We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ! defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with \Sigma b 1 -induction over PV ! and use this to reestablish that the provably total functions in this system are in polynomial time computable. Finally, we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. 1 Introduction Cook and Urquhart's system PV ! [3] is a simply-typed lambda calculus providing constants to denote natural numbers and an operator for bounded recursion on notation like in Cobham's characterisation of polynomial-time computability. 1 Although functionals of arbitrary type can be defined in this system one can show that thei...

Recently explicit type parameter passing has been studied as an attractive approach to utilizing type information in ML. An important issue in this approach is to develop an efficient method of type parameter passing. Tolmach proposed a method based on lazy substitution on types and demonstrated the feasibility of his method in his implementation of tag-free garbage collection. However, there are still some significant costs associated with runtime construction of type parameters in his method. In this paper we propose a refinement of his method based on the transformation which completely eliminates runtime construction of type parameters. The transformation lifts type parameters that cause runtime construction so that they can be constructed statically. We present our transformation as a type preserving translation and prove the correctness of the translation. Furthermore, we describe an implementation of a compiler for Core Standard ML based on our method and compare our method to T...

data types; F.3.1 [Logics and Meaning of Programs] Specifying and verifying and reasoning about programs --- logics of programs. F.3.2 [Logics and Meanings of Programs] Semantics of Programming Languages --- algebraic approaches to semantics, denotational sematics. Submitted to the European Conference on Object-Oriented Programming, ECOOP '93. c fl Krishna Kishore Dhara and Gary T. Leavens, 1992. All rights reserved. Department of Computer Science 226 Atanasoff Hall Iowa Sate University Ames, Iowa 50011-1040, USA Subtyping for mutable types in object-oriented programming languages Krishna Kishore Dhara and Gary T. Leavens 3 Department of Computer Science, 226 Atanasoff Hall Iowa State University, Ames, Iowa 50011-1040 USA dhara@cs.iastate.edu and leavens@cs.iastate.edu November 24, 1992 Abstract Subtype relationships in object-oriented programming languages are studied to aid code reuse and reasoning about programs that use subtype polymorphism. We define what it means...

A model theory for proving correctness of abstract data types is developed within the framework of the behavior-realization adjunction. To allow for incomplete specifications, proof-of-correctness ...

We propose several calculi for explicit type passing that enable us to formalize compilation of polymorphic programming languages like MLasphasesoftypepreserving translations. In our calculi various manipulations for type parameters can be expressed without typing problems|this is impossible in the polymorphic-calculi. Furthermore, we develop the translation from an explicit typed source calculus similar to Core-XML to one of the proposed calculi which completely eliminates runtime construction type parameters. We proposeanintermediate language based on this calculus, and discuss an implementation of a compiler for Core Standard ML. 1.

ion: [[H B x : u: M 0 : u ! v]]ae is the function from [[u]] to [[v]] given by d 7! [[H; x : u B M 0 : v]](ae[x 7! d]), that is, the function f defined by f(d) = [[H; x : u B M 0 : v]](ae[x 7! d]): 22 Carl A. Gunter ffl Application: [[H B L(N ) : t]]ae is the value obtained by applying the function [[H B L : s ! t]]ae to argument [[H B N : s]]ae where s is the unique type such that H ` L : s ! t and H ` N : s. It will save us quite a bit of ink to drop the parentheses that appear as part of expressions such as [[H; x : u B M 0 : v]](ae[x 7! d]) and simply write [[H; x : u B M 0 : v]]ae[x 7! d]. Doing so appears to violate the convention of associating applications to the left, but there is little chance of confusion in the case of expressions such as these. Hence, we will adopt the convention that the postfix update operator binds more tightly than general application. It can be shown that this assignment of meanings respects our equational rules. This is the soundness ...

Abstract. The correspondence between natural deduction proofs and λ-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) λ-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed λ-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors

. The correspondence between natural deduction proofs and -terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) -terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed -calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors !, \Theta, +, 8, 9, and ? (falsity) (with or without j-like rules). This research was partially supported by ONR Grant NOOO14-88-K-0593. Contents 1 Introduction 3 2 Natural Deduction, Simply-Typed -Calculus 5 3 Adding Conjunction, Negation, and Disjunction 11 4 First-Order Quantifiers 14 5 P-Candidates for the Arrow Type Constructor ! 18 6 Adding Product and Sum Types \Theta and + 23 7 Adding the Absurdity Type ? 28 8 Adding First-Order Quantifiers 8 and 9 35 9 Adding j-like Reduction Rules 52 1 Introductio...

. The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this...

Recursive functions are representable as lambda terms, and de nability in the calculus may be regarded as a de nition of computability. This forms part of the standard foundations of computer science. Lambda calculus is the commonly accepted basis of functional programming languages � and it is folklore that the calculus is the prototypical functional language in puri ed form. The course investigates the syntax and semantics of lambda calculus both as a theory of functions from a foundational point of view, and as a minimal programming language. Synopsis Formal theory, xed point theorems, combinatory logic: combinatory completeness, translations between lambda calculus and combinatory logic � reduction: Church-Rosser theorem � Bohm's theorem and applications � basic recursion theory � lambda calculi considered as programming languages � simple type theory and pcf: correspondence between operational and denotational semantics � current developments. Relationship with other courses Basic knowledge of logic and computability in paper B1 is assumed.