The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed -calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Lof's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools such as proof editors and proof checkers can be constructed.

this paper and the LF. In particular the idea of having an operator T : Prop ! Type appears already in De Bruijn's earlier work, as does the idea of having several judgements. The paper [24] describes the basic features of the LF. In this paper we are going to provide a broader illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal presentation) of a system carried out through encoding often illuminates the system itself. This paper will also deal with this phenomenon.

We describe Elf, a metalanguage for proof manipulation environments that are independent of any particular logical system. Elf is intended for meta-programs such as theorem provers, proof transformers, or type inference programs for programming languages with complex type systems. Elf unifies logic definition (in the style of LF, the Edinburgh Logical Framework) with logic programming (in the style of Prolog). It achieves this unification by giving types an operational interpretation, much the same way that Prolog gives certain formulas (Horn-clauses) an operational interpretation. Novel features of Elf include: (1) the Elf search process automatically constructs terms that can represent object-logic proofs, and thus a program need not construct them explicitly, (2) the partial correctness of meta-programs with respect to a given logic can be expressed and proved in Elf itself, and (3) Elf exploits Elliott's unification algorithm for a -calculus with dependent types. This research was...

A formal methodology is presented for the systematic evolution of modular Standard ML programs from specifications by means of verified refinement steps, in the framework of the Extended ML specification language. Program development proceeds via a sequence of design (modular decomposition), coding and refinement steps. For each of these three kinds of steps, conditions are given which ensure the correctness of the result. These conditions seem to be as weak as possible under the constraint of being expressible as "local" interface matching requirements. Interfaces are only required to match up to behavioural equivalence, which is seen as vital to the use of data abstraction in program development. Copyright c fl 1989 by D. Sannella and A. Tarlecki. All rights reserved. An extended abstract of this paper will appear in Proc. Colloq. on Current Issues in Programming Languages, Joint Conf. on Theory and Practice of Software Development (TAPSOFT), Barcelona, Springer LNCS (1989)....

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. Set constraints are inclusion relations between expressions denoting sets of ground terms over a ranked alphabet. They are the main ingredient in set-based program analysis. In this paper we provide a Gentzen-style axiomatization for sequents \Phi ` \Psi , where \Phi and \Psi are finite sets of set constraints, based on the axioms of termset algebra. Sequents of the restricted form \Phi ` ? correspond to positive set constraints, and those of the more general form \Phi ` \Psi correspond to systems of mixed positive and negative set constraints. We show that the deductive system is (i) complete for the restricted sequents \Phi ` ? over standard models, (ii) incomplete for general sequents \Phi ` \Psi over standard models, but (iii) complete for general sequents over set-theoretic termset algebras. 1 Introduction Set constraints are inclusions between expressions denoting sets of ground terms. They have been used extensively in program analysis and type inference for many years [AM91...

. Attribute grammars are a valuable tool for constructing compilers and building user interfaces. This paper reports on a system we are developing, called AML (for Attribution in ML), which is an attribute grammar toolkit for building such applications as language-based programming environments using SML. This system builds on the proven technology of efficient attribute evaluation, while using a higher-level foundation for the implementation of interactive systems. It supports a general and uniform platform for building applications that can manipulate attributed terms and allow access to attribute values. We describe the design of the AML system, its current implementation status, and our plans for the future. CR Classification: D.1.1, D.2.6, D.3.4 Key words: attribute grammars, attribute evaluation, functional programming, program generator, programming environments 1. Introduction Attribute grammars provide a formalism for assigning meaning to parse trees of a context-free languag...