Extended Frege Proofs Research Articles

Feasibly constructive proofs of succinct weak circuit lower bounds

We ask for feasibly constructive proofs of known circuit lower bounds for explicit functions on bit strings of length n. In 1995 Razborov showed that many can be proved in PV1, a bounded arithmetic formalizing polynomial time reasoning. He formalized circuit lower bound statements for small n of doubly logarithmic order. It is open whether PV1 proves known lower bounds in succinct formalizations for n of logarithmic order. We give such proofs in APC1, an extension of PV1 formalizing probabilistic polynomial time reasoning: for parity and AC0, for mod q and AC0[p] (only for n slightly smaller than logarithmic), and for k-clique and monotone circuits. We also formalize Razborov and Rudich's natural proof barrier.We ask for short propositional proofs of circuit lower bounds expressed succinctly by propositional formulas of size nO(1) or at least much smaller than the 2O(n) size of the common “truth table” formula. We discuss two such expressions: one via feasible functions witnessing errors of circuits, and one via the anticheckers of Lipton and Young 1994. Our APC1 formalizations yield conditional upper bounds for the succinct formulas obtained by witnessing: we get short Extended Frege proofs from general circuit lower bounds expressed by the common “truth-table” formulas. We also show how to construct in quasipolynomial time propositional proofs of quasipolynomial size tautologies expressing AC0[p] quasipolynomial size lower bounds; these proofs are in Jeřábek's system WF.

Short proofs of the Kneser–Lovász coloring principle

We prove that propositional translations of the Kneser–Lovász theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs for all fixed values of k. We present a new counting-based combinatorial proof of the Kneser–Lovász theorem based on the Hilton–Milner theorem; this avoids the topological arguments of prior proofs for all but finitely many base cases. We introduce new “truncated Tucker lemma” principles, which are miniaturizations of the octahedral Tucker lemma. The truncated Tucker lemma implies the Kneser–Lovász theorem. We show that the k=1 case of the truncated Tucker lemma has polynomial size extended Frege proofs.

The NP Search Problems of Frege and Extended Frege Proofs

We study consistency search problems for Frege and extended Frege proofs—namely the NP search problems of finding syntactic errors in Frege and extended Frege proofs of contradictions. The input is a polynomial time function, or an oracle, describing a proof of a contradiction; the output is the location of a syntactic error in the proof. The consistency search problems for Frege and extended Frege systems are shown to be many-one complete for the provably total NP search problems of the second-order bounded arithmetic theories U 1 2 and V 1 2 , respectively.

Quasipolynomial size proofs of the propositional pigeonhole principle

Cook and Reckhow proved in 1979 that the propositional pigeonhole principle has polynomial size extended Frege proofs. Buss proved in 1987 that it also has polynomial size Frege proofs; these Frege proofs used a completely different proof method based on counting. This paper shows that the original Cook and Reckhow extended Frege proofs can be formulated as quasipolynomial size Frege proofs. The key point is that st-connectivity can be used to define the Cook–Reckhow construction.

Substitution Frege and extended Frege proof systems in non-classical logics

We investigate the substitution Frege ( SF ) proof system and its relationship to extended Frege ( EF ) in the context of modal and superintuitionistic (si) propositional logics. We show that EF is p-equivalent to tree-like SF , and we develop a “normal form” for SF -proofs. We establish connections between SF for a logic L , and EF for certain bimodal expansions of L . We then turn attention to specific families of modal and si logics. We prove p-equivalence of EF and SF for all extensions of KB , all tabular logics, all logics of finite depth and width, and typical examples of logics of finite width and infinite depth. In most cases, we actually show an equivalence with the usual EF system for classical logic with respect to a naturally defined translation. On the other hand, we establish exponential speed-up of SF over EF for all modal and si logics of infinite branching, extending recent lower bounds by P. Hrubeš. We develop a model-theoretical characterization of maximal logics of infinite branching to prove this result.

Berkowitz's algorithm and clow sequences

A combinatorial interpretation of Berkowitz's algorithm is presented. Berkowitz's algorithm is the fastest known parallel algorithm for computing the characteristic polynomial of a matrix. The combinatorial interpretation is based on loop covers introduced by Valiant, and sequences, defined by Mahajan and Vinay. Clow sequences turn out to capture very succinctly the computations performed by Berkowitz's algorithm, which otherwise are quite difficult to analyze. The main contribution of this paper is a proof of correctness of Berkowitz's algorithm in terms of clow sequences. multiple of log 2 n. There are two other fast parallel algorithms for computing the coefficients of the characteristic polynomial of a matrix: Chistov's algorithm and Csanky's algorithm. Chistov's algorithm is more difficult to formalize, and Csanky's algorithm works only for fields of char 0; see (8, section 13.4) for details about these two algorithms. The author's original motivation for studying Berkowitz's algorithm was the proof complexity of linear algebra. Proof Complexity deals with the complexity of formal mathematical derivations, and it has applications in lower bounds and automated theorem proving. In particular, the author was interested in the complexity of deriva- tions of matrix identities such as AB = I → BA = I (right matrix inverses are left inverses). These identities have been proposed by Cook as candidates for separating the Frege and Extended Frege proof systems. Proving (or disproving) this separation is one of the outstanding problems in propositional proof complexity; see (3) for a comprehensive exposition of this area of research. Thus, we were interested in an algorithm that could compute inverses of matrices, of the lowest complexity possible. Berkowitz's algorithm is ideal for our purposes for several reasons: • as was mentioned above, it has the lowest known complexity for computing

Some Consequences of Cryptographical Conjectures forS12and EF

We show that there is a pair of disjoint N P -sets, whose disjointness is provable in S 1 2 and which cannot be separated by a set in P /poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 1 2 admitting an N P -definition of primes about which it can prove that no number satisfying the definition is composite. As a corollary we obtain that the Extended Frege (EF) proof system does not admit a feasible interpolation theorem unless the RSA cryptosystem is not secure and that an extension of EF by tautologies τ p ( p primes), formalizing that p is not composite, as additional axioms do not admit a feasible interpolation theorem unless factoring and the discrete logarithm are in P /poly.

Some remarks on lengths of propositional proofs

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(md+1) symbols. We show that renaming Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed.

Propositional consistency proofs

Partial consistency statements can be expressed as polynomial-size propositional formulas. Frege proof systems have polynomial-size partial self-consistency proofs. Frege proof systems have polynomial-size proofs of partial consistency of extended Frege proof systems if and only if Frege proof systems polynomially simulate extended Frege proof systems. We give a new proof of Reckhow's theorem that any two Frege proof systems p-simulate each other. The proofs depend on polynomial size propositional formulas defining the truth of propositional formulas. These are already known to exist since the Boolean formula value problem is in alternating logarithmic time; this paper presents a proof of this fact based on a construction which is somewhat simpler than the prior proofs of Buss and of Buss-Cook- Gupta-Ramachandran.