We give a recursion-theoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 |x||y| ) of Cobham.

In order to define models of simply typed functional programming languages being closer to the operational semantics of these languages, the notions of sequentiality, stability and seriality were introduced. These works originated from the definability problem for PCF, posed in [Sco72], and the full abstraction problem for PCF, raised in [Plo77]. The presented computation model, forming the class of hereditarily sequential functionals, is based on a game in which each play describes the interaction between a functional and its arguments during a computation. This approach is influenced by the work of Kleene [Kle78], Gandy [Gan67], Kahn and Plotkin [KP78], Berry and Curien [BC82, Cur86, Cur92], and Cartwright and Felleisen [CF92]. We characterize the computable elements in this model in two different ways: (a) by recursiveness requirements for the game, and (b) as definability with the schemata (S1)-- (S8), (S11), which is related to definability in PCF. It turns out that both definitio...

The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this first-order system relates to the second-order system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...

. K. Mehlhorn introduced a class of polynomial time computable operators in order to study poly time reducibilities between functions. This class is defined using a generalization of A. Cobham's definition of feasibility for type 1 functions to type 2 functionals. Cobham's feasible functions are equivalent to the familiar poly time functions. We generalize this equivalence to type 2 functionals. This requires a definition of the notion `poly time in the length of type 1 inputs'. The proof of this equivalence is not a simple generalization of the proof for type 1 functions; it depends on the fact that Mehlhorn's class is closed under a strong form of simultaneous limited recursion on notation, and requires an analysis of the structure of oracle queries in time bounded computations. Key words. type 2 computability, polynomial time, notational recursion, oracle Turing machine AMS subject classifications. 68Q05,68Q15,03D65,03D20 1. Introduction. A type 1 function is a mapping from N to ...

It is shown how to restrict recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types !oe, and by adding linear concepts to the lambda calculus. 1 Introduction Recursion in all finite types was introduced by Hilbert [9] and later became known as the essential part of Godel's system T [8]. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time. Simmons [16] showed that ramification can be used to characterize the primitive recursive functions by higher type recursion, and Leivant and Marion [14] showed that another form of ramification can be used to restrict higher type recursion to PSPACE. However, to characterize the much smaller class of polynomial-time computable functions by higher type recursion, it seems that an additional principle is required. By introducing linear...

An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in type theory, and consequently such concepts are not available in applications of type theory where they are greatly needed. It is also not clear how to provide a practical and adequate account in programming logics based on set theory. This paper provides a practical theory supporting all these concepts in the setting of constructive type theories. We first introduce an extensional theory of partial computable functions in type theory. We then add support for intensional reasoning about programs by explicitly reflecting the essential properties of the underlying computation system. We use the resulting intensional reasoning tools to justify computational induction and to define computationa...

Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: self-application is meaningful, but not necessarily total. It has turned out that theories with self-application provide a natural setting for studying notions of abstract computability, especially from a proof-theoretic perspective.

This paper investigates analogs of the Kreisel-Lacombe-Shoenfield Theorem in the context of the type-2 basic feasible functionals. We develop a direct, polynomial-time analog of effective operation in which the time boundingon computations is modeled after Kapron and Cook's scheme for their basic polynomial-time functionals. We show that if P = NP, these polynomial-time effective operations are strictly more powerful on R (the class of recursive functions) than the basic feasible functions. We also consider a weaker notion of polynomial-time effective operation where the machines computing these functionals have access to the computations of their procedural parameter, but not to its program text. For this version of polynomial-time effective operations, the analog of the Kreisel-Lacombe-Shoenfield is shown to hold---their power matches that of the basic feasible functionals on R.

A type 1 function is a total mapping from N to N. We will denote the set of all such functions by N N. A type 2 functional is a total mapping from ( N N) k \Theta N l to N, for some k; l. More specifically, we will call a mapping of this sort a functional with rank (k; l). For type 1 functions, there is a well established notion of computational feasibility. Namely a function is feasible if it is computable in polynomial time on a Turing machine. More specifically, a function f is poly time if there is a TM M and a polynomial p such that for all x, M with input x computes f(x) and runs in time p(n), where n = jxj, and for x 2 N, jxj denotes the length of the binary notation of x, that is dlog(x + 1)e. This notion of f...

In this paper we study (self-)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on W = f0; 1g are exactly the polynomial time computable functions.