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Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

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Abstract

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.

BibTeX - Entry

@InProceedings{li_et_al:LIPIcs:2015:5058,
  author =	{Fu Li and Iddo Tzameret and Zhengyu Wang},
  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 =	{David Zuckerman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5058},
  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}
}

Keywords: Proof complexity, algebraic complexity, arithmetic circuits, Frege, non-commutative formulas
Seminar: 30th Conference on Computational Complexity (CCC 2015)
Issue date: 2015
Date of publication: 2015


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