A meta programming language must be able to represent and manipulate such syntactic structures as programs, formulas, types, and proofs. A common characteristic of all these structures is that they involve notions of abstractions, scope, bound and free variables, substitution instances, and equality up to alphabetic changes of bound variables.

Unification problems are identified with conjunctions of equations between simply typed λ-terms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are presented. In this setting where variables of functional type can be quantified and not all types contain closed terms, the naive generalization of firstorder Skolemization has several technical problems that are addressed. The method of searching for pre-unifiers described by Huet is easily extended to the mixed prefix setting, although solving flexible-flexible unification problems is undecidable since types may be empty. Unification problems may have numerous incomparable unifiers. Occasionally, unifiers share common factors and several of these are presented. Various optimizations on the general unification search problem are as discussed. 1.

Soft type systems provide the benefits of static type checking for dynamically typed languages without rejecting untypable programs. A soft type checker infers types for variables and expressions and inserts explicit run-time checks to transform untypable programs to typable form. We describe a practical soft type system for R4RS Scheme. Our type checker uses a representation for types that is expressive, easy to interpret, and supports efficient type inference. Soft Scheme supports all of R4RS Scheme, including procedures of fixed and variable arity, assignment, continuations, and top-level definitions. Our implementation is available by anonymous FTP. The first author was supported in part by the United States Department of Defense under a National Defense Science and Engineering Graduate Fellowship. y The second author was supported by NSF grant CCR-9122518 and the Texas Advanced Technology Program under grant 003604-014. 1 Introduction Dynamically typed languages like Scheme...

We present a new method for characterizing the interpretive possibilities generated by elliptical constructions in natural language. Unlike previous analyses, which postulate ambiguity of interpretation or derivation in the full clause source of the ellipsis, our analysis requires no such hidden ambiguity. Further, the analysis follows relatively directly from an abstract statement of the ellipsis interpretation problem. It predicts correctly a wide range of interactions between ellipsis and other semantic phenomena such as quantifier scope and bound anaphora. Finally, although the analysis itself is stated nonprocedurally, it admits of a direct computational method for generating interpretations. This article is available through the Computation and Language E-Print Archive as cmp-lg/9503008, and also appears in Linguistics and Philosophy 14(4):399–452. cmp-lg/9503008 Ellipsis and Higher-Order Unification 1

Functional logic languages with a sound and complete operational semantics are mainly based on narrowing. Due to the huge search space of simple narrowing, steadily improved narrowing strategies have been developed in the past. Needed narrowing is currently the best narrowing strategy for first-order functional logic programs due to its optimality properties w.r.t. the length of derivations and the number of computed solutions. In this paper, we extend the needed narrowing strategy to higher-order functions and λ-terms as data structures. By the use of definitional trees, our strategy computes only incomparable solutions. Thus, it is the first calculus for higher-order functional logic programming which provides for such an optimality result. Since we allow higher-order logical variables denoting λ-terms, applications go beyond current functional and logic programming languages.

We consider the problem of mechanically constructing abstract machines from operational semantics, producing intermediate-level specifications of evaluators guaranteed to be correct with respect to the operational semantics. We construct these machines by repeatedly applying correctness-preserving transformations to operational semantics until the resulting specifications have the form of abstract machines. Though not automatable in general, this approach to constructing machine implementations can be mechanized, providing machine-verified correctness proofs. As examples we present the transformation of specifications for both call-by-name and call-by-value evaluation of the untyped λ-calculus into abstract machines that implement such evaluation strategies. We also present extensions to the call-by-value machine for a language containing constructs for recursion, conditionals, concrete data types, and built-in functions. In all cases, the correctness of the derived abstract machines follows from the (generally transparent) correctness of the initial operational semantic specification and the correctness of the transformations applied. 1.

Abstract. We present five axioms of name-carrying lambda-terms identified up to alpha-conversion—that is, up to renaming of bound variables. We assume constructors for constants, variables, application and lambdaabstraction. Other constants represent a function Fv that returns the set of free variables in a term and a function that substitutes a term for a variable free in another term. Our axioms are (1) equations relating Fv and each constructor, (2) equations relating substitution and each constructor, (3) alpha-conversion itself, (4) unique existence of functions on lambda-terms defined by structural iteration, and (5) construction of lambda-abstractions given certain functions from variables to terms. By building a model from de Bruijn’s nameless lambda-terms, we show that our five axioms are a conservative extension of HOL. Theorems provable from the axioms include distinctness, injectivity and an exhaustion principle for the constructors, principles of structural induction and primitive recursion on lambda-terms, Hindley and Seldin’s substitution lemmas and

In this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a bi-rewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→

Many computer programs have the property that they work correctly on a variety of types of input; such programs are called polymorphic. Polymorphic type systems support polymorphism by allowing programs to be given multiple types. In this way, programs are permitted greater flexibility of use, while still receiving the benefits of strong typing. One especially successful polymorphic type system is the system of Hindley, Milner, and Damas, which is used in the programming language ML. This type system allows programs to be given universally quantified types as a means of expressing polymorphism. It has two especially nice properties. First, every well-typed program has a “best ” type, called the principal type, that captures all the possible types of the program. Second, principal types can be inferred, allowing programs to be written without type declarations. However, two useful kinds of polymorphism cannot be expressed in this type system: overloading and subtyping. Overloading is the kind of polymorphism exhibited by a function like addition, whose types cannot be captured by a single universally quantified type formula.

this paper we present a non-standard logic for our structures. It is a type-free intensional logic, and is also in the tradition of Curry's illative logic [HS86]; see also [AczN, FM87, Smi84, MA88]. The logic has two judgments: that an object is a fact and that an object is a state-of-a#airs (cf. truth and proposition). Objects are given using a variant of the traditional situation theory notation which is more standard, logically speaking, with explicit negation and quantification (see also [Bar87]). No metalinguistic apparatus is employed