Abstract not found

not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a meta-logical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, and experimental results related to the areas of programming languages, type theory, and logics. Design: The meta-logical framework extends the logical framework LF [HHP93] by a meta-logic M + 2. This design is novel and unique since it allows higher-order encodings of deductive systems and induction principles to coexist. On the one hand, higher-order representation techniques lead to concise and direct encodings of programming languages and logic calculi. Inductive de nitions on the other hand allow the formalization of properties about deductive systems, such as the proof that an operational semantics preserves types or the proof that a logic is is a proof calculus whose proof terms are recursive functions that may be consistent.M +

Abstract. Higher-order encodings use functions provided by one language to represent variable binders of another. They lead to concise and elegant representations, which historically have been difficult to analyze and manipulate. In this paper we present the ∇-calculus, a calculus for defining general recursive functions over higher-order encodings. To avoid problems commonly associated with using the same function space for representations and computations, we separate one from the other. The simply-typed λ-calculus plays the role of the representation-level. The computationlevel contains not only the usual computational primitives but also an embedding of the representation-level. It distinguishes itself from similar systems by allowing recursion under representation-level λ-binders while permitting a natural style of programming which we believe scales to other logical frameworks. Sample programs include bracket abstraction, parallel reduction, and an evaluator for a simple language with first-class continuations. 1

this paper we describe a new, categorical approach to normalization in typed -

Term rewriting has proven to be an important technique in theorem proving. In this paper, we illustrate that rewrite systems and strategies for higher-order term rewriting, which includes the usual notion of first-order rewriting, can be naturally specified and implemented in a higher-order logic programming language. We adopt a notion of higher-order rewrite system which uses the simply typed -calculus as the language for expressing rules, with a restriction on the occurrences of free variables on the left hand sides of rules so that matching of terms with rewrite templates is decidable. The logic programming language contains an implementation of the simply-typed lambda calculus including fij- conversion and higher-order unification. In addition, universal quantification in queries and the bodies of clauses is permitted. For higher-order rewriting, we show how these operations implemented at the meta-level provide elegant mechanisms for the object-level operations of descending thro...

It is well known that confluence and strong normalization are preserved when combining left-linear algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual extensional rule for j, or recursion together with the usual contraction rule for surjective pairing. We show that confluence and normalization are modular properties for the combination of left-linear algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for j and surjective pairing. For that, we use a translation technique allowing to simulate expansions without expansion rules. We also show that confluence is maintained in a modular way when adding fixpoints. This result is also obtained by a simple translation technique allowing to simulate bounded recursion with fi reduction. 1 Introduction Confluence and strong normalization for the combination of lambda calculus and algebraic rewriting systems have...

. This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in [12]. This formalism combines the computational aspects of term rewriting and simply typed lambda calculus. The result is a proof technique for the termination of a HRS, similar to the proof technique "Termination by interpretation in a wellfounded monotone algebra", described in [8, 19]. The resulting technique is as follows: Choose a higher-order algebra with operations for each function symbol in the HRS, equipped with some well-founded partial ordering. The operations must be strictly monotonic in this ordering. This choice generates a model for the HRS. If the choice can be made in such a way that for each rule the interpretation of the left hand side is greater than the interpretation of the right hand side, then the HRS is terminating. At the end of the paper some applications of this technique are given, which show that this technique is natural and can easily be applied. 1 Introdu...

Consider Adam and Eve. Count generations starting from them. Supposing that there will always be people, then it's true that for any generation X, eventually there will be people belonging to the next generation X + 1. In this paper the same result is established for the class of higher order pattern rewriting systems. 1 Introduction Consider a set of structures and a set of transformations on them specifying how a structure may be transformed into another one. Suppose the transformations are of the following form: first a structure is decomposed into substructures, next some substructure is replaced by another one, and finally the substructures are composed into a structure again. (destroy) The parts of the initial structure eliminated in the course of the transformation (i.e. the parts of the replaced substructure as well as the parts eliminated in the initial decomposition) can be thought of as being destroyed . (create) The parts of the final structure introduced in the cou...

In this note we present a simple proof of a result of Toyama which states that the disjoint union of confluent term rewriting systems is confluent. 1985 Mathematics Subject Classification: 68Q50 1987 CR Categories: F.4.2 Key Words and Phrases: theory of computation, term rewriting systems, modularity, confluence Introduction The topic of modularity of properties of term rewriting systems has caught much attention recently. An introduction to this area can be found in Klop [6]. For an early survey one may consult Middeldorp [7]. Moreover, the topic has received a fruitful offspring in the study of the conservation of properties when adding algebraic rewrite rules to various (typed) lambda calculi, see e.g. Breazu-Tannen and Gallier [1, 2] and Jouannaud and Okada [5]. 5 Partially supported by ESPRIT Basic Research Action 3020, INTEGRATION. 6 Partially supported by ESPRIT Basic Research Action 3074, SEMAGRAPH. 7 Partially supported by grants from NWO, Vrije Universiteit Amsterdam...

Let E be a first-order equational theory. A translation of typed higher-order E-unification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modification of ordinary narrowing is shown to be a complete method for enumerating higher-order E-unifiers. In fact, we treat a more general problem, in which the types of terms contain type variables. 1 Introduction Investigation of the interaction between first-order and higher-order equational reasoning has emerged as an active line of research. The collective import of a recent series of papers, originating with [Bre88] and including (among others) [Bar90], [BG91a], [BG91b], [Dou92], [JO91] and [Oka89], is that when various typed -calculi are enriched by first-order equational theories, the validity problem is well-behaved, and furthermore that the respective computational approaches to ...