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Documents authored by Sofronova, Anastasia

Document
DOI: 10.4230/LIPIcs.CCC.2022.13
A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates

Authors: Ivan Mihajlin and Anastasia Sofronova

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

Abstract
We prove that a modification of Andreev’s function is not computable by (3 + α - ε) log(n) depth De Morgan formula with (2α - ε)log{n} layers of AND gates at the top for any 0 < α < 1/5 and any constant ε > 0. In order to do this, we prove a weak variant of Karchmer-Raz-Wigderson conjecture. To be more precise, we prove the existence of two functions f : {0,1}ⁿ → {0,1} and g : {0,1}ⁿ → {0,1}ⁿ such that f(g(x) ⊕ y) is not computable by depth (1 + α - ε) n formulas with (2 α - ε) n layers of AND gates at the top. We do this by a top-down approach, which was only used before for depth-3 model. Our technical contribution includes combinatorial insights into structure of composition with random boolean function, which led us to introducing a notion of well-mixed sets. A set of functions is well-mixed if, when composed with a random function, it does not have subsets that agree on large fractions of inputs. We use probabilistic method to prove the existence of well-mixed sets.
Cite as

Ivan Mihajlin and Anastasia Sofronova. A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mihajlin_et_al:LIPIcs.CCC.2022.13,
  author =	{Mihajlin, Ivan and Sofronova, Anastasia},
  title =	{{A Better-Than-3log(n) Depth Lower Bound for De Morgan Formulas with Restrictions on Top Gates}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{13:1--13:15},
  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.2022.13},
  URN =		{urn:nbn:de:0030-drops-165755},
  doi =		{10.4230/LIPIcs.CCC.2022.13},
  annote =	{Keywords: formula complexity, communication complexity, Karchmer-Raz-Wigderson conjecture, De Morgan formulas}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.17
Branching Programs with Bounded Repetitions and Flow Formulas

Authors: Anastasia Sofronova and Dmitry Sokolov

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

Abstract
Restricted branching programs capture various complexity measures like space in Turing machines or length of proofs in proof systems. In this paper, we focus on the application in the proof complexity that was discovered by Lovasz et al. [László Lovász et al., 1995] who showed the equivalence between regular Resolution and read-once branching programs for "unsatisfied clause search problem" (Search_φ). This connection is widely used, in particular, in the recent breakthrough result about the Clique problem in regular Resolution by Atserias et al. [Albert Atserias et al., 2018]. We study the branching programs with bounded repetitions, so-called (1,+k)-BPs (Sieling [Detlef Sieling, 1996]) in application to the Search_φ problem. On the one hand, it is a natural generalization of read-once branching programs. On the other hand, this model gives a powerful proof system that can efficiently certify the unsatisfiability of a wide class of formulas that is hard for Resolution (Knop [Alexander Knop, 2017]). We deal with Search_φ that is "relatively easy" compared to all known hard examples for the (1,+k)-BPs. We introduce the first technique for proving exponential lower bounds for the (1,+k)-BPs on Search_φ. To do it we combine a well-known technique for proving lower bounds on the size of branching programs [Detlef Sieling, 1996; Detlef Sieling and Ingo Wegener, 1994; Stasys Jukna and Alexander A. Razborov, 1998] with the modification of the "closure" technique [Michael Alekhnovich et al., 2004; Michael Alekhnovich and Alexander A. Razborov, 2003]. In contrast with most Resolution lower bounds, our technique uses not only "local" properties of the formula, but also a "global" structure. Our hard examples are based on the Flow formulas introduced in [Michael Alekhnovich and Alexander A. Razborov, 2003].
Cite as

Anastasia Sofronova and Dmitry Sokolov. Branching Programs with Bounded Repetitions and Flow Formulas. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{sofronova_et_al:LIPIcs.CCC.2021.17,
  author =	{Sofronova, Anastasia and Sokolov, Dmitry},
  title =	{{Branching Programs with Bounded Repetitions and Flow Formulas}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{17:1--17:25},
  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.2021.17},
  URN =		{urn:nbn:de:0030-drops-142915},
  doi =		{10.4230/LIPIcs.CCC.2021.17},
  annote =	{Keywords: proof complexity, branching programs, bounded repetitions, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.49
Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs

Authors: Nicola Galesi, Dmitry Itsykson, Artur Riazanov, and Anastasia Sofronova

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2^{tw(G)^{Omega(1/d)}} in depth-d Frege systems for d<(K log n)/(log log n), where tw(G) is the treewidth of G. This extends Håstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2^{tw(G)^{O(1/d)}} poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.
Cite as

Nicola Galesi, Dmitry Itsykson, Artur Riazanov, and Anastasia Sofronova. Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{galesi_et_al:LIPIcs.MFCS.2019.49,
  author =	{Galesi, Nicola and Itsykson, Dmitry and Riazanov, Artur and Sofronova, Anastasia},
  title =	{{Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{49:1--49:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.49},
  URN =		{urn:nbn:de:0030-drops-109932},
  doi =		{10.4230/LIPIcs.MFCS.2019.49},
  annote =	{Keywords: Tseitin formula, treewidth, AC0-Frege}
}
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