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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...

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Stephen A. Cook and Bruce M. Kapron Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 1 Introduction Functionals are functions which take natural numbers and other functionals as arguments and return natural numbers as values. The class of "feasible" functionals of finite type was introduced in [6] via the typed lambda calculus, and used to interpret certain formal systems of arithmetic: systems capturing the notion of "feasibly constructive proof" (we equate feasibility with polynomial time computability) . Here we name the functionals of [6] the basic feasible functionals and justify the designation by presenting results which include two programming style characterizations of the class. We also give examples of both feasible and infeasible functionals, and argue that the notion plays a natural role in complexity theory. Type 2 functionals take numbers and ordinary numerical functions as arguments. When these argument functions are 0-1 valued (i.e. sets) ...

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.

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. A formal approach to feasible numbers, as well as to middle and small numbers, is introduced, based on ideas of Parikh (1971) and improving his formalization. The "vague" set F of feasible numbers intuitively satisfies the axioms 0 2 F , F + 1 ` F and 2 1000 62 F , where the latter is stronger than a condition considered by Parikh, and seems to be treated rigorously here for the first time. Our technical considerations, though quite simple, have some unusual consequences. A discussion of methodological questions and of relevance to the foundations of mathematics and of computer science is an essential part of the paper. 1 Introduction How to formalize the intuitive notion of feasible numbers? To see what feasible numbers are, let us start by counting: 0,1,2,3, and so on. At this point, A.S. Yesenin-Volpin (in his "Analysis of potential feasibility", 1959) asks: "What does this `and so on' mean?" "Up to what extent `and so on'?" And he answers: "Up to exhaustion!" Note that by cos...

) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different ways. Recursion-theoretic, proof-theoretic and machine-theoretic characterizations are given for various classes, providing evidence of their naturalness. In this section, we motivate the approach of our work. In proof theory, primitive recursive functionals of higher type were introduced in Godel's Dialectica [13] paper, where they were used to "witness" the truth of arithmetic formulas. For instance, a function f witnesses the formula 8x9y\Phi(x; y), where \Phi is quantifier-free, provided that 8x\Phi(x; f(x)); while a type 2 functional F witnesses the formula 8x9y8u9v\Phi(x; y; u; v), provided that 8x8u\Phi(x; f(x); u; F (x; f(x); u)): Godel's formal system T is a variant of the finit...

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The basic concepts of type theory are fundamental to computer science, logic and mathematics. Indeed, the language of type theory connects these regions of science. It plays a role in computing and information science akin to that of set theory in pure mathematics. There are many excellent accounts of the basic ideas of type theory, especially at the interface of computer science and logic — specifically, in the literature of programming languages, semantics, formal methods and automated reasoning. Most of these are very technical, dense with formulas, inference rules, and computation rules. Here we follow the example of the mathematician Paul Halmos, who in 1960 wrote a 104-page book called Naïve Set Theory intended to make the subject accessible to practicing mathematicians. His book served many generations well. This article follows the spirit of Halmos ’ book and introduces type theory without recourse to precise axioms and inference rules, and with a minimum of formalism. I start by paraphrasing the preface to Halmos ’ book. The sections of this article follow his chapters closely. Every computer scientist agrees that every computer scientist must know some type theory; the disagreement begins in trying to decide how much is some. This article contains my partial answer to that question. The purpose of the article is to tell the beginning student of advanced computer science the basic type theoretic facts of life, and to do so with a minimum of philosophical discourse and logical formalism. The point throughout is that of a prospective computer scientist eager to study programming languages, or database systems, or computational complexity theory, or distributed systems or information discovery. In type theory, “naïve ” and “formal ” are contrasting words. The present treatment might best be described as informal type theory from a naïve point of view. The concepts are very general and very abstract; therefore they may