ECCC-Report TR15-134https://eccc.weizmann.ac.il/report/2015/134Comments and Revisions published for TR15-134en-usWed, 19 Aug 2015 20:53:59 +0300
Paper TR15-134
| Characterizing Propositional Proofs as Non-Commutative Formulas |
Fu Li,
Iddo Tzameret,
Zhengyu Wang
https://eccc.weizmann.ac.il/report/2015/134Does 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 [Tza11] that considered adding the commutator as an axiom in algebraic propositional proof systems. This also 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].Wed, 19 Aug 2015 20:53:59 +0300https://eccc.weizmann.ac.il/report/2015/134