
Do Hard SATRelated Reasoning Tasks Become Easier in the Krom Fragment?
Many reasoning problems are based on the problem of satisfiability (SAT)...
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Percolation and Phase Transition in SAT
Erdös and Rényi proved in 1960 that a drastic change occurs in a large r...
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Predicative proof theory of PDL and basic applications
Propositional dynamic logic (PDL) is presented in Schüttestyle mode as ...
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Iterated multiplication in VTC^0
We show that VTC^0, the basic theory of bounded arithmetic corresponding...
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Approximate counting and NP search problems
We study a new class of NP search problems, those which can be proved to...
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On the Complexity of Branching Proofs
We consider the task of proving integer infeasibility of a bounded conve...
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A Constraint Propagation Algorithm for SumsofSquares Formulas over the Integers
Sumsofsquares formulas over the integers have been studied extensively...
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Separation of bounded arithmetic using a consistency statement
This paper proves Buss's hierarchy of bounded arithmetics S^1_2 ⊆ S^2_2 ⊆...⊆ S^i_2 ⊆... does not entirely collapse. More precisely, we prove that, for a certain D, S^1_2 ⊊ S^2D+5_2 holds. Further, we can allow any finite set of true quantifier free formulas for the BASIC axioms of S^1_2, S^2_2, .... By Takeuti's argument, this implies P≠NP. Let Ax be a certain formulation of BASIC axioms. We prove that S^1_2 Con(PV^_1(D) + Ax) for sufficiently large D, while S^2D+7_2 Con(PV^_1(D) + Ax) for a system PV^_1(D), a fragment of the system PV^_1, induction free first order extension of Cook's PV, of which proofs contain only formulas with less than D connectives. S^1_2 Con(PV^_1(D) + Ax) is proved by straightforward adaption of the proof of PVCon(PV^) by Buss and Ignjatović. S^2D+5_2 Con(PV^_1(D) + Ax) is proved by S^2D+7_2 Con(PV^_q(D+2) + Ax), where PV^_q is a quantifieronly extension of PV^. The later statement is proved by an extension of a technique used for Yamagata's proof of S^2_2 Con(PV^), in which a kind of satisfaction relation Sat is defined. By extending Sat to formulas with less than Dquantifiers, S^2D+3_2 Con(PV^_q(D) + Ax) is obtained in a straightforward way.
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