6 Search Results for "Krajicek, Jan"

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-dev.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
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.101
Learning Algorithms Versus Automatability of Frege Systems

Authors: Ján Pich and Rahul Santhanam

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We connect learning algorithms and algorithms automating proof search in propositional proof systems: for every sufficiently strong, well-behaved propositional proof system P, we prove that the following statements are equivalent, - Provable learning. P proves efficiently that p-size circuits are learnable by subexponential-size circuits over the uniform distribution with membership queries. - Provable automatability. P proves efficiently that P is automatable by non-uniform circuits on propositional formulas expressing p-size circuit lower bounds. Here, P is sufficiently strong and well-behaved if I.-III. holds: I. P p-simulates Jeřábek’s system WF (which strengthens the Extended Frege system EF by a surjective weak pigeonhole principle); II. P satisfies some basic properties of standard proof systems which p-simulate WF; III. P proves efficiently for some Boolean function h that h is hard on average for circuits of subexponential size. For example, if III. holds for P = WF, then Items 1 and 2 are equivalent for P = WF. The notion of automatability in Item 2 is slightly modified so that the automating algorithm outputs a proof of a given formula (expressing a p-size circuit lower bound) in p-time in the length of the shortest proof of a closely related but different formula (expressing an average-case subexponential-size circuit lower bound). If there is a function h ∈ NE∩ coNE which is hard on average for circuits of size 2^{n/4}, for each sufficiently big n, then there is an explicit propositional proof system P satisfying properties I.-III., i.e. the equivalence of Items 1 and 2 holds for P.
Cite as

Ján Pich and Rahul Santhanam. Learning Algorithms Versus Automatability of Frege Systems. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 101:1-101:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{pich_et_al:LIPIcs.ICALP.2022.101,
  author =	{Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{Learning Algorithms Versus Automatability of Frege Systems}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{101:1--101:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.101},
  URN =		{urn:nbn:de:0030-drops-164427},
  doi =		{10.4230/LIPIcs.ICALP.2022.101},
  annote =	{Keywords: learning algorithms, automatability, proof complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.19
Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Authors: Fedor Part and Iddo Tzameret

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted Res(lin_R), this refutation system operates with disjunctions of linear equations with boolean variables over a ring R, to refute unsatisfiable sets of such disjunctions. Beginning in the work of [Ran Raz and Iddo Tzameret, 2008], through the work of [Dmitry Itsykson and Dmitry Sokolov, 2014] which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf. [Jan Krajícek, 2017; Dmitry Itsykson and Dmitry Sokolov, 2014; Jan Krajícek and Igor Carboni Oliveira, 2018; Michal Garlik and Lezsek Kołodziejczyk, 2018]) made it evident that establishing lower bounds against general Res(lin_R) refutations is a challenging and interesting task since the system captures a "minimal" extension of resolution with counting gates for which no super-polynomial lower bounds are known to date. We provide the first super-polynomial size lower bounds on general (dag-like) resolution over linear equations refutations in the large characteristic regime. In particular we prove that the subset-sum principle 1+ x_1 + ̇s +2^n x_n = 0 requires refutations of exponential-size over ℚ. Our proof technique is nontrivial and novel: roughly speaking, we show that under certain conditions every refutation of a subset-sum instance f=0, where f is a linear polynomial over ℚ, must pass through a fat clause containing an equation f=α for each α in the image of f under boolean assignments. We develop a somewhat different approach to prove exponential lower bounds against tree-like refutations of any subset-sum instance that depends on n variables, hence also separating tree-like from dag-like refutations over the rationals. We then turn to the finite fields regime, showing that the work of Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014] who obtained tree-like lower bounds over ?_2 can be carried over and extended to every finite field. We establish new lower bounds and separations as follows: (i) for every pair of distinct primes p,q, there exist CNF formulas with short tree-like refutations in Res(lin_{?_p}) that require exponential-size tree-like Res(lin_{?_q}) refutations; (ii) random k-CNF formulas require exponential-size tree-like Res(lin_{?_p}) refutations, for every prime p and constant k; and (iii) exponential-size lower bounds for tree-like Res(lin_?) refutations of the pigeonhole principle, for every field ?.
Cite as

Fedor Part and Iddo Tzameret. Resolution with Counting: Dag-Like Lower Bounds and Different Moduli. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 19:1-19:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{part_et_al:LIPIcs.ITCS.2020.19,
  author =	{Part, Fedor and Tzameret, Iddo},
  title =	{{Resolution with Counting: Dag-Like Lower Bounds and Different Moduli}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{19:1--19:37},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.19},
  URN =		{urn:nbn:de:0030-drops-117041},
  doi =		{10.4230/LIPIcs.ITCS.2020.19},
  annote =	{Keywords: Proof complexity, concrete lower bounds, resolution, satisfiability, combinatorics}
}
Document
DOI: 10.4230/LIPIcs.CCC.2019.27
Hardness Magnification near State-Of-The-Art Lower Bounds

