {Rubinstein, David}, - title = {Sensitivity vs. Block Sensitivity of {Boolean} Functions}, - journal = {Combinatorica}, - volume = {15}, - number = {2}, - pages = {297--299}, - year = {1995}, - doi = {10.1007/BF01200762} -} - @Book{ Sangiorgi2011, location = {Cambridge}, title = {Introduction to Bisimulation and Coinduction}, @@ -4980,14 +3397,6 @@ @misc{Schlesinger2026Sensitivity url = {https://github.com/SamuelSchlesinger/complexitylib-v1/tree/add0aca8ee0d71e68969f2cb45cf91554a50f787/Complexitylib/BooleanAnalysis/Sensitivity} } -@misc{Schlesinger2026Sensitivity, - author = {Schlesinger, Samuel}, - title = {{Sensitivity} separation in {Complexitylib}}, - year = {2026}, - note = {Source of ported Lean proofs, revision add0aca8ee0d71e68969f2cb45cf91554a50f787}, - url = {https://github.com/SamuelSchlesinger/complexitylib-v1/tree/add0aca8ee0d71e68969f2cb45cf91554a50f787/Complexitylib/BooleanAnalysis/Sensitivity} -} - @article{Schnorr1974, author = {Schnorr, Claus-Peter}, title = {Zwei lineare untere {S}chranken f{\"u}r die {K}omplexit{\"a}t {B}oolescher {F}unktionen}, @@ -5112,28 +3521,6 @@ @article{Sheppard1899 doi = {10.1098/rsta.1899.0003} } -@article{Sheppard1899, - author = {Sheppard, W. F.}, - title = {On the Application of the Theory of Error to Cases of Normal Distribution and Normal - Correlation}, - journal = {Philosophical Transactions of the Royal Society of London. Series A}, - volume = {192}, - year = {1899}, - pages = {101--167}, - doi = {10.1098/rsta.1899.0003} -} - -@article{Sholomov1969, - author = {Sholomov, L. A.}, - title = {On the Realization of Incompletely Defined {B}oolean Functions by Circuits of - Functional Elements}, - journal = {Problemy Kibernetiki}, - volume = {21}, - pages = {215--226}, - year = {1969}, - note = {In Russian} -} - @article{Sholomov1969, author = {Sholomov, L. A.}, title = {On the Realization of Incompletely Defined {B}oolean Functions by Circuits of @@ -5163,28 +3550,6 @@ @article{Singleton1964 doi = {10.1109/TIT.1964.1053661} } -@article{Singleton1964, - author = {Singleton, Richard C.}, - title = {Maximum Distance $q$-Nary Codes}, - journal = {IEEE Transactions on Information Theory}, - volume = {10}, - number = {2}, - pages = {116--118}, - year = {1964}, - doi = {10.1109/TIT.1964.1053661} -} - -@article{Sion1958, - author = {Sion, Maurice}, - title = {On General Minimax Theorems}, - journal = {Pacific Journal of Mathematics}, - volume = {8}, - number = {1}, - pages = {171--176}, - year = {1958}, - doi = {10.2140/pjm.1958.8.171} -} - @article{Sion1958, author = {Sion, Maurice}, title = {On General Minimax Theorems}, @@ -5205,34 +3570,13 @@ @inproceedings{Sipser1983 doi = {10.1145/800061.808762}, url = {https://doi.org/10.1145/800061.808762} } - -@inproceedings{Sipser1983, - author = {Sipser, Michael}, - title = {A Complexity Theoretic Approach to Randomness}, - booktitle = {Proceedings of the 15th Annual ACM Symposium on Theory of Computing}, - pages = {330--335}, - year = {1983}, - doi = {10.1145/800061.808762}, - url = {https://doi.org/10.1145/800061.808762} -} - -@book{Sipser2013, - author = {Sipser, Michael}, - title = {Introduction to the Theory of Computation}, - edition = {3rd}, - publisher = {Cengage Learning}, - year = {2013} -} - -@article{SkyumValiant1985, - author = {Skyum, Sven and Valiant, Leslie G.}, - title = {A Complexity Theory Based on {B}oolean Algebra}, - journal = {Journal of the ACM}, - volume = {32}, - number = {2}, - pages = {484--502}, - year = {1985}, - doi = {10.1145/3149.3158} + +@book{Sipser2013, + author = {Sipser, Michael}, + title = {Introduction to the Theory of Computation}, + edition = {3rd}, + publisher = {Cengage Learning}, + year = {2013} } @article{SkyumValiant1985, @@ -5287,28 +3631,6 @@ @inproceedings{StearnsHartmanisLewis1965 doi = {10.1109/FOCS.1965.11} } -@inproceedings{StearnsHartmanisLewis1965, - author = {Stearns, Richard E. and Hartmanis, Juris and Lewis, II, Philip M.