We present a type theory for higher-order modules that accounts for many central issues in module system design, including translucency, applicativity, generativity, and modules as first-class values. Our type system harmonizes design elements from previous work, resulting in a simple, economical account of modular programming. The main unifying principle is the treatment of abstraction mechanisms as computational effects. Our language is the first to provide a complete and practical formalization of all of these critical issues in module system design.

Girard and Reynolds independently invented System F (a.k.a. the second-order polymorphically typed lambda calculus) to handle problems in logic and computer programming language design, respectively. Viewing F in the Curry style, which associates types with untyped lambda terms, raises the questions of typability and type checking . Typability asks for a term whether there exists some type it can be given. Type checking asks, for a particular term and type, whether the term can be given that type. The decidability of these problems has been settled for restrictions and extensions of F and related systems and complexity lower-bounds have been determined for typability in F, but this report is the rst to resolve whether these problems are decidable for System F. This report proves that type checking in F is undecidable, by a reduction from semiuni cation, and that typability in F is undecidable, by a reduction from type checking. Because there is an easy reduction from typability to typ...

Tree descriptions based on dominance constraints are popular in several areas of computational linguistics including syntax, semantics, and discourse. Tree descriptions in the language of context unification have attracted some interest in unification and rewriting theory. Recently, dominance constraints and context unification have both been used in different underspecified approaches to the semantics of scope, parallelism, and their interaction. This raises the question whether both description languages are related. In this paper, we show for a first time that dominance constraints can be expressed in context unification. We also prove that dominance constraints extended with parallelism constraints are equal in expressive power to context unification.

Abstract. Bounded Second-Order Unification is the problem of deciding, for a given second-order equation t? = u and a positive integer m, whether there exists a unifier σ such that, for every second-order variable F, the terms instantiated for F have at most m occurrences of every bound variable. It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-Order Unification is undecidable. We prove that Bounded Second-Order Unification is NP-complete, provided that m is given in unary encoding, by proving that a size-minimal solution can be represented in polynomial space, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time. 1

Abstract. The Curry form of a term, like f(a, b), allows us to write it, using just a single binary function symbol, as @(@(f,a),b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one binary symbol is enough. By currying variable applications, like X(a), as

It is shown that unifiability of terms in the simply typed lambda calculus with beta and eta rules becomes decidable if there is a bound on the number of bound variables and lambdas in a unifier in eta-expanded beta-normal form.

Explicit substitution calculi are extensions of the lambda-calculus where the substitution mechanism is internalized into the theory. This feature makes them suitable for implementation and theoretical study of logic based tools as strongly typed programming languages and proof assistant systems. In this paper we explore new developments on two of the most successful styles of explicit substitution calculi: the lambdasigma- and lambda_se-calculi.

A unication method based on the se -style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure -calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the -style of explicit substitution. Correctness and completeness properties of the proposed se-unication method are shown and its advantages, inherited from the qualities of the se -calculus, are pointed out. Our method needs only one sort of objects: terms. And in contrast to the HOU approach based on the -calculus, it avoids the use of substitution objects. This makes our method closer to the syntax of the -calculus. Furthermore, detection of redices depends on the search for solutions of simple arithmetic constraints which makes our method more operational than the one based on the -style of explicit substitution. Keywords: Higher order unication, explicit substitution, lambda-calculi. 1

A higher order unification (HOU) method based on the ...-style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...-style of explicit substitution. Correctness and completeness properties of the proposed approach are shown and advantages of the method, inherited from the qualities of the ... calculus, are pointed out.

We show that rigid reachability, the non-symmetric form of rigid E-unication, is already undecidable in the case of a single constraint. From this we infer the undecidability of a new and rather restricted kind of second-order unication. We also show that certain decidable subclasses of the problem which are P-complete in the equational case become EXPTIME-complete when symmetry is absent. By applying automatatheoretic methods, simultaneous monadic rigid reachability with ground rules is shown to be PSPACE-complete. Moreover, we identify two decidable non-monadic fragments that are complete for EXPTIME. 1. Introduction Rigid reachability is the problem, given a rewrite system R and two terms s and t, whether there exists a ground substitution such that s rewrites via R to t. The term \rigid" refers to the fact that for no rule more than one instance can be used in the rewriting process. Simultaneous rigid reachability (SRR) is the problem in which a substitution is soug...