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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2015.412
URN: urn:nbn:de:0030-drops-50585
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5058/
Li, Fu ;
Tzameret, Iddo ;
Wang, Zhengyu
Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs
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}
}
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Keywords: |
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Proof complexity, algebraic complexity, arithmetic circuits, Frege, non-commutative formulas |
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Collection: |
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30th Conference on Computational Complexity (CCC 2015) |
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Issue Date: |
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2015 |
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Date of publication: |
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06.06.2015 |
