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Document

**Published in:** LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

This paper is a contribution to our understanding of the relationship between uniform and nonuniform proof complexity. The latter studies the lengths of proofs in various propositional proof systems such as Frege and bounded-depth Frege systems, and the former studies the strength of the corresponding logical theories such as VNC1 and V0 in [Cook/Nguyen, 2010]. A superpolynomial lower bound on the length of proofs in a propositional proof system for a family of tautologies expressing a result like the pigeonhole principle implies that the result is not provable in the theory associated with the propositional proof system.
We define a new class of bounded arithmetic theories n^epsilon-ioV^\infinity for epsilon < 1 and show that they correspond to complexity classes AltTime(O(1),O(n^epsilon)), uniform classes of subexponential-size bounded-depth circuits DepthSize(O(1),2^O(n^epsilon)). To accomplish this we introduce the novel idea of using types to control the amount of composition in our bounded arithmetic theories. This allows our theories to capture complexity classes that have weaker closure properties and are not closed under composition. We show that the proofs of Sigma^B_0-theorems in our theories translate to subexponential-size bounded-depth Frege proofs.
We use these theories to formalize the complexity theory result that problems in uniform NC1 circuits can be computed by uniform subexponential bounded-depth circuits in [Allender/Koucky, 2010]. We prove that our theories contain a variation of the theory VNC1 for the complexity class NC1. We formalize Buss's proof in [Buss, 1993] that the (unbalanced) Boolean Formula Evaluation problem is in NC1 and use it to prove the soundness of Frege systems. As a corollary, we obtain an alternative proof of [Filmus et al, ICALP, 2011] that polynomial-size Frege proofs can be simulated by subexponential-size bounded-depth Frege proofs.
Our results can be extended to theories corresponding to other nice complexity classes inside NTimeSpace(n^O(1), n^o(1)) such as NL. This is achieved by essentially formalizing the containment
NTimeSpace(n^O(1), n^o(1)) \subseteq AltTime(O(1), O(n^epsilon))
for all epsilon > 0.

Kaveh Ghasemloo and Stephen A. Cook. Theories for Subexponential-size Bounded-depth Frege Proofs. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 296-315, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{ghasemloo_et_al:LIPIcs.CSL.2013.296, author = {Ghasemloo, Kaveh and Cook, Stephen A.}, title = {{Theories for Subexponential-size Bounded-depth Frege Proofs}}, booktitle = {Computer Science Logic 2013 (CSL 2013)}, pages = {296--315}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-60-6}, ISSN = {1868-8969}, year = {2013}, volume = {23}, editor = {Ronchi Della Rocca, Simona}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.296}, URN = {urn:nbn:de:0030-drops-42044}, doi = {10.4230/LIPIcs.CSL.2013.296}, annote = {Keywords: Computational Complexity Theory, Proof Complexity, Bounded Arithmetic, NC1-Frege, AC0- Frege} }

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Invited Talk

**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

This is a survey talk explaining the connection between the three items mentioned in the title.

Stephen A. Cook. Connecting Complexity Classes, Weak Formal Theories, and Propositional Proof Systems (Invited Talk). In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 9-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{cook:LIPIcs.CSL.2012.9, author = {Cook, Stephen A.}, title = {{Connecting Complexity Classes, Weak Formal Theories, and Propositional Proof Systems}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {9--11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.9}, URN = {urn:nbn:de:0030-drops-36594}, doi = {10.4230/LIPIcs.CSL.2012.9}, annote = {Keywords: Complexity Classes, Weak Formal Theories, Propositional Proof Systems} }

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**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Subramanian defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem. He proved that several other problems are complete for CC, including the stable marriage problem, and finding the lexicographical first maximal matching in a bipartite graph. We suggest alternative definitions of CC based on different reducibilities and introduce a two-sorted theory VCC* based on one of them. We sharpen and simplify Subramanian's completeness proofs for the above two problems and formalize them in VCC*.

