DROPS

Document

DOI: 10.4230/LIPIcs.CCC.2024.9

Lifting Dichotomies

Authors: Yaroslav Alekseev, Yuval Filmus, and Alexander Smal

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure A for some function f, we compose f with a carefully chosen gadget function g and get essentially the same lower bound on a complexity measure B for the lifted function f ⋄ g. Lifting theorems have a number of applications in many different areas such as circuit complexity, communication complexity, proof complexity, etc. One of the main question in the context of lifting is how to choose a suitable gadget g. Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of f, and it is unclear whether it can be improved to a constant size gadget. This motivates us to identify the properties of gadgets that make lifting possible. In this paper, we systematically study the question "For which gadgets does the lifting result hold?" in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there is no intermediate cases - for every gadget there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as f ⋄ OR ⋄ XOR for some function f. In this extended abstract, the proofs are omitted. Full proofs are given in the full version [Yaroslav Alekseev et al., 2024].

Cite as

Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alekseev_et_al:LIPIcs.CCC.2024.9,
  author =	{Alekseev, Yaroslav and Filmus, Yuval and Smal, Alexander},
  title =	{{Lifting Dichotomies}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.9},
  URN =		{urn:nbn:de:0030-drops-204051},
  doi =		{10.4230/LIPIcs.CCC.2024.9},
  annote =	{Keywords: decision trees, log-rank conjecture, lifting, parity decision trees}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.15

Quantum Automating TC⁰-Frege Is LWE-Hard

Authors: Noel Arteche, Gaia Carenini, and Matthew Gray

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate TC⁰-Frege. This extends the line of results of Krajíček and Pudlák (Information and Computation, 1998), Bonet, Pitassi, and Raz (FOCS, 1997), and Bonet, Domingo, Gavaldà, Maciel, and Pitassi (Computational Complexity, 2004), who showed that Extended Frege, TC⁰-Frege and AC⁰-Frege, respectively, cannot be weakly automated by classical algorithms if either the RSA cryptosystem or the Diffie-Hellman key exchange protocol are secure. To the best of our knowledge, this is the first interaction between quantum computation and propositional proof search.

Cite as

Noel Arteche, Gaia Carenini, and Matthew Gray. Quantum Automating TC⁰-Frege Is LWE-Hard. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 15:1-15:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arteche_et_al:LIPIcs.CCC.2024.15,
  author =	{Arteche, Noel and Carenini, Gaia and Gray, Matthew},
  title =	{{Quantum Automating TC⁰-Frege Is LWE-Hard}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{15:1--15:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.15},
  URN =		{urn:nbn:de:0030-drops-204117},
  doi =		{10.4230/LIPIcs.CCC.2024.15},
  annote =	{Keywords: automatability, post-quantum cryptography, feasible interpolation}
}

Document

DOI: 10.4230/LIPIcs.CCC.2024.22

Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP

Authors: Theodoros Papamakarios

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract

We show that for any integer d > 0, depth-d Frege systems are NP-hard to automate. Namely, given a set S of depth-d formulas, it is NP-hard to find a depth-d Frege refutation of S in time polynomial in the size of the shortest such refutation. This extends the result of Atserias and Müller [JACM, 2020] for the non-automatability of resolution - a depth-1 Frege system - to Frege systems of any depth d > 0.

Cite as

Theodoros Papamakarios. Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{papamakarios:LIPIcs.CCC.2024.22,
  author =	{Papamakarios, Theodoros},
  title =	{{Depth-d Frege Systems Are Not Automatable Unless 𝖯 = NP}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.22},
  URN =		{urn:nbn:de:0030-drops-204187},
  doi =		{10.4230/LIPIcs.CCC.2024.22},
  annote =	{Keywords: Proof complexity, Automatability, Bounded-depth Frege}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.12

