A logical calculus for polynomial-time realizability

A logical calculus, not unlike Gentzen's sequent calculus for intuitionist logic, is described which is sound for polynomial-time realizability as deened by Crossley and Remmel. The sequent calculus admits cut elimination, thus giving a decision procedure for the propositional fragment. 0 Introduction In 4], a restricted notion of realizability is introduced, a special case of which is polynomial-time realizability: this is like Kleene's original realizability, save for three features. First, closed atomic formull are realized by realizers that give a measure of the resources required to establish the formula, unlike Kleene's system which only reeects the fact that the formula is provable. Second, open formull are treated as the corresponding closed formull with all free variables universally quantiied simultaneously. (There is a diierence between the quantiiers 8h; i and 88.) And third, the realizers code polynomial-time (\p-time") functions, rather than arbitrary recursive functions. In 4], only the p-time realizability of single formull is discussed|in this paper we shall extend these notions to logical rules, to give a calculus that is sound for p-time realizability. This will be a sequent calculus, much like Gentzen's formulation for intuitionist logic, with three main points of diierence. First, a sequent of the form A; B ?! C is interpreted as if it were a formula A (B C), rather than (A ^ B) C (which would be the Gentzen interpretation). As was shown in 4], these are not equivalent. Indeed, if k? (A ^ B) C then k? A (B C), but not conversely. (k? A means \A is realizable"; similarly e k? A means \e realizes A".) Second, among the structure rules we keep thinning, but drop exchange and contraction, (roughly as in Girard's linear logic 7]). Again, it was shown in 4] that one could have k? A (B C) without having k? B (A C). For example, in 4] there is an example that occurs frequently. In the notation of this paper, it can be given in these three avours: (x :); (y :) ?! x y+1 = x y x (1) (y :); (x :) ?! x y+1 = x y x (2) (x; y :) ?! x y+1 = x y x (3) (These examples may also be presented as quantiied formull.) Of course, there is no way that Examples 1, 3 can be realizable in polynomial time, but Example 2 is realizable in polynomial time, essentially because Partially …