Stringent Relativization ∗ — A New Approach for Studying Complexity Classes —

A new notion of relativization—stringent relativization—has been proposed recently [CW06] for discussing collapsing relations of complexity classes, with which we hope to open a new approach for studying complexity classes. Starting with the motivation of this notion, we discuss the meaning and implication of collapsing relations under the stringent relativization. 1 Introduction to Stringent Relativization The notion of relativization was introduced to demonstrate the difficulty of proving certain relations among complexity classes. For example, regarding the most famous complexity relation, that of P and NP, Baker, Gill and Solovay [BGS75] showed that we can relativize it in both ways; that is, there exist two oracles A and B such that P A = NP A (the collapsing) hold

Technical Reports on Mathematical and Computing Sciences: TR-C168

For relativized arguments, we propose to restrict oracle queries to "stringent" ones; for comparing the power of two machine models relative to some oracle set, stringent relativization is to restrict machines of both types to ask queries on the same segment of the oracle. In particular, for investigating polynomial-time (or polynomial-size) computability, we propose polynomial stringency, bounding query length to any fixed polynomial of input length. Under such stringent oracle access, we show an oracle G such that BPP