From Proof Complexity to Circuit Complexity via Interactive Protocols

Authors: Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

Folklore in complexity theory suspects that circuit lower bounds against NC¹ or P/poly, currently out of reach, are a necessary step towards proving strong proof complexity lower bounds for systems like Frege or Extended Frege. Establishing such a connection formally, however, is already daunting, as it would imply the breakthrough separation NEXP ⊈ P/poly, as recently observed by Pich and Santhanam [Pich and Santhanam, 2023]. We show such a connection conditionally for the Implicit Extended Frege proof system (iEF) introduced by Krajíček [Krajíček, 2004], capable of formalizing most of contemporary complexity theory. In particular, we show that if iEF proves efficiently the standard derandomization assumption that a concrete Boolean function is hard on average for subexponential-size circuits, then any superpolynomial lower bound on the length of iEF proofs implies #P ⊈ FP/poly (which would in turn imply, for example, PSPACE ⊈ P/poly). Our proof exploits the formalization inside iEF of the soundness of the sum-check protocol of Lund, Fortnow, Karloff, and Nisan [Lund et al., 1992]. This has consequences for the self-provability of circuit upper bounds in iEF. Interestingly, further improving our result seems to require progress in constructing interactive proof systems with more efficient provers.

Cite as

Noel Arteche, Erfan Khaniki, Ján Pich, and Rahul Santhanam. From Proof Complexity to Circuit Complexity via Interactive Protocols. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arteche_et_al:LIPIcs.ICALP.2024.12,
  author =	{Arteche, Noel and Khaniki, Erfan and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{From Proof Complexity to Circuit Complexity via Interactive Protocols}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.12},
  URN =		{urn:nbn:de:0030-drops-201557},
  doi =		{10.4230/LIPIcs.ICALP.2024.12},
  annote =	{Keywords: proof complexity, circuit complexity, interactive protocols}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.116

One-Way Communication Complexity of Partial XOR Functions

Authors: Vladimir V. Podolskii and Dmitrii Sluch

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

Boolean function F(x,y) for x,y ∈ {0,1}ⁿ is an XOR function if F(x,y) = f(x⊕ y) for some function f on n input bits, where ⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing the Fourier analytic technique. For total XOR functions, it is known that deterministic communication complexity of F is closely related to parity decision tree complexity of f. Montanaro and Osbourne (2009) observed that one-way communication complexity D_{cc}^{→}(F) of F is exactly equal to non-adaptive parity decision tree complexity NADT^{⊕}(f) of f. Hatami et al. (2018) showed that unrestricted communication complexity of F is polynomially related to parity decision tree complexity of f. We initiate the study of a similar connection for partial functions. We show that in the case of one-way communication complexity whether these measures are equal, depends on the number of undefined inputs of f. More precisely, if D_{cc}^{→}(F) = t and f is undefined on at most O((2^{n-t})/(√{n-t})) inputs, then NADT^{⊕}(f) = t. We also provide stronger bounds in extreme cases of small and large complexity. We show that the restriction on the number of undefined inputs in these results is unavoidable. That is, for a wide range of values of D_{cc}^{→}(F) and NADT^{⊕}(f) (from constant to n-2) we provide partial functions (with more than Ω((2^{n-t})/(√{n-t})) undefined inputs, where t = D_{cc}^{→}) for which D_{cc}^{→}(F) < NADT^{⊕}(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-way communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on the Boolean Fourier analysis and thus, the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

Cite as

Vladimir V. Podolskii and Dmitrii Sluch. One-Way Communication Complexity of Partial XOR Functions. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 116:1-116:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{podolskii_et_al:LIPIcs.ICALP.2024.116,
  author =	{Podolskii, Vladimir V. and Sluch, Dmitrii},
  title =	{{One-Way Communication Complexity of Partial XOR Functions}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{116:1--116:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.116},
  URN =		{urn:nbn:de:0030-drops-202591},
  doi =		{10.4230/LIPIcs.ICALP.2024.116},
  annote =	{Keywords: Partial functions, XOR functions, communication complexity, decision trees, covering codes}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.117

Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle

Authors: Aaron Potechin and Aaron Zhang

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

We show that the minimum total coefficient size of a Nullstellensatz proof of the pigeonhole principle on n+1 pigeons and n holes is 2^{Θ(n)}. We also investigate the ordering principle and construct an explicit Nullstellensatz proof for the ordering principle on n elements with total coefficient size 2ⁿ - n.

