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CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS

  • JAN KRAJÍČEK (a1)

Abstract

We consider sets ${\it\Gamma}(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF with a short refutation in extended R, ER, can be easily reduced to an instance of ${\it\Gamma}(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that ${\it\Gamma}(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V_{1}^{1}$ . We use the ideas of implicit proofs from Krajíček [J. Symbolic Logic, 69 (2) (2004), 387–397; J. Symbolic Logic, 70 (2) (2005), 619–630] to define from ${\it\Gamma}(0,s,k)$ a nonrelativized NP search problem $i{\it\Gamma}$ and we show that it is complete among all such problems provably total in bounded arithmetic theory $V_{2}^{1}$ . The reductions are definable in theory $S_{2}^{1}$ . We indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

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[7] Krajíček, J., ‘Exponentiation and second-order bounded arithmetic’, Ann. Pure Appl. Logic 48(3) (1990), 261276.
[8] Krajíček, J., Bounded Arithmetic, Propositional Logic, and Complexity Theory, Encyclopedia of Mathematics and Its Applications, 60 (Cambridge University Press, Cambridge, 1995).
[9] Krajíček, J., ‘Implicit proofs’, J. Symbolic Logic 69(2) (2004), 387397.
[10] Krajíček, J., ‘Structured pigeonhole principle, search problems and hard tautologies’, J. Symbolic Logic 70(2) (2005), 619630.
[11] Krajíček, J., Forcing with Random Variables and Proof Complexity, London Mathematical Society Lecture Note Series, 382 (Cambridge University Press, Cambridge, 2011).
[12] Papadimitriou, C. H., ‘On the complexity of the parity argument and other inefficient proofs of existence’, J. Comput. Syst. Sci. 48(3) (1994), 498532.
[13] Tseitin, G. S., ‘On the complexity of derivations in propositional calculus’, inStudies in Mathematics and Mathematical Logic (ed. Slisenko, A. O.) Part II (Springer US, New York, 1968), 115125.
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  • EISSN: 2050-5094
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