Authors: Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam

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

Abstract
This work continues the development of hardness magnification. The latter proposes a new strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful. We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs to distinguish instances (strings or truth-tables) of complexity <= s_1(N) from instances of complexity >= s_2(N), and N = 2^n denotes the input length. In MCSP, complexity is measured by circuit size, while in MKtP one considers Levin’s notion of time-bounded Kolmogorov complexity. (In our results, the parameters s_1(N) and s_2(N) are asymptotically quite close, and the problems almost coincide with their standard formulations without a gap.) We establish that for Gap-MKtP[s_1,s_2] and Gap-MCSP[s_1,s_2], a marginal improvement over the state-of-the-art in unconditional lower bounds in a variety of computational models would imply explicit super-polynomial lower bounds. Theorem. There exists a universal constant c >= 1 for which the following hold. If there exists epsilon > 0 such that for every small enough beta > 0 (1) Gap-MCSP[2^{beta n}/c n, 2^{beta n}] !in Circuit[N^{1 + epsilon}], then NP !subseteq Circuit[poly]. (2) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in TC^0[N^{1 + epsilon}], then EXP !subseteq TC^0[poly]. (3) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in B_2-Formula[N^{2 + epsilon}], then EXP !subseteq Formula[poly]. (4) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in U_2-Formula[N^{3 + epsilon}], then EXP !subseteq Formula[poly]. (5) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in BP[N^{2 + epsilon}], then EXP !subseteq BP[poly]. (6) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in (AC^0[6])[N^{1 + epsilon}], then EXP !subseteq AC^0[6]. These results are complemented by lower bounds for Gap-MCSP and Gap-MKtP against different models. For instance, the lower bound assumed in (1) holds for U_2-formulas of near-quadratic size, and lower bounds similar to (3)-(5) hold for various regimes of parameters. We also identify a natural computational model under which the hardness magnification threshold for Gap-MKtP lies below existing lower bounds: U_2-formulas that can compute parity functions at the leaves (instead of just literals). As a consequence, if one managed to adapt the existing lower bound techniques against such formulas to work with Gap-MKtP, then EXP !subseteq NC^1 would follow via hardness magnification.
Cite as

Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. Hardness Magnification near State-Of-The-Art Lower Bounds. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 27:1-27:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.27,
  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)},
  pages =	{27:1--27:29},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.27},
  URN =		{urn:nbn:de:0030-drops-108494},
  doi =		{10.4230/LIPIcs.CCC.2019.27},
  annote =	{Keywords: Circuit Complexity, Minimum Circuit Size Problem, Kolmogorov Complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.32
Randomness and Intractability in Kolmogorov Complexity

Authors: Igor Carboni Oliveira

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin’s notion [Leonid A. Levin, 1984] of Kolmogorov complexity. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability. This complexity measure gives rise to a decision problem over strings: MrKtP (The Minimum rKt Problem). We explore ideas from pseudorandomness to prove that MrKtP and its variants cannot be solved in randomized quasi-polynomial time. This exhibits a natural string compression problem that is provably intractable, even for randomized computations. Our techniques also imply that there is no n^{1 - epsilon}-approximate algorithm for MrKtP running in randomized quasi-polynomial time. Complementing this lower bound, we observe connections between rKt, the power of randomness in computing, and circuit complexity. In particular, we present the first hardness magnification theorem for a natural problem that is unconditionally hard against a strong model of computation.
Cite as

Igor Carboni Oliveira. Randomness and Intractability in Kolmogorov Complexity. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{oliveira:LIPIcs.ICALP.2019.32,
  author =	{Oliveira, Igor Carboni},
  title =	{{Randomness and Intractability in Kolmogorov Complexity}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{32:1--32:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.32},
  URN =		{urn:nbn:de:0030-drops-106087},
  doi =		{10.4230/LIPIcs.ICALP.2019.32},
  annote =	{Keywords: computational complexity, randomness, circuit lower bounds, Kolmogorov complexity}
}
Document
DOI: 10.4230/DagRep.4.10.51
Optimal algorithms and proofs (Dagstuhl Seminar 14421)

Authors: Olaf Beyersdorff, Edward A. Hirsch, Jan Krajicek, and Rahul Santhanam

Published in: Dagstuhl Reports, Volume 4, Issue 10 (2015)

Abstract
This report documents the programme and the outcomes of the Dagstuhl Seminar 14421 "Optimal algorithms and proofs". The seminar brought together researchers working in computational and proof complexity, logic, and the theory of approximations. Each of these areas has its own, but connected notion of optimality; and the main aim of the seminar was to bring together researchers from these different areas, for an exchange of ideas, techniques, and open questions, thereby triggering new research collaborations across established research boundaries.
Cite as

Olaf Beyersdorff, Edward A. Hirsch, Jan Krajicek, and Rahul Santhanam. Optimal algorithms and proofs (Dagstuhl Seminar 14421). In Dagstuhl Reports, Volume 4, Issue 10, pp. 51-68, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Article{beyersdorff_et_al:DagRep.4.10.51,
  author =	{Beyersdorff, Olaf and Hirsch, Edward A. and Krajicek, Jan and Santhanam, Rahul},
  title =	{{Optimal algorithms and proofs (Dagstuhl Seminar 14421)}},
  pages =	{51--68},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2015},
  volume =	{4},
  number =	{10},
  editor =	{Beyersdorff, Olaf and Hirsch, Edward A. and Krajicek, Jan and Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.4.10.51},
  URN =		{urn:nbn:de:0030-drops-48923},
  doi =		{10.4230/DagRep.4.10.51},
  annote =	{Keywords: computational complexity, proof complexity, approximation algorithms, optimal algorithms, optimal proof systems, speedup theorems}
}
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