}, - title = {Hierarchies of Memory Limited Computations}, - booktitle = {Proceedings of the 6th Annual Symposium on Switching Circuit Theory and Logical - Design}, - series = {SWCT 1965}, - pages = {179--190}, - year = {1965}, - doi = {10.1109/FOCS.1965.11} -} - -@article{Stockmeyer1976, - author = {Stockmeyer, Larry J.}, - title = {The Polynomial-Time Hierarchy}, - journal = {Theoretical Computer Science}, - volume = {3}, - number = {1}, - pages = {1--22}, - year = {1976}, - doi = {10.1016/0304-3975(76)90061-X} -} - @article{Stockmeyer1976, author = {Stockmeyer, Larry J.}, title = {The Polynomial-Time Hierarchy}, @@ -5328,23 +3650,6 @@ @inproceedings{Stockmeyer1983 year = {1983} } -@inproceedings{Stockmeyer1983, - author = {Stockmeyer, Larry J.}, - title = {The Complexity of Approximate Counting}, - booktitle = {Proceedings of the 15th Annual ACM Symposium on Theory of Computing}, - pages = {118--126}, - year = {1983} -} - -@inproceedings{StockmeyerMeyer1973, - author = {Stockmeyer, Larry J. and Meyer, Albert R.}, - title = {Word Problems Requiring Exponential Time (Preliminary Report)}, - booktitle = {Proceedings of the Fifth Annual {ACM} Symposium on Theory of Computing (STOC 1973)}, - pages = {1--9}, - year = {1973}, - doi = {10.1145/800125.804029} -} - @inproceedings{StockmeyerMeyer1973, author = {Stockmeyer, Larry J. and Meyer, Albert R.}, title = {Word Problems Requiring Exponential Time (Preliminary Report)}, @@ -5375,28 +3680,6 @@ @article{Sudan1997 doi = {10.1006/jcom.1997.0439} } -@article{Sudan1997, - author = {Sudan, Madhu}, - title = {Decoding of {Reed Solomon} Codes beyond the Error-Correction Bound}, - journal = {Journal of Complexity}, - volume = {13}, - number = {1}, - pages = {180--193}, - year = {1997}, - doi = {10.1006/jcom.1997.0439} -} - -@article{SudanTrevisanVadhan2001, - author = {Sudan, Madhu and Trevisan, Luca and Vadhan, Salil}, - title = {Pseudorandom Generators without the {XOR} Lemma}, - journal = {Journal of Computer and System Sciences}, - volume = {62}, - number = {2}, - pages = {236--266}, - year = {2001}, - doi = {10.1006/jcss.2000.1730} -} - @article{SudanTrevisanVadhan2001, author = {Sudan, Madhu and Trevisan, Luca and Vadhan, Salil}, title = {Pseudorandom Generators without the {XOR} Lemma}, @@ -5428,17 +3711,6 @@ @article{Talagrand1994 doi = {10.1214/aop/1176988612} } -@article{Talagrand1994, - author = {Talagrand, Michel}, - title = {On {Russo}'s Approximate Zero-One Law}, - journal = {The Annals of Probability}, - volume = {22}, - number = {3}, - pages = {1576--1587}, - year = {1994}, - doi = {10.1214/aop/1176988612} -} - @article{Tardos1989, author = {Tardos, G{\'a}bor}, title = {Query complexity, or why is it difficult to separate {NP}$^A \cap$ co{NP}$^A$ from @@ -5480,15 +3752,6 @@ @misc{ThresholdCircuitRepresentations2026 url = {https://github.com/SamuelSchlesinger/threshold-circuits/blob/791fa2749317fbcc7b5c5b030a1e6fadd28cc44c/threshold_circuit_representations.tex} } -@misc{ThresholdCircuitRepresentations2026, - title = {Threshold Circuits as Decision Trees and Finite-State Programs}, - year = {2026}, - howpublished = {Research draft, second pass}, - note = {October 18, 2026; repository revision - 791fa2749317fbcc7b5c5b030a1e6fadd28cc44c}, - url = {https://github.com/SamuelSchlesinger/threshold-circuits/blob/791fa2749317fbcc7b5c5b030a1e6fadd28cc44c/threshold_circuit_representations.tex} -} - @article{Toda1991, author = {Toda, Seinosuke}, title = {{PP} Is as Hard as the Polynomial-Time Hierarchy}, @@ -5553,17 +3816,6 @@ @article{Turing1936 doi = {10.1112/plms/s2-42.1.230} } -@article{Turing1936, - author = {Turing, Alan M.}, - title = {On Computable Numbers, with an Application to the {Entscheidungsproblem}}, - journal = {Proceedings of the London Mathematical Society}, - volume = {s2-42}, - number = {1}, - pages = {230--265}, - year = {1936}, - doi = {10.1112/plms/s2-42.1.230} -} - @incollection{Uhlig1992, author = {Uhlig, Dietmar}, title = {Networks Computing {Boolean} Functions for Multiple Input Values}, @@ -5601,17 +3853,6 @@ @article{Vadhan2012 doi = {10.1561/0400000010} } -@article{Vadhan2012, - author = {Vadhan, Salil P.