Abstract not found

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. Depth d Frege proofs of m lines can be transformed into depth d proofs of O(m^(d+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.

Abstract Frege systems with the deduction rule produce at most quadratic speedup over Frege systems using as a measure of length the number of symbols in the proof. We study whether that speedup is in reality smaller. We show that the speedup is linear when the Frege proofs are tree-like. Also, two groups of formulas, permutation formulas and transitive closure formulas, that seemed most likely to produce an almost quadratic speedup when using the deduction rule, are shown to produce only log n and log 2 n factors respectively. 1 Introduction A Frege proof system is an inference system for propositional logic in which the only rule of inference is Modus Ponens.

Abstract not found

Abstract. Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the so-called reflected fragments, which still contain complete information about the respective justification logics, are known to be in NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound. 1 Introduction and Main Definitions Justification Logic is an emerging field that studies provability, knowledge, and belief via explicit proofs or justifications that are part of the language. A justification

. We investigate the complexity of proofs in Frege (F), Substitution Frege (sF) and Renaming Frege (rF) systems. Starting from a recent work of Urquhart and using Kolmogorov Complexity we give a more general framework to obtain superlogarithmic lower bounds for the number of lines in both tree-like and dag-like sF. We show the previous known lower bound, extend it to the tree-like case and, for another class of tautologies, we give new lower bounds that in the dag-like case slightly improve the previous one. Also we show optimality of Urquhart's lower bounds giving optimal proofs. Finally we give the following two simulation results: (1) tree-like sF p-simulates dag-like sF; (2) Tree-like F p-simulates tree-like rF . 1 Introduction Since the work of Cook and Reckhow [CR], the study of complexity of proofs in propositional logic is viewed as related to main questions like NP 6= coNP in Complexity Theory. The main open problem is whether for all propositional proof systems th...

Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the so-called reflected fragments, which still contain complete information about the respective justification logics, are known to be in NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound. The proof method is then extended to provide a uniform proof that the corresponding full pure justification logics areΠ p 2-hard, reproving and generalizing an earlier result by Milnikel.