2 Time - Complexity Semantics for Feasible Affine Recursions

The authors' ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs run in type-2 polynomial-time and all standard type-2 basic feasible functionals are ATR-deenable (ATR types are connned to levels 0, 1, and 2). A limitation of the original version of ATR is that the only directly expressible recursions are tail-recursions. Here we extend ATR so that a broad range of aane recursions are directly expressible. In particular, the revised ATR can fairly naturally express the classic insertion-and selection-sort algorithms, thus overcoming a sticking point of most prior implicit-complexity-based formalisms. The paper's main work is in extending and simplifying the original time-complexity semantics for ATR to develop a set of tools for extracting and solving the higher-type recurrences arising from feasible aane recursions. 1. Two algorithms in search of a type-system As Hofmann [9] has noted, a problem with implicit characterizations of complexity classes is that they often fail to capture many natural algorithms|usually because the complexity-theoretic types used to control primitive recursion impose draconian restrictions on programming. Here is an example. In Bellantoni and Cook's [3] and Leivant's [11] well-known characterizations of the polynomial time computable functions, a recursively-computed value is prohibited from driving another recursion. But, for instance, the recursion clause of insertion-sort has the form ins sort(cons(a; l)) = insert(a; ins sort(l)), where insert is deened by recursion on its second argument; selection-sort presents analogous problems. Hofmann [9, 8] addresses this problem by noting that the output of a non-size-increasing program (such as ins sort) should be permitted to drive another recursion, as it cannot cause the sort of complexity blow-up the B-C-L restrictions guard against. To incorporate such recursions, Hof-mann deenes a higher-order language with typical rst-order types and a special type through which functions deened recursively must \pay" for any use of size-increasing constructors, in eeect guaranteeing that there is no size increase. Through this scheme Hofmann is able to implement many natural algorithms while still ensuring that any typable program is non-size-increasing polynomial time computable (Aehlig and Schwichtenberg [1] sketch an extension that captures all of polynomial-time). Our earlier paper [5, 6], hereafter referred to as ATS, takes a diierent approach to constructing a usable programming language with guaranteed resource usage. We introduce a type-2 programming formalism called ATR (for AAne Tail Recursion, which we rechristen in this paper as AAne Tiered Recursion) based on PCF. …