Are there Hard Examples for Frege Systems?

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

On the proof complexity of logics of bounded branching

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

The Hajós calculus is a simple, nondeterministic procedure that generates the class of non-3-colorable graphs. Mansfield and Welsch posed the question of whether there exist graphs that require exponential-sized Hajós constructions. Unless ${\text{NP}} \ne {\text{coNP}}$, there must exist graphs that require exponential-sized constructions, but to date, little progress has been made on this question, despite considerable effort. In this paper, we prove that the Hajós calculus generates polynomial-sized constructions for all non-3-colorable graphs if and only if extended Frege systems are polynomially bounded. Extended Frege systems are a very powerful family of proof systems for proving tautologies, and proving superpolynomial lower bounds for these systems is a long-standing, important problem in logic and complexity theory. We also establish a relationship between a complete subsystem of the Hajós calculus and bounded-depth Frege systems; this enables us to prove exponential lower bounds on this subsystem of the Hajós calculus.

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

A picture over a finite alphabet \(\varGamma \) is a matrix whose entries are drawn from \(\varGamma \). Let \(\pi :\varSigma \rightarrow \varGamma \) be a function between finite alphabets \(\varSigma \) and \(\varGamma \), and let \(V,H\subseteq \varSigma \times \varSigma \) be binary relations over \(\varSigma \). Given a picture \(N\) over \(\varGamma \), the picture satisfiability problem consists in determining whether there exists a picture M over \(\varSigma \) such that \(\pi (M_{ij})=N_{ij}\), and such that the constraints imposed by V and H on adjacent vertical and horizontal positions of M are respectively satisfied. This problem can be easily shown to be NP-complete. In this work we introduce the notion of s-smooth picture. Our main result states the satisfiability problem for s-smooth pictures can be solved in time polynomial on s and on the size of the input picture. With each picture \(N\), one can naturally associate a CNF formula \(F(N)\) which is satisfiable if and only if \(N\) is satisfiable. In our second result, we define an infinite family of unsatisfiable pictures which intuitively encodes the pigeonhole principle. We show that this family of pictures is polynomially smooth. In contrast we show that the formulas which naturally arise from these pictures are hard for bounded-depth Frege proof systems. This shows that there are families of pictures for which our algorithm for the satisfiability for smooth pictures performs exponentially better than certain classical variants of SAT solvers based on the technique of conflict-driven clause-learning (CDCL).

We prove that there is a constant K such that Tseitin formulas for a connected graph G requires proofs of size 2tw(G)Ω(1/d) in depth-d Frege systems for d<Klog⁡nlog⁡log⁡n, where tw(G) is the treewidth of G. This extends Håstad's recent lower bound from grid graphs to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)O(1/d)poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.