京都大学数理解析研究所の佐藤です。

10月6日11:00から、産業技術総合研究所の山形頼之氏に 以下の講演をしていただくことになりましたので、 ご連絡いたします。どうぞお気軽にお越しください。 ========== Time: 11:00-12:00, 6 Oct, 2016 Place: Rm 478, Research Building 2, Main Campus, Kyoto University 京都大学 本部構内 総合研究2号館 4階478号室 http://www.kyoto-u.ac.jp/en/access/yoshida/main.html (Building 34) http://www.kyoto-u.ac.jp/ja/access/campus/map6r_y.htm (34番の建物)

Speaker: Yoriyuki Yamagata (National Institute of Advanced Industrial Science and Technology) (Website: http://staff.aist.go.jp/yoriyuki.yamagata/en/)

Title: Consistency proof of an arithmetic with substitution inside a bounded arithmetic

Abstract: Bounded arithmetics are systems of arithmetic in which induction is restricted to bounded formulas. Bounded formulas are formulas in which all quantifiers are bounded by some terms. Among many such systems, Buss's hierarchy of bounded arithmetics have particular interest, because their provably total functions exactly correspond polynomial-time hierarchy.

A major open problem on bounded arithmetics is whether Buss's hierarchy collapses or not. If it collapses, for example, it implies P=NP. One strategy to differentiate the hierarchy is to use Gödel sentence of some theories. In this talk, we consider a theory PV by Cook and Urquhart and its restrictions. PV is an equational system, which has roughly the same strength to Buss's S12. Beckmann proved that when induction and substitution are removed from PV, the consistency of the system can be proved by S12. On the other hand, Buss and Ignjatović proved that S12 cannot prove the consistency of the system which obtained by removing just induction, but adding propositional logic and BASIC axioms, from PV.

In this talk, we improve the Beckmann by proving that S22 can prove the consistency of the system which obtained by removing just induction from PV. By delineating bounded arithmetics which can prove consistency of various restrictions of PV, we hope to prove non-collapse of Buss's hierarchy, and relatedly, polynomial-time hierarchy.