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

Fu Li, Iddo Tzameret, Zhengyu Wang, Marc Herbstritt
2015 Computational Complexity Conference
Does every Boolean tautology have a short propositional-calculus proof? Here, a propositionalcalculus (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.
more » ... tive 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 , 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  that introduced a propositional proof system in which proofs are arithmetic circuits, and the work in  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 , 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. The field of propositional proof complexity aims to understand and analyze the computational resources required to prove propositional statements. The problems the field poses are fundamental, difficult and go back to the work of Cook and Reckhow  , who showed the immediate relevance of these problems to the NP vs. coNP problem (and thus the P vs. NP problem). Among the major unsolved questions in proof complexity, is whether the standard propositional logic calculus, either in the form of the Sequent Calculus, or equivalently, in the axiomatic form of Hilbert proofs (i.e., Frege proofs), is polynomially bounded; that is, whether every propositional tautology (or unsatisfiable formula) has a proof whose size is polynomially bounded (refutation, resp.) in the size of the formula proved. Here, we consider the size of proofs as the number of symbols it takes to write them down, where each formula in the proof is written as a Boolean formula (in other words we count the total number of logical gates appearing in the proof where each proof-line is a formula). It is known  that all Frege proof-systems (formally, a Frege proof system is any propositional proof system with a fixed number of axiom schemes and sound derivation rules that is also implicationally complete, and in which proof-lines are written as propositional formulas (see e.g.,  and Definition 2.4 below)) as well as the Gentzen sequent calculus (with the cut rule) are polynomially equivalent to each other, and hence it does not matter precisely which rules, axioms, and logical-connectives we use. Complexity-wise, the Frege proof system is considered a very strong system alas a poorly understood one. The qualification strong here has several meanings: first, that no superpolynomial lower bound is known for Frege proofs. Second, that there are not even good hard candidates for the Frege system (see [4, 17, 18] for a further discussion on hard proof complexity candidates). Third, that for most hard instances (e.g., the pigeonhole principle and Tseitin tautologies) that are known to be had for weaker systems (e.g., resolution, cutting planes, etc.), there are known polynomial bounds on Frege proofs. Fourth, that proving super-polynomial lower bounds on Frege proofs seems to a certain extent out of reach of current techniques. And finally, that by the common (mainly informal) correspondence between circuits and proofs -namely, the correspondence between a circuit-class C and a proof system in which every proof-line is written as a circuit from C (to be more precise, one has to associate a circuit class C with a proof system in which a family of proofs is written such that every proof-line in the family is a circuit family from C) -Frege system corresponds to the circuit class of polynomial-size log(n)-depth circuits denoted NC 1 (equivalently, of polynomial-size formulas ), considered to be a strong computational model for which no (explicit) super-polynomial lower bounds are currently known. Accordingly, proving lower bounds on Frege proofs is considered an extremely hard task. In fact, the best lower bound known today is only quadratic  , which uses a fairly simple syntactic argument. If we put further impeding restrictions on Frege proofs, like restricting the depth of each formula appearing in a proof to a certain fixed constant, exponential lower bounds can be obtained [1, 21, 21] . Although these constant-depth Frege exponential-size lower bounds go back to Ajtai's result from 1988, they are still in some sense the state-of-theart in proof complexity lower bounds (beyond the important developments on weaker proof systems, such as resolution and its weak extensions). Constant-depth Frege lower bounds use quite involved probabilistic arguments, mainly specialized switching lemmas tailored for specific tautologies (namely, counting tautologies, most notable of which are the Pigeonhole C C C 2 0 1 5