The provably total search problems of bounded arithmetic

0369680 - MÚ 2012 RIV GB eng J - Journal Article
Skelley, A. - Thapen, Neil
The provably total search problems of bounded arithmetic.
Proceedings of the London Mathematical Society. Roč. 103, č. 1 (2011), s. 106-138. ISSN 0024-6115. E-ISSN 1460-244X
R&D Projects: GA MŠMT LC505
Institutional research plan: CEZ:AV0Z10190503
Keywords : Pigeonhole principle * polynomial hierarchy * local search
Subject RIV: BA - General Mathematics
Impact factor: 1.324, year: 2011
http://plms.oxfordjournals.org/content/103/1/106

We give combinatorial principles GI(k), based on k-turn games, which are complete for the class of NP search problems provably total at the kth level T(2)(k) of the bounded arithmetic hierarchy and hence characterize the for all Sigma(b)(1) consequences of T(2)(k). Our argument uses a translation of first-order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that for all(Sigma) over cap (b)(1)(alpha) conservativity of T(2)(i+1) (alpha) over T(2)(i)(alpha) already implies. for all(Sigma) over cap (b)(1)(a) conservativity of T(2)(alpha) over T(2)(i)(alpha). We translate this into propositional form and give a polylogarithmic width CNF (GI(3)) over bar such that if (GI(3)) over bar has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for (GI(3)) over bar.
Permanent Link: http://hdl.handle.net/11104/0203689