Consistency proof of a feasible arithmetic inside a bounded arithmetic

Ever since Buss showed the relation of his hierarchy of bounded arithmetic S 2, i = 1, 2, . . . to polynomial-time hierarchy of computational complexity [2], the question of whether his hierarchy is collapsed at some i = n or not, has become one of the central questions of bounded arithmetic. This is because the collapse of Buss’s hierarchy implies the collapse of polynomial-time hierarchy. In particular, S 2 = ⋃ i=1,2,... S i 2 (the right hand side will be denoted by S2.) implies P = NP , because S 2 characterizes polynomial time-computable functions P . A classical way to prove the separation of theories is the use of the second incompleteness theorem of Gödel. If it is proved that S proves the consistency of S 2 , S 1 2 6= S2 is obtained, because S 2 cannot prove its own consistency. Unfortunately, Wilkie and Paris showed that S2 cannot prove the consistency of Robinson arithmetic Q [8], a much weaker system. Although this result stems more from the free use of unbounded quantifiers than from the power of arithmetic, Pudlák showed that S2 cannot prove the consistency of bounded proofs (proofs of which the formulas only have bounded quantifiers) of S 2 [6]. The result was refined by Takeuti [7] and Buss and Ignjatović [3], who showed that, even if the induction were to be removed from S 2 , S2 would still not be able to prove the consistency of its bounded proofs. Thus, it would be interesting to delineate theories which can be proven to be consistent in S2 and S 1 2 , to find a theory T that can be proven to be consistent in S2 but not in S 1 2 . In particular, we focus on Cook and Urquhart’s system PV, which is essentially an equational version of S 2 . Buss and Ignjatović stated that PV cannot prove the consistency of PV−, a system based on PV from which induction has been removed. On the other hand, Beckmann [1] later proved that S 2 can prove the consistency of a theory which is obtained from PV− by removing the substitution rule. In this talk, I will present a proof that S 2 is capable of proving the consistency of purely equational PV− with the substitution rule. This result apparently contradicts that of Buss and Ignjatović. However, their proof actually shows that PV cannot prove the consistency of the extension of PV− that contains propositional logic and BASIC axioms. On the other hand, our PV− is strictly equational, which is a property on which our proof relies. The consistency of PV− can be proven by using the following strategy. Beckmann uses a rewriting system to prove consistency of PV− excluding the substitution rule. According to the terminology of programming language theory, the use of a rewriting system to define the evaluation of terms, is referred to as small-step semantics (referred to as structural operational semantics in [5]).