Graded Multicategories of Polynomial-time Realizers

We present a logical calculus which imposes a grading on a sequent-style calculus to account for the runtime of the programmes represented by the sequents. This system is sound for a notion of polynomial-time realizability. An extension of the grading is also considered, giving a notion of \dependant grades", which is also sound. Furthermore, we deene a notion of closed graded multicategory, and show how the structure of polynomial-time realizers has that structure. 0 Introduction In 4], a restricted notion of realizability is deened, a special case of which is polynomial-time realizability: this is like Kleene's original realizability, save for three features. First, closed atomic formulae are realized only by realizers that express a reason for the \truth" (or provability) of the formula, unlike Kleene's system which only reeects the fact that the formula is provable. Second, open formulae are treated as the corresponding closed for-mulae 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 formulae is discussed|in 5] these notions are extended to logical rules, to give a sequent calculus that is sound for p-time realiz-ability. This sequent calculus is much like Gentzen's formulation for intuitionist logic, with three main points of diierence, which we summarize again here. 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. and the author, who wishes to express his thanks to those two for their many kindnesses during his visit.