Cite as

Aaron Potechin and Aaron Zhang. Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 117:1-117:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{potechin_et_al:LIPIcs.ICALP.2024.117,
  author =	{Potechin, Aaron and Zhang, Aaron},
  title =	{{Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{117:1--117:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.117},
  URN =		{urn:nbn:de:0030-drops-202604},
  doi =		{10.4230/LIPIcs.ICALP.2024.117},
  annote =	{Keywords: Proof complexity, Nullstellensatz, pigeonhole principle, coefficient size}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2024.48

Proving Unsatisfiability with Hitting Formulas

Authors: Yuval Filmus, Edward A. Hirsch, Artur Riazanov, Alexander Smal, and Marc Vinyals

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract

A hitting formula is a set of Boolean clauses such that any two of the clauses cannot be simultaneously falsified. Hitting formulas have been studied in many different contexts at least since [Iwama, 1989] and, based on experimental evidence, Peitl and Szeider [Tomás Peitl and Stefan Szeider, 2022] conjectured that unsatisfiable hitting formulas are among the hardest for resolution. Using the fact that hitting formulas are easy to check for satisfiability we make them the foundation of a new static proof system {{rmHitting}}: a refutation of a CNF in {{rmHitting}} is an unsatisfiable hitting formula such that each of its clauses is a weakening of a clause of the refuted CNF. Comparing this system to resolution and other proof systems is equivalent to studying the hardness of hitting formulas. Our first result is that {{rmHitting}} is quasi-polynomially simulated by tree-like resolution, which means that hitting formulas cannot be exponentially hard for resolution and partially refutes the conjecture of Peitl and Szeider. We show that tree-like resolution and {{rmHitting}} are quasi-polynomially separated, while for resolution, this question remains open. For a system that is only quasi-polynomially stronger than tree-like resolution, {{rmHitting}} is surprisingly difficult to polynomially simulate in another proof system. Using the ideas of Raz-Shpilka’s polynomial identity testing for noncommutative circuits [Raz and Shpilka, 2005] we show that {{rmHitting}} is p-simulated by {{rmExtended {{rmFrege}}}}, but we conjecture that much more efficient simulations exist. As a byproduct, we show that a number of static (semi)algebraic systems are verifiable in deterministic polynomial time. We consider multiple extensions of {{rmHitting}}, and in particular a proof system {{{rmHitting}}(⊕)} related to the {{{rmRes}}(⊕)} proof system for which no superpolynomial-size lower bounds are known. {{{rmHitting}}(⊕)} p-simulates the tree-like version of {{{rmRes}}(⊕)} and is at least quasi-polynomially stronger. We show that formulas expressing the non-existence of perfect matchings in the graphs K_{n,n+2} are exponentially hard for {{{rmHitting}}(⊕)} via a reduction to the partition bound for communication complexity. See the full version of the paper for the proofs. They are omitted in this Extended Abstract.

Cite as

Yuval Filmus, Edward A. Hirsch, Artur Riazanov, Alexander Smal, and Marc Vinyals. Proving Unsatisfiability with Hitting Formulas. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{filmus_et_al:LIPIcs.ITCS.2024.48,
  author =	{Filmus, Yuval and Hirsch, Edward A. and Riazanov, Artur and Smal, Alexander and Vinyals, Marc},
  title =	{{Proving Unsatisfiability with Hitting Formulas}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{48:1--48:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.48},
  URN =		{urn:nbn:de:0030-drops-195762},
  doi =		{10.4230/LIPIcs.ITCS.2024.48},
  annote =	{Keywords: hitting formulas, polynomial identity testing, query complexity}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.17

Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds

Authors: Erfan Khaniki

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

A map g:{0,1}ⁿ → {0,1}^m (m > n) is a hard proof complexity generator for a proof system P iff for every string b ∈ {0,1}^m ⧵ Rng(g), formula τ_b(g) naturally expressing b ∉ Rng(g) requires superpolynomial size P-proofs. One of the well-studied maps in the theory of proof complexity generators is Nisan-Wigderson generator. Razborov [A. A. {Razborov}, 2015] conjectured that if A is a suitable matrix and f is a NP∩CoNP function hard-on-average for 𝖯/poly, then NW_{f, A} is a hard proof complexity generator for Extended Frege. In this paper, we prove a form of Razborov’s conjecture for AC⁰-Frege. We show that for any symmetric NP∩CoNP function f that is exponentially hard for depth two AC⁰ circuits, NW_{f,A} is a hard proof complexity generator for AC⁰-Frege in a natural setting. As direct applications of this theorem, we show that: 1) For any f with the specified properties, τ_b(NW_{f,A}) (for a natural formalization) based on a random b and a random matrix A with probability 1-o(1) is a tautology and requires superpolynomial (or even exponential) AC⁰-Frege proofs. 2) Certain formalizations of the principle f_n ∉ (NP∩CoNP)/poly requires superpolynomial AC⁰-Frege proofs. These applications relate to two questions that were asked by Krajíček [J. {Krajíček}, 2019].

