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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional-calculus (i.e. Frege) proof is any proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of sound derivation rules. Establishing any super-polynomial size lower bound on Frege proofs (in terms of the size of the formula proved) is a major open problem in proof complexity, and among a handful of fundamental hardness questions in complexity theory by and large. Non-commutative arithmetic formulas, on the other hand, constitute a quite weak computational model, for which exponential-size lower bounds were shown already back in 1991 by Nisan [STOC 1991], using a particularly transparent argument.
In this work we show that Frege lower bounds in fact follow from corresponding size lower bounds on non-commutative formulas computing certain polynomials (and that such lower bounds on non-commutative formulas must exist, unless NP=coNP). More precisely, we demonstrate a natural association between tautologies T to non-commutative polynomials p, such that:
(*) if T has a polynomial-size Frege proof then p has a polynomial-size non-commutative arithmetic formula; and conversely, when T is a DNF, if p has a polynomial-size non-commutative arithmetic formula over GF(2) then T has a Frege proof of quasi-polynomial size.
The argument is a characterization of Frege proofs as non-commutative formulas: we show that the Frege system is (quasi-)polynomially equivalent to a non-commutative Ideal Proof System (IPS), following the recent work of Grochow and Pitassi [FOCS 2014] that introduced a propositional proof system in which proofs are arithmetic circuits, and the work in [Tzameret 2011] that considered adding the commutator as an axiom in algebraic propositional proof systems. This gives a characterization of propositional Frege proofs in terms of (non-commutative) arithmetic formulas that is tighter than (the formula version of IPS) in Grochow and Pitassi [FOCS 2014], in the following sense:
(i) The non-commutative IPS is polynomial-time checkable - whereas the original IPS was checkable in probabilistic polynomial-time; and
(ii) Frege proofs unconditionally quasi-polynomially simulate the non-commutative IPS - whereas Frege was shown to efficiently simulate IPS only assuming that the decidability of PIT for (commutative) arithmetic formulas by polynomial-size circuits is efficiently provable in Frege.

Fu Li, Iddo Tzameret, and Zhengyu Wang. Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 412-432, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{li_et_al:LIPIcs.CCC.2015.412, author = {Li, Fu and Tzameret, Iddo and Wang, Zhengyu}, title = {{Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {412--432}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.412}, URN = {urn:nbn:de:0030-drops-50585}, doi = {10.4230/LIPIcs.CCC.2015.412}, annote = {Keywords: Proof complexity, algebraic complexity, arithmetic circuits, Frege, non-commutative formulas} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5391, Algebraic and Numerical Algorithms and Computer-assisted Proofs (2006)

We present a computational enclosure method for the solution of a class of nonlinear complementarity problems.
The procedure also delivers a proof for the uniqueness of the solution.

Götz Alefeld and Zhengyu Wang. Verification of Solutions for Almost Linear Complementarity Problems. In Algebraic and Numerical Algorithms and Computer-assisted Proofs. Dagstuhl Seminar Proceedings, Volume 5391, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{alefeld_et_al:DagSemProc.05391.9, author = {Alefeld, G\"{o}tz and Wang, Zhengyu}, title = {{Verification of Solutions for Almost Linear Complementarity Problems}}, booktitle = {Algebraic and Numerical Algorithms and Computer-assisted Proofs}, pages = {1--21}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2006}, volume = {5391}, editor = {Bruno Buchberger and Shin'ichi Oishi and Michael Plum and Sigfried M. Rump}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05391.9}, URN = {urn:nbn:de:0030-drops-4431}, doi = {10.4230/DagSemProc.05391.9}, annote = {Keywords: Complementarity problems, verification of solutions} }

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