KLAIM is an experimental programming language that supports a programming paradigm where both processes and data can be moved across di#erent computing environments. This paper presents the mathematical foundations of the KLAIM type system; this system permits checking access rights violations of mobile agents. Types are used to describe the intentions (read, write, execute, :::) of processes relative to the di#erent localities with which they are willing to interact, or to which they want to migrate. Type checking then determines whether processes comply with the declared intentions, and whether they have been assigned the necessary rights to perform the intended operations at the speci#ed localities. The KLAIM type system encompasses both subtyping and recursively de#ned types. The former occurs naturally when considering hierarchies of access rights, while the latter is needed to model migration of recursive processes. c 2000 Elsevier Science B.V. All rights reserved.

Many polyvariant program analyses have been studied in the 1990s, including k-CFA, polymorphic splitting, and the cartesian product algorithm. The idea of polyvariance is to analyze functions more than once and thereby obtain better precision for each call site. In this paper we present an equivalence theorem which relates a co-inductively defined family of polyvariant ow analyses and a standard type system. The proof embodies a way of understanding polyvariant flow information in terms of union and intersection types, and, conversely, a way of understanding union and intersection types in terms of polyvariant flow information. We use the theorem as basis for a new flow-type system in the spirit of the CIL -calculus of Wells, Dimock, Muller, and Turbak, in which types are annotated with flow information. A flow-type system is useful as an interface between a owanalysis algorithm and a program optimizer. Derived systematically via our equivalence theorem, our flow-type system should be a g...

Over the last several years, much interesting work has been done in modelling object-oriented programming languages in terms of extensions of the bounded second-order lambda calculus, F . Unfortunately, it has recently been shown by Pierce ([Pie92]) that type checking F is undecidable. Moreover, he showed that the undecidability arises in the seemingly simpler problem of determining whether one type is a subtype of another. In [Bru93a, Bru93b], the first author introduced a statically-typed, functional, object-oriented programming language, TOOPL, which supports classes, objects, methods, instance variables, subtypes, and inheritance. The semantics of TOOPL is based on F , so the question arises whether type checking in this language is decidable. In this paper we show that type checking for TOOPLE, a minor variant of TOOPL (Typed Object-Oriented Programming Language), is decidable. The proof proceeds by showing that subtyping is decidable, that all terms of TOOPLE have minimum types...

Algorithms for checking subtyping between recursive types lie at the core of many programming language implementations. But the fundamental theory of these algorithms and how they relate to simpler declarative specifications is not widely understood, due in part to the difficulty of the available introductions to the area. This tutorial paper offers an "end-to-end" introduction to recursive types and subtyping algorithms, from basic theory to efficient implementation, set in the unifying mathematical framework of coinduction. 1. INTRODUCTION Recursively defined types in programming languages and lambda-calculi come in two distinct varieties. Consider, for example, the type X described by the equation X = Nat!(Nat\ThetaX): An element of X is a function that maps a number to a pair consisting of a number and a function of the same form. This type is often written more concisely as X.Nat!(Nat\ThetaX). A variety of familiar recursive types such as lists and trees can be defined analogou...

A subtyping 0 is entailed by a set of subtyping constraints C, written C j= 0 , if every valuation (mapping of type variables to ground types) that satisfies C also satisfies 0 . We study the complexity of subtype entailment for simple types over lattices of base types. We show that: ffl deciding C j= 0 is coNP-complete. ffl deciding C j= ff fi for consistent, atomic C and ff; fi atomic can be done in linear time. The structural lower (coNP-hardness) and upper (membership in coNP) bounds as well as the optimal algorithm for atomic entailment are new. The coNP-hardness result indicates that entailment is strictly harder than satisfiability, which is known to be in PTIME for lattices of base types. The proof of coNP-completeness gives an improved algorithm for deciding entailment and puts a precise complexitytheoretic marker on the intuitive "exponential explosion" in the algorithm. Central to our results is a novel characterization of C j= ff fi for atomic, co...

A constrained type consists of both a standard type and a constraint set. Such types enable efficient type inference for objectoriented languages with polymorphism and subtyping, as demonstrated by Eifrig, Smith, and Trifonov. Until now, it has been unclear how expressive constrained types are. In this paper we prove that for a language without polymorphism, constrained types accept the same programs as the type system of Amadio and Cardelli with subtyping and recursive types. This result gives a precise connection between constrained types and the standard notion of type. 1 Introduction A constrained type consists of both a standard type and a constraint set. For example, x:xx : (v ! w) n fv v ! wg Here, v and w are type variables. This typing says that the -term x:xx has every type of the form v ! w where v; w satisfy the constraint v v ! w. Jens Palsberg, Laboratory for Computer Science, Massachusetts Institute of Technology, NE43-340, 545 Technology Square, Cambridg...

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We show that the first-order theory of structural subtyping of non-recursive types is decidable. Let Σ be a language consisting of function symbols (representing type constructors) and C a decidable structure in the relational language L containing a binary relation ≤. C represents primitive types; ≤ represents a subtype ordering. We introduce the notion of Σ-term-power of C, which generalizes the structure arising in structural subtyping. The domain of the Σ-term-power of C is the set of Σ-terms over the set of elements of C. We show that the decidability of the first-order theory of C implies the decidability of the first-order theory of the Σterm-power of C. This result implies the decidability of the first-order theory of structural subtyping of non-recursive types.

. We study entailment of structural and nonstructural recursive subtyping constraints. Constraints are formal inequalities between type expressions, interpreted over an ordered set of possibly infinite labeled trees. The nonstructural ordering on trees is the one introduced by Amadio and Cardelli for subtyping with recursive types. The structural ordering compares only trees with common shape. A constraint set entails an inequality if every assignment of meanings (trees) to type expressions that satisfies all the constraints also satisfies the inequality. In this paper we prove that nonstructural subtype entailment is PSPACEhard, both for finite trees (simple types) and infinite trees (recursive types). For the structural ordering we prove that subtype entailment over infinite trees is PSPACE-complete, when the order on trees is generated from a lattice of type constants. Since structural subtype entailment over finite trees has been shown to be coNP-complete these are the first comple...

Recent work h& s h wn equivalences between various type systems and flow logics. Ideally, th translations upon wh= h such equivalences are basedshd&@ be faithful in th sense the information is not lost in round-trip translations from flows to types and back or from types to flows and back. Building on t h work of Nielson Nielson and of Palsberg Pavlopoulou, we present t h firstfaithT# translations between a class of finitary polyvariant flow analyses and a type system supporting polymorph@@ in th form of intersection and union types. Additionally, our flow/type correspondence solves several open problems posed by Palsberg Pavlopoulou: (1) it expresses call-string based polyvariance (such as k-CFA) as well as argument based polyvariance; (2) it enjoys a subject reduction property for flows as well as for types; and (3) it supports a flow-oriented perspectiverath# thh a type-oriented one. 1