Preprints of Proceedings of GWAI-92

This is a preprint of the proceedings of the German Workshop on Artificial Intelligence (GWAI) 1992. The final version will appear in the Lecture Notes in Artificial Intelligence

Multi-level Contextual Type Theory

Contextual type theory distinguishes between bound variables and meta-variables to write potentially incomplete terms in the presence of binders. It has found good use as a framework for concise explanations of higher-order unification, characterize holes in proofs, and in developing a foundation for programming with higher-order abstract syntax, as embodied by the programming and reasoning environment Beluga. However, to reason about these applications, we need to introduce meta^2-variables to characterize the dependency on meta-variables and bound variables. In other words, we must go beyond a two-level system granting only bound variables and meta-variables. In this paper we generalize contextual type theory to n levels for arbitrary n, so as to obtain a formal system offering bound variables, meta-variables and so on all the way to meta^n-variables. We obtain a uniform account by collapsing all these different kinds of variables into a single notion of variabe indexed by some level k. We give a decidable bi-directional type system which characterizes beta-eta-normal forms together with a generalized substitution operation.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

A resolution principle for clauses with constraints

We introduce a general scheme for handling clauses whose variables are constrained by an underlying constraint theory. In general, constraints can be seen as quantifier restrictions as they filter out the values that can be assigned to the variables of a clause (or an arbitrary formulae with restricted universal or existential quantifier) in any of the models of the constraint theory. We present a resolution principle for clauses with constraints, where unification is replaced by testing constraints for satisfiability over the constraint theory. We show that this constrained resolution is sound and complete in that a set of clauses with constraints is unsatisfiable over the constraint theory if we can deduce a constrained empty clause for each model of the constraint theory, such that the empty clauses constraint is satisfiable in that model. We show also that we cannot require a better result in general, but we discuss certain tractable cases, where we need at most finitely many such empty clauses or even better only one of them as it is known in classical resolution, sorted resolution or resolution with theory unification

Polymorphic Rewriting Conserves Algebraic Strong Normalization and Confluence

We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants. We show that if a many-sorted algebraic rewrite system R is strongly normalizing (terminating, noetherian), then R + β + η + type-β + type-η rewriting of mixed terms is also strongly normalizing. We obtain this results using a technique which generalizes Girard\u27s candidats de reductibilité , introduced in the original proof of strong normalization for the polymorphic lambda calculus. We also show that if a many-sorted algebraic rewrite system R has the Church-Rosser property (is confluent), then R + β + type-β + type-η rewriting of mixed terms has the Church- Rosser property too. Combining the two results, we conclude that if R is canonical (complete) on algebraic terms, then R + β + type-β + type-η is canonical on mixed terms. η reduction does not commute with a1gebraic reduction, in general. However, using long β- normal forms, we show that if R is canonical then R + β + η + type-β + type-η convertibility is still decidable

Nominal Logic Programming

Nominal logic is an extension of first-order logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, alpha-equivalence). This article investigates logic programming based on nominal logic. We describe some typical nominal logic programs, and develop the model-theoretic, proof-theoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via examples.Comment: 46 pages; 19 page appendix; 13 figures. Revised journal submission as of July 23, 200

Inheritance hierarchies: Semantics and unification

Inheritance hierarchies are introduced as a means of representing taxonomicallyorganized data. The hierarchies are built up from so-called feature types that are ordered by subtyping and whose elements are records. Every feature type comes with a set of features prescribing fields of its record elements. So-called feature terms are available to denote subsets of feature types. Feature unification is introduced as an operation that decides whether two feature terms have a nonempty intersection and computes a feature term denoting the intersection.We model our inheritance hierarchies as algebraic specifications in ordersortedequational logic using initial algebra semantics. Our framework integrates feature types whose elements are obtained as records with constructor types whose elements are obtained by constructor application. Unification in these hierarchies combines record unification with order-sorted term unification and is presented as constraint solving. We specify a unitary unification algorithm by a set of simplification rules and prove its soundness and completeness with respect to the model-theoretic semantics

From LCF to Isabelle/HOL

Interactive theorem provers have developed dramatically over the past four decades, from primitive beginnings to today's powerful systems. Here, we focus on Isabelle/HOL and its distinctive strengths. They include automatic proof search, borrowing techniques from the world of first order theorem proving, but also the automatic search for counterexamples. They include a highly readable structured language of proofs and a unique interactive development environment for editing live proof documents. Everything rests on the foundation conceived by Robin Milner for Edinburgh LCF: a proof kernel, using abstract types to ensure soundness and eliminate the need to store proofs. Compared with the research prototypes of the 1970s, Isabelle is a practical and versatile tool. It is used by system designers, mathematicians and many others

Order-Sorted Equational Computation

The expressive power of many-sorted equational logic can be greatly enhanced by allowing for subsorts and multiple function declarations. In this paper we study some computational aspects of such a logic. We start with a self-contained introduction to order-sorted equational logic including initial algebra semantics and deduction rules. We then present a theory of order-sorted term rewriting and show that the key results for unsorted rewriting extend to sort decreasing rewriting. We continue with a review of order-sorted unification and prove the basic results. In the second part of the paper we study hierarchical order-sorted specifications with strict partial functions. We define the appropriate homomorphisms for strict algebras and show that every strict algebra is base isomorphic to a strict algebra with at most one error element. For strict specifications, we show that their categories of strict algebras have initial objects. We validate our approach to partial functions by proving that completely defined total functions can be defined as partial without changing the initial algebra semantics. Finally, we provide decidable sufficient criteria for the consistency and strictness of ground confluent rewriting systems