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12
EXPONENTIAL LOWER BOUNDS FOR The Pigeonhole Principle
, 1993
"... In this paper we prove an exponential lower bound on the size of boundeddepth Frege proofs for the pigeonhole principle (PHP). We also obtain an f~(log log n)depth lower bound for any polynomialsized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact c ..."
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Cited by 122 (27 self)
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In this paper we prove an exponential lower bound on the size of boundeddepth Frege proofs for the pigeonhole principle (PHP). We also obtain an f~(log log n)depth lower bound for any polynomialsized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact complexity of the PHP, as S. Buss has constructed polynomialsize, log ndepth Frege proofs for the PHP. The main 1emma in our proof can be viewed as a general Hs Switching Lemma for restrictions that are partial matchings. Our lower bounds for the pigeonhole principle improve on previous superpolynomial lower bounds.
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 11 (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
, 2004
"... 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 o ..."
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Cited by 6 (0 self)
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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 log² n factors respectively.
The NPCompleteness of Reflected Fragments of Justification Logics
, 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 just ..."
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Cited by 4 (3 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.
Proofgraphs for Minimal Implicational Logic
, 2014
"... It is wellknown that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective nonnormal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We pre ..."
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Cited by 2 (2 self)
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It is wellknown that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective nonnormal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We present the formalism of proofgraphs for purely implicational logic, which are graphs of a specific shape that are intended to capture the logical structure of a deduction. The advantage of this formalism is that formulas can be shared in the reduced proof. In the present paper we give a precise definition of proofgraphs for the minimal implicational logic, together with a normalization procedure for these proofgraphs. In contrast to standard treelike formalisms, our normalization does not increase the number of nodes, when applied to the corresponding minimal proofgraph representations.
The Deduction Theorem for Strong Propositional Proof Systems (Extended 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 p ..."
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Cited by 2 (2 self)
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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.
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.
On the Deduction Theorem and Complete Disjoint NPPairs
, 2006
"... In this paper we ask the question whether the extended Frege proof system EF satisfies a weak version of the deduction theorem. We prove that if this is the case, then complete disjoint NPpairs exist. On the other hand, if EF is an optimal proof system, then the weak deduction theorem holds for E ..."
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In this paper we ask the question whether the extended Frege proof system EF satisfies a weak version of the deduction theorem. We prove that if this is the case, then complete disjoint NPpairs exist. On the other hand, if EF is an optimal proof system, then the weak deduction theorem holds for EF. Hence the weak deduction property for EF is a natural intermediate condition between the optimality of EF and the completeness of its canonical pair. We also exhibit two conditions that imply the completeness of the canonical pair of Frege systems.