}, - title = {Pseudorandomness}, - journal = {Foundations and Trends in Theoretical Computer Science}, - volume = {7}, - number = {1--3}, - pages = {1--336}, - year = {2012}, - doi = {10.1561/0400000010} -} - @inproceedings{Valiant1977, author = {Valiant, Leslie G.}, title = {Graph-Theoretic Arguments in Low-Level Complexity}, @@ -5635,28 +3876,6 @@ @article{Valiant1979 doi = {10.1016/0304-3975(79)90044-6} } -@article{Valiant1979, - author = {Valiant, Leslie G.}, - title = {The Complexity of Computing the Permanent}, - journal = {Theoretical Computer Science}, - volume = {8}, - number = {2}, - pages = {189--201}, - year = {1979}, - doi = {10.1016/0304-3975(79)90044-6} -} - -@inproceedings{Valiant1979Algebra, - author = {Valiant, Leslie G.}, - title = {Completeness Classes in Algebra}, - booktitle = {Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing}, - series = {STOC '79}, - pages = {249--261}, - publisher = {ACM}, - year = {1979}, - doi = {10.1145/800135.804419} -} - @inproceedings{Valiant1979Algebra, author = {Valiant, Leslie G.}, title = {Completeness Classes in Algebra}, @@ -5692,16 +3911,6 @@ @article{ValiantVazirani1986 doi = {10.1016/0304-3975(86)90135-0} } -@article{ValiantVazirani1986, - author = {Valiant, Leslie G. and Vazirani, Vijay V.}, - title = {{NP} is as Easy as Detecting Unique Solutions}, - journal = {Theoretical Computer Science}, - volume = {47}, - pages = {85--93}, - year = {1986}, - doi = {10.1016/0304-3975(86)90135-0} -} - @article{VanEmdeBoasKaasZijlstra1977, author = {van Emde Boas, Peter and Kaas, R. and Zijlstra, E.}, title = {Design and Implementation of an Efficient Priority Queue}, @@ -5732,22 +3941,6 @@ @article{Varshamov1957 year = {1957} } -@article{Varshamov1957, - author = {Varshamov, R. R.}, - title = {Estimate of the Number of Signals in Error Correcting Codes}, - journal = {Doklady Akademii Nauk SSSR}, - volume = {117}, - pages = {739--741}, - year = {1957} -} - -@phdthesis{Vazirani1986, - author = {Vazirani, Umesh V.}, - title = {Randomness, Adversaries and Computation}, - school = {University of California, Berkeley}, - year = {1986} -} - @phdthesis{Vazirani1986, author = {Vazirani, Umesh V.}, title = {Randomness, Adversaries and Computation}, @@ -5788,17 +3981,6 @@ @article{vonNeumann1928 doi = {10.1007/BF01448847} } -@article{vonNeumann1928, - author = {von Neumann, John}, - title = {Zur {Theorie} der {Gesellschaftsspiele}}, - journal = {Mathematische Annalen}, - volume = {100}, - number = {1}, - pages = {295--320}, - year = {1928}, - doi = {10.1007/BF01448847} -} - @article{Wagner1986, author = {Wagner, Klaus W.}, title = {The Complexity of Combinatorial Problems with Succinct Input Representation}, @@ -5819,15 +4001,6 @@ @techreport{Watson2012 url = {https://eccc.weizmann.ac.il/report/2012/070/} } -@techreport{Watson2012, - author = {Watson, Thomas}, - title = {The Complexity of Estimating Min-Entropy}, - institution = {Electronic Colloquium on Computational Complexity}, - number = {TR12-070}, - year = {2012}, - url = {https://eccc.weizmann.ac.il/report/2012/070/} -} - @book{Wegener1987, author = {Wegener, Ingo}, title = {The Complexity of {Boolean} Functions}, @@ -5847,24 +4020,6 @@ @article{WegmanCarter1981 doi = {10.1016/0022-0000(81)90033-7} } -@article{WegmanCarter1981, - author = {Wegman, Mark N. and Carter, J. Lawrence}, - title = {New Hash Functions and Their Use in Authentication and Set Equality}, - journal = {Journal of Computer and System Sciences}, - volume = {22}, - number = {3}, - pages = {265--279}, - year = {1981}, - doi = {10.1016/0022-0000(81)90033-7} -} - -@misc{WelchBerlekamp1986, - author = {Welch, Lloyd R. and Berlekamp, Elwyn R.}, - title = {Error Correction for Algebraic Block Codes}, - note = {U.S. Patent 4,633,470}, - year = {1986} -} - @misc{WelchBerlekamp1986, author = {Welch, Lloyd R. and Berlekamp, Elwyn R.