A foundational delineation of computational feasibility

A new recursion-theoretic characterization of the polytime functions

“…We formulate the result using computation on non-negative integers, but the same proof carries over to computation on binary strings as in Leivant [17], replacing 0 with . We write |x| for the binary length log 2 (x + 1) of integer x; and the terms "predecessor" and "successor" refer to binary notation.…”

“…As Nelson [21] points out and Leivant [17] emphasises, definitions of N are impredicative because they are justified by inductive means, which presuppose an understanding of N . In other words, the validity of defining a function by recursion requires an impredicatively defined concept, namely N .…”

“…Leivant's recent elegant characterization of polynomial time [17] suggests that one might be able to dispense with these features controlling the growth rate of functions. Leivant proves that a function is polytime if and only if it can be proved convergent in the logical system L 2 (QF + ) using the function's recursion equations and a "surjective" principle.…”

“…The result can be read independently of the results in Leivant's paper [17]. Our closure operations are essentially composition and recursion on notation, syntactically restricted in ways which have no direct connection to bounds on growth rates.…”

“…Motivating Example. As a motivating example, consider the definition of the function (similar to the smash function of Buss and others) in [17]. The definition uses recursion on binary strings as follows: (w, sv) = ⊕(w, (w, v)) for s ∈ {0, 1}, where ⊕ (concatenation) has been defined by recursion on its first input, not its second.…”