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Some Remarks on Lengths of Propositional Proofs
, 2002
"... 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 lengt ..."
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Cited by 10 (1 self)
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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 pequivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed.
Number of symbols in Frege proofs with and without the deduction rule
 in Arithmetic, Proof Theory and Computational Complexity, P. Clote and
, 1993
"... 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 treelike. Also, two ..."
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Cited by 6 (0 self)
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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 treelike. 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.
The NPCompleteness of Reflected Fragments of Justification Logics
"... 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 w ..."
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Cited by 4 (3 self)
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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 socalled 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 NPcomplete, 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
The Deduction Theorem for Strong Propositional Proof Systems (Extended Abstract)
"... Abstract. This paper focuses on the deduction theorem for propositional logic. We define and investigate different deduction properties and show that the presence of these deduction properties for strong proof systems is powerful enough to characterize the existence of optimal and even polynomially ..."
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Cited by 2 (2 self)
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Abstract. This paper focuses on the deduction theorem for propositional logic. We define and investigate different deduction properties and show that the presence of these deduction properties for strong proof systems is powerful enough to characterize the existence of optimal and even polynomially bounded proof systems. We also exhibit a similar, but apparently weaker condition that implies the existence of complete disjoint NPpairs. In particular, this yields a sufficient condition for the completeness of the canonical pair of Frege systems and provides a general framework for the search for complete NPpairs. 1
Lower Complexity Bounds in Justification Logic
, 2009
"... 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 justif ..."
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Cited by 1 (0 self)
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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 socalled 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 NPcomplete, 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 2hard, reproving and generalizing an earlier result by Milnikel.
Linear lower bounds and simulations in Frege systems with substitutions
, 1997
"... . 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 treelike ..."
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. 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 treelike and daglike sF. We show the previous known lower bound, extend it to the treelike case and, for another class of tautologies, we give new lower bounds that in the daglike 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) treelike sF psimulates daglike sF; (2) Treelike F psimulates treelike 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...
Logical Closure Properties of Propositional Proof Systems
"... Abstract. In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of E ..."
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Abstract. In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic. 1