Cite as

Erfan Khaniki. Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{khaniki:LIPIcs.CCC.2022.17,
  author =	{Khaniki, Erfan},
  title =	{{Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.17},
  URN =		{urn:nbn:de:0030-drops-165799},
  doi =		{10.4230/LIPIcs.CCC.2022.17},
  annote =	{Keywords: Proof complexity, Bounded arithmetic, Bounded depth Frege, Nisan-Wigderson generators, Meta-complexity, Lower bounds}
}

Document

DOI: 10.4230/LIPIcs.CCC.2021.40

The Power of Negative Reasoning

Authors: Susanna F. de Rezende, Massimo Lauria, Jakob Nordström, and Dmitry Sokolov

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract

Semialgebraic proof systems have been studied extensively in proof complexity since the late 1990s to understand the power of Gröbner basis computations, linear and semidefinite programming hierarchies, and other methods. Such proof systems are defined alternately with only the original variables of the problem and with special formal variables for positive and negative literals, but there seems to have been no study how these different definitions affect the power of the proof systems. We show for Nullstellensatz, polynomial calculus, Sherali-Adams, and sums-of-squares that adding formal variables for negative literals makes the proof systems exponentially stronger, with respect to the number of terms in the proofs. These separations are witnessed by CNF formulas that are easy for resolution, which establishes that polynomial calculus, Sherali-Adams, and sums-of-squares cannot efficiently simulate resolution without having access to variables for negative literals.

Cite as

Susanna F. de Rezende, Massimo Lauria, Jakob Nordström, and Dmitry Sokolov. The Power of Negative Reasoning. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 40:1-40:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2021.40,
  author =	{de Rezende, Susanna F. and Lauria, Massimo and Nordstr\"{o}m, Jakob and Sokolov, Dmitry},
  title =	{{The Power of Negative Reasoning}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{40:1--40:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.40},
  URN =		{urn:nbn:de:0030-drops-143140},
  doi =		{10.4230/LIPIcs.CCC.2021.40},
  annote =	{Keywords: Proof complexity, Polynomial calculus, Nullstellensatz, Sums-of-squares, Sherali-Adams}
}

Document

DOI: 10.4230/LIPIcs.CCC.2020.28

Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs

Authors: Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract

We show exponential lower bounds on resolution proof length for pigeonhole principle (PHP) formulas and perfect matching formulas over highly unbalanced, sparse expander graphs, thus answering the challenge to establish strong lower bounds in the regime between balanced constant-degree expanders as in [Ben-Sasson and Wigderson '01] and highly unbalanced, dense graphs as in [Raz '04] and [Razborov '03, '04]. We obtain our results by revisiting Razborov’s pseudo-width method for PHP formulas over dense graphs and extending it to sparse graphs. This further demonstrates the power of the pseudo-width method, and we believe it could potentially be useful for attacking also other longstanding open problems for resolution and other proof systems.

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Susanna F. de Rezende, Jakob Nordström, Kilian Risse, and Dmitry Sokolov. Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 28:1-28:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2020.28,
  author =	{de Rezende, Susanna F. and Nordstr\"{o}m, Jakob and Risse, Kilian and Sokolov, Dmitry},
  title =	{{Exponential Resolution Lower Bounds for Weak Pigeonhole Principle and Perfect Matching Formulas over Sparse Graphs}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{28:1--28:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.28},
  URN =		{urn:nbn:de:0030-drops-125804},
  doi =		{10.4230/LIPIcs.CCC.2020.28},
  annote =	{Keywords: proof complexity, resolution, weak pigeonhole principle, perfect matching, sparse graphs}
}

Document

DOI: 10.4230/LIPIcs.CCC.2019.18

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

Authors: Susanna F. de Rezende, Jakob Nordström, Or Meir, and Robert Robere

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Abstract

We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t+1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.

