On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies

We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. ˝ We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles

Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's

On the relative proof complexity of deep inference via atomic flows

We consider the proof complexity of the minimal complete fragment, KS, of standard deep inference systems for propositional logic. To examine the size of proofs we employ atomic flows, diagrams that trace structural changes through a proof but ignore logical information. As results we obtain a polynomial simulation of versions of Resolution, along with some extensions. We also show that these systems, as well as bounded-depth Frege systems, cannot polynomially simulate KS, by giving polynomial-size proofs of certain variants of the propositional pigeonhole principle in KS.Comment: 27 pages, 2 figures, full version of conference pape

The Cook-Reckhow definition

The Cook-Reckhow 1979 paper defined the area of research we now call Proof Complexity. There were earlier papers which contributed to the subject as we understand it today, the most significant being Tseitin's 1968 paper, but none of them introduced general notions that would allow to make an explicit and universal link between lengths-of-proofs problems and computational complexity theory. In this note we shall highlight three particular definitions from the paper: of proof systems, p-simulations and the pigeonhole principle formula, and discuss their role in defining the field. We will also mention some related developments and open problems

Reasons for Hardness in QBF Proof Systems

We aim to understand inherent reasons for lower bounds for QBF proof systems, and revisit and compare two previous approaches in this direction. The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS\u2716). Here we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) `genuine\u27 QBF lower bounds. The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ICALP\u2716). We prove a strong lower bound for relaxing QU-Res, which also exhibits significant shortcomings of that model. Prompted by this we propose an alternative, improved version, allowing more flexible oracle queries in proofs. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction

Monotone Proofs of the Pigeon Hole Principle

Lecture Notes in Computer Science. Geneva, Switzerland, July 9-15

On the pigeonhole and related principles in deep inference and monotone systems

International audienceWe construct quasipolynomial-size proofs of the propositional pigeonhole principle in the deep inference system KS, addressing an open problem raised in previous works and matching the best known upper bound for the more general class of monotone proofs. We make significant use of monotone formulae computing boolean threshold functions, an idea previously considered in works of Atserias et al. The main construction, monotone proofs witnessing the symmetry of such functions, involves an implementation of merge-sort in the design of proofs in order to tame the structural behaviour of atoms, and so the complexity of normalization. Proof transformations from previous work on atomic flows are then employed to yield appropriate KS proofs. As further results we show that our constructions can be applied to provide quasipolynomial-size KS proofs of the parity principle and the generalized pigeonhole principle. These bounds are inherited for the class of monotone proofs, and we are further able to construct n^O(log log n) -size monotone proofs of the weak pigeonhole principle with (1 + ε)n pigeons and n holes for ε = 1/ polylog n, thereby also improving the best known bounds for monotone proofs