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Quantified propositional calculus and a second-order theory for NC1

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Abstract

Let H be a proof system for quantified propositional calculus (QPC). We define the Σq j -witnessing problem for H to be: given a prenex Σq j -formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq 1 -witnessing problems for the systems G* 1 and G1 are complete for polynomial time and PLS (polynomial local search), respectively.

We introduce and study the systems G* 0 and G0, in which cuts are restricted to quantifier-free formulas, and prove that the Σq j -witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G* 0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them.

We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb 0 -CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G* 0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID.

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Authors and Affiliations

  1. Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, Ontario, M5S 3G4, Canada

    Stephen Cook

  2. Hitachi Systems Development Laboratory,  , 292 Yoshida-cho, Totsuka-ku, Yokohama-city, 244-0817, Japan

    Tsuyoshi Morioka

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  1. Stephen Cook
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  2. Tsuyoshi Morioka
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Correspondence to Stephen Cook.

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This research was carried while this author was a student at the University of Toronto.

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Cook, S., Morioka, T. Quantified propositional calculus and a second-order theory for NC1. Arch. Math. Logic 44, 711–749 (2005). https://doi.org/10.1007/s00153-005-0282-2

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  • DOI: https://doi.org/10.1007/s00153-005-0282-2

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