An Improved Separation of Regular Resolution from Pool Resolution and Clause Learning

We prove that the graph tautology principles of Alekhnovich, Johannsen, Pitassi and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses and without degenerate resolution inferences. We also prove that these graph tautology principles can be refuted by polynomial size DPLL proofs with clause learning, even when restricted to greedy, unit-propagating DPLL search

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

Resolution Lower Bounds for Refutation Statements

For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three applications. (1) An open question in (Atserias, M\"uller 2019) asks whether a certain natural propositional encoding of the above statement is hard for Resolution. We answer by giving an exponential size lower bound. (2) We show exponential resolution size lower bounds for reflection principles, thereby improving a result in (Atserias, Bonet 2004). (3) We provide new examples of CNFs that exponentially separate Res(2) from Resolution (an exponential separation of these two proof systems was originally proved in (Segerlind, Buss, Impagliazzo 2004))

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

A switching lemma for small restrictions and lower bounds for k-DNF resolution

We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to formulas in disjunctive normal form (DNFs) with small terms. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with k-DNFs instead of clauses. We also obtain an exponential separation between depth d circuits of bottom fan-in k and depth d circuits of bottom fan-in k+1. Our results for Res(k) are as follows: 1. The 2n to n weak pigeonhole principle requires exponential size to refute in Res(k) for krootlog n/log log n. 2. For each constant k, there exists a constant w>k so that random w-CNFs require exponential size to refute in Res(k). 3. For each constant k, there are sets of clauses which have polynomial size Res(k+1) refutations but which require exponential size Res(k) refutations