}, title = {Error Correction for Algebraic Block Codes}, @@ -5892,17 +4047,6 @@ @article{Williams2005 doi = {10.1016/j.tcs.2005.09.023} } -@article{Williams2005, - author = {Williams, Ryan}, - title = {A New Algorithm for Optimal 2-Constraint Satisfaction and Its Implications}, - journal = {Theoretical Computer Science}, - volume = {348}, - number = {2--3}, - pages = {357--365}, - year = {2005}, - doi = {10.1016/j.tcs.2005.09.023} -} - @article{Williams2014, author = {Williams, Ryan}, title = {Nonuniform {ACC} Circuit Lower Bounds}, @@ -5950,28 +4094,6 @@ @article{Wrathall1976 doi = {10.1016/0304-3975(76)90062-1} } -@article{Wrathall1976, - author = {Wrathall, Celia}, - title = {Complete Sets and the Polynomial-Time Hierarchy}, - journal = {Theoretical Computer Science}, - volume = {3}, - number = {1}, - pages = {23--33}, - year = {1976}, - doi = {10.1016/0304-3975(76)90062-1} -} - -@article{Yablonskii1954, - author = {Yablonskii, S. V.}, - title = {Realization of a linear function in the class of $\pi$-schemes}, - journal = {Doklady Akademii Nauk SSSR}, - volume = {94}, - number = {5}, - pages = {805--806}, - year = {1954}, - note = {In Russian} -} - @article{Yablonskii1954, author = {Yablonskii, S. V.}, title = {Realization of a linear function in the class of $\pi$-schemes}, @@ -5993,26 +4115,6 @@ @inproceedings{Yao1977 doi = {10.1109/SFCS.1977.24} } -@inproceedings{Yao1977, - author = {Yao, Andrew Chi-Chih}, - title = {Probabilistic Computations: Toward a Unified Measure of Complexity}, - booktitle = {Proceedings of the 18th Annual Symposium on Foundations of Computer Science}, - series = {FOCS 1977}, - pages = {222--227}, - year = {1977}, - doi = {10.1109/SFCS.1977.24} -} - -@inproceedings{Yao1982, - author = {Yao, Andrew C.}, - title = {Theory and Application of Trapdoor Functions}, - booktitle = {Proceedings of the 23rd Annual Symposium on Foundations of Computer Science}, - series = {FOCS 1982}, - pages = {80--91}, - year = {1982}, - doi = {10.1109/SFCS.1982.45} -} - @inproceedings{Yao1982, author = {Yao, Andrew C.}, title = {Theory and Application of Trapdoor Functions}, @@ -6029,10 +4131,3 @@ @misc{Zhao2023 howpublished = {Lecture notes, MIT 18.226}, year = {2023} } - -@misc{Zhao2023, - author = {Zhao, Yufei}, - title = {Probabilistic Methods in Combinatorics}, - howpublished = {Lecture notes, MIT 18.226}, - year = {2023} -} From 38288cd91ae94d14997b223d136c92dcf1c9e1da Mon Sep 17 00:00:00 2001 From: Samuel Giuseppe Schlesinger Date: Sat, 24 Oct 2026 23:15:13 -0400 Subject: [PATCH 5/5] docs(GKR): name the existing wiring-predicate lemma in the module doc The Main results list named eval_multilinearExtension_addPred and eval_multilinearExtension_multPred, which do not exist; the closed form is eval_multilinearExtension_wiringPred, of which addPred and multPred are instances. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_01Dvi6MU29zU9t8zYfeU51sB --- Complexity/Computability/Interactive/GKR/Layered.lean | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/Complexity/Computability/Interactive/GKR/Layered.lean b/Complexity/Computability/Interactive/GKR/Layered.lean index e152887ddc..b8ed089100 100644 --- a/Complexity/Computability/Interactive/GKR/Layered.lean +++ b/Complexity/Computability/Interactive/GKR/Layered.lean @@ -38,9 +38,8 @@ wiring predicates and the gate values of the layer below ## Main results -* `Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_addPred`, - `Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_multPred`: closed forms of the - multilinear extensions of the wiring predicates. +* `Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_wiringPred`: the closed form of + the multilinear extensions of the wiring predicates (`addPred` and `multPred` are its two cases). * `Cslib.Interactive.GKR.LayeredCircuit.eval_multilinearExtension_step`: the GKR identity. ## References