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Susanna F. de Rezende, Jakob Nordström, Or Meir, and Robert Robere. Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{derezende_et_al:LIPIcs.CCC.2019.18,
  author =	{de Rezende, Susanna F. and Nordstr\"{o}m, Jakob and Meir, Or and Robere, Robert},
  title =	{{Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.18},
  URN =		{urn:nbn:de:0030-drops-108403},
  doi =		{10.4230/LIPIcs.CCC.2019.18},
  annote =	{Keywords: proof complexity, Nullstellensatz, pebble games, trade-offs, size, degree}
}

Document

DOI: 10.4230/LIPIcs.STACS.2019.50

Lifting Theorems for Equality

Authors: Bruno Loff and Sagnik Mukhopadhyay

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract

We show a deterministic simulation (or lifting) theorem for composed problems f o Eq_n where the inner function (the gadget) is Equality on n bits. When f is a total function on p bits, it is easy to show via a rank argument that the communication complexity of f o Eq_n is Omega(deg(f) * n). However, there is a surprising counter-example of a partial function f on p bits, such that any completion f' of f has deg(f') = Omega(p), and yet f o Eq_n has communication complexity O(n). Nonetheless, we are able to show that the communication complexity of f o Eq_n is at least D(f) * n for a complexity measure D(f) which is closely related to the AND-query complexity of f and is lower-bounded by the logarithm of the leaf complexity of f. As a corollary, we also obtain lifting theorems for the set-disjointness gadget, and a lifting theorem in the context of parity decision-trees, for the NOR gadget. As an application, we prove a tight lower-bound for the deterministic communication complexity of the communication problem, where Alice and Bob are each given p-many n-bit strings, with the promise that either all of the strings are distinct, or all-but-one of the strings are distinct, and they wish to know which is the case. We show that the complexity of this problem is Theta(p * n).

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Bruno Loff and Sagnik Mukhopadhyay. Lifting Theorems for Equality. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{loff_et_al:LIPIcs.STACS.2019.50,
  author =	{Loff, Bruno and Mukhopadhyay, Sagnik},
  title =	{{Lifting Theorems for Equality}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{50:1--50:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.50},
  URN =		{urn:nbn:de:0030-drops-102892},
  doi =		{10.4230/LIPIcs.STACS.2019.50},
  annote =	{Keywords: Communication complexity, Query complexity, Simulation theorem, Equality function}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2017.38

Cumulative Space in Black-White Pebbling and Resolution

Authors: Joël Alwen, Susanna F. de Rezende, Jakob Nordström, and Marc Vinyals

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract

We study space complexity and time-space trade-offs with a focus not on peak memory usage but on overall memory consumption throughout the computation. Such a cumulative space measure was introduced for the computational model of parallel black pebbling by [Alwen and Serbinenko 2015] as a tool for obtaining results in cryptography. We consider instead the nondeterministic black-white pebble game and prove optimal cumulative space lower bounds and trade-offs, where in order to minimize pebbling time the space has to remain large during a significant fraction of the pebbling. We also initiate the study of cumulative space in proof complexity, an area where other space complexity measures have been extensively studied during the last 10-15 years. Using and extending the connection between proof complexity and pebble games in [Ben-Sasson and Nordström 2008, 2011], we obtain several strong cumulative space results for (even parallel versions of) the resolution proof system, and outline some possible future directions of study of this, in our opinion, natural and interesting space measure.

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Joël Alwen, Susanna F. de Rezende, Jakob Nordström, and Marc Vinyals. Cumulative Space in Black-White Pebbling and Resolution. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 38:1-38:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{alwen_et_al:LIPIcs.ITCS.2017.38,
  author =	{Alwen, Jo\"{e}l and de Rezende, Susanna F. and Nordstr\"{o}m, Jakob and Vinyals, Marc},
  title =	{{Cumulative Space in Black-White Pebbling and Resolution}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{38:1--38:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.38},
  URN =		{urn:nbn:de:0030-drops-81918},
  doi =		{10.4230/LIPIcs.ITCS.2017.38},
  annote =	{Keywords: pebble game, pebbling, proof complexity, space, cumulative space, clause space, resolution, parallel resolution}
}

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