Dai Tri Man Lê, Stephen A. Cook, and Yuli Ye. A Formal Theory for the Complexity Class Associated with the Stable Marriage Problem. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 381-395, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{le_et_al:LIPIcs.CSL.2011.381, author = {L\^{e}, Dai Tri Man and Cook, Stephen A. and Ye, Yuli}, title = {{A Formal Theory for the Complexity Class Associated with the Stable Marriage Problem}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {381--395}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.381}, URN = {urn:nbn:de:0030-drops-32440}, doi = {10.4230/LIPIcs.CSL.2011.381}, annote = {Keywords: bounded arithmetic, complexity theory, comparator circuits} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We prove exponential lower bounds on the size of semantic read-once 3-ary nondeterministic branching programs. Prior to our result the best that was known was for D-ary branching programs with |D| >= 2^{13}.

Stephen Cook, Jeff Edmonds, Venkatesh Medabalimi, and Toniann Pitassi. Lower Bounds for Nondeterministic Semantic Read-Once Branching Programs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cook_et_al:LIPIcs.ICALP.2016.36, author = {Cook, Stephen and Edmonds, Jeff and Medabalimi, Venkatesh and Pitassi, Toniann}, title = {{Lower Bounds for Nondeterministic Semantic Read-Once Branching Programs}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {36:1--36:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.36}, URN = {urn:nbn:de:0030-drops-63166}, doi = {10.4230/LIPIcs.ICALP.2016.36}, annote = {Keywords: Branching Programs, Semantic, Non-deterministic, Lower Bounds} }

Document

**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We study the branching program complexity of the {\em tree evaluation problem},
introduced in \cite{BrCoMcSaWe09} as a candidate for separating \nl\ from\logcfl. The input to the problem is a rooted, balanced $d$-ary tree of height$h$, whose internal nodes are labelled with $d$-ary functions on$[k]=\{1,\ldots,k\}$, and whose leaves are labelled with elements of $[k]$.Each node obtains a value in $[k]$ equal to its $d$-ary function applied to the values of its $d$ children. The output is the value of the root.
Deterministic $k$-way branching programs as related to black pebbling algorithms have been studied in \cite{BrCoMcSaWe09}. Here we introduce the notion of {\em fractional pebbling} of graphs to study non-deterministicbranching program size. We prove that this yields non-deterministic branching
programs with $\Theta(k^{h/2+1})$ states solving the Boolean problem ``determine whether the root has value 1'' for binary trees - this isasymptotically better than the branching program size corresponding toblack-white pebbling. We prove upper and lower bounds on the fractionalpebbling number of $d$-ary trees, as well as a general result relating thefractional pebbling number of a graph to the black-white pebbling number.
We introduce a simple semantic restriction called {\em thrifty} on $k$-way branching programs solving tree evaluation problems and show that the branchingprogram size bound of $\Theta(k^h)$ is tight (up to a constant factor) for all
$h\ge 2$ for deterministic thrifty programs. We show that thenon-deterministic branching programs that correspond to fractional pebbling are
thrifty as well, and that the bound of $\Theta(k^{h/2+1})$ is tight for
non-deterministic thrifty programs for $h=2,3,4$. We hypothesise that thrifty
branching programs are optimal among $k$-way branching programs solving the
tree evaluation problem - proving this for deterministic programs would
separate \lspace\ from \logcfl\, and proving it for non-deterministic programs
would separate \nl\ from \logcfl.

Mark Braverman, Stephen Cook, Pierre McKenzie, Rahul Santhanam, and Dustin Wehr. Fractional Pebbling and Thrifty Branching Programs. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 109-120, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{braverman_et_al:LIPIcs.FSTTCS.2009.2311, author = {Braverman, Mark and Cook, Stephen and McKenzie, Pierre and Santhanam, Rahul and Wehr, Dustin}, title = {{Fractional Pebbling and Thrifty Branching Programs}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {109--120}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2311}, URN = {urn:nbn:de:0030-drops-23111}, doi = {10.4230/LIPIcs.FSTTCS.2009.2311}, annote = {Keywords: Branching programs, space complexity, tree evaluation, pebbling} }

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