The pumping lemma for context-free languages is a result about pushdown automata which is strikingly similar to the well-known pumping lemma for regular languages. However, though the lemma for regular languages is simply proved by using the pigeonhole principle on deterministic automata, the lemma for pushdown automata is proven through an equivalence with context-free languages and through th...

In this paper we study the complexity of Cutting Planes (CP) refutations, and tree-like CP refutations. Tree-like CP proofs are natural and still quite powerful. In particular, the propositional pigeonhole principle (PHP) has been shown to have polynomial-sized tree-like CP proofs. Our main result shows that a family of tautologies, introduced in this paper requires exponential-sized tree-like ...

Semi-algebraic proof systems were introduced in [1] as extensions of Lovász-Schrijver proof systems [2,3]. These systems are very strong; in particular, they have short proofs of Tseitin’s tautologies, the pigeonhole principle, the symmetric knapsack problem and the cliquecoloring tautologies [1]. In this paper we study static versions of these systems. We prove an exponential lower bound on th...

Abstract. 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 th...

We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the KneserLovász Theorem. This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k = 1). We show, for all fixed k, 2Ω(n) lower bounds on resolution complexity and exponential lower bou...

A basic pigeonhole principle insures an existence of two objects of the same type if the number of objects is larger than the number of types. Can such a principle be extended to a more complex combinatorial structure? Here, we address such a question for graphs. We call two disjoint subsets A,B of vertices twins if they have the same cardinality and induce subgraphs of the same size. Let t(G) ...

Studying the complexity of mathematical proofs is important not only for automated theorem proving, but also for Mathematics as a whole. Each significant result in this direction would potentially have a great impact on Foundations of mathematics. Surprisingly enough, the general Proof Complexity is closely related to Propositional Proof Complexity. The latter area was founded by Cook and Reckh...

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the runni...

In Arai (1996), we introduced a new inference rule called permutation to propositional calculus and showed that cut-free Gentzen system LK (GCNF) with permutation (1) satis es the feasible subformula property, and (2) proves pigeonhole principle and k-equipartition polynomially. In this paper, we survey more properties of our system. First, we prove that cut-free LK + permutation has polynomial...

It was recently argued that the pigeonhole principle, which states if three pigeons are put into two pigeonholes then at least one must contain more than pigeon, is violated in quantum systems [Y. Aharonov et al., PNAS 113, 532 (2016)]. An experimental verification of this effect reported [M.-C. Chen 116, 1549 (2019)]. In another recent work, it entities were observed to exchange properties wit...