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 ...

. We show how to reduce the unification problem modulo fij- conversion and a first-order equational theory E, into a first-order unification problem in a union of two non-disjoint equational theories including E and a calculus of explicit substitutions. A rule-based unification procedure in this combined theory is described and may be viewed as an extension of the one initially designed by G. Dowek, T. Hardin and C. Kirchner for performing unification of simply typed -terms in a first-order setting via the oe-calculus of explicit substitutions. Additional rules are used to deal with the interaction between E and oe. 1 Introduction Unification modulo an equational theory plays an important role in automated deduction and in logic programming systems. For example, Prolog[NM88] is based on higher-order unification, ie. unification modulo the fij-conversion. In order to design more expressive higher-order logic programming systems enhanced with a first-order equational theory E,...

: We consider the following problem in proving observational congruences in functional languages: given a call-by-name language based on the simplytyped -calculus with algebraic operations axiomatized by algebraic equations E, is the set of observational congruences between terms exactly those provable from (fi), (j), and E? We find conditions for determining whether fijE-equational reasoning is complete for proving the observational congruences between such terms. We demonstrate the power and generality of the theorems by presenting a number of easy corollaries for particular algebras. 1 Introduction The (fi) and (j) axioms form the basis for proving equations in call-by-name functional languages. In these languages, (fi) and (j) yield sound program optimizations. For example, consider a version of the call-by-name language PCF [11, 15] which is described in Appendix A. Our version of PCF includes simplytyped -calculus, numerals 0; 1; 2; : : :, successor and predecessor, addition, ...

. Higher-order E-unification, i.e. the problem of finding substitutions that make two simply typed -terms equal modulo fi or fij- equivalence and a given equational theory, is undecidable. We propose to rigidify it, to get a resource-bounded decidable unification problem (with arbitrary high bounds), providing a complete higher-order E-unification procedure. The techniques are inspired from Gallier's rigid E-unification and from Dougherty and Johann's use of combinatory logic to solve higher-order E-unification problems. We improve their results by using general equational theories, and by defining optimizations such as higherorder rigid E-preunification, where flexible terms are used, gaining much efficiency, as in the non-equational case due to Huet. 1 Introduction Higher-order E-unification is the problem of finding complete sets of unifiers of two simply typed -terms modulo fi or fij-equivalence, and modulo an equational theory E . This problem has applications in higher-order a...

. Modular higher-order E-unification, as described in [6], produces numerous redundant solutions in many practical cases. We present a refined algorithm for finitary theories with a finite number of non-free constants that avoids most redundant solutions, analyse it theoretically, and give first experimental results. In comparison to [6], the description of the E-unification algorithm is enriched by a definition of the translation process between first and higher-order terms and an explicit handling of new variables. This explicit handling gives deeper insight into the reasons for redundant solutions and thus provides a method for their avoidance. In order to study the efficiency of this method and the performance of the modular approach in general, some benchmark examples are presented and an interpretation of their empirical evaluation is given. 1 Introduction There is considerable practical potential in studying higher-order E-unification. First, E-unification is a key algorithm f...

Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 Special Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Semantics of Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.1 Semantics of Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.2 Action Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.7 Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.1 Early Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.2 Recent Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . 55 4.7.3 The Common Framework Initiative. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Methodology 57 5.1 Development Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.1 Applica...

This techreport presents two approaches to primitive equality treatment in higher-order (HO) automated theorem proving: a calculus EP adapting traditional first-order (FO) paramodulation [RW69] , and a calculus ERUE adapting FO RUE-Resolution [Dig79] to HO logic (based on Church's simply typed -calculus). EP and ERUE extend the extensional HO resolution approach ER [BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both calculi employ new, positive extensionality rules analogously to the respective negative ones provided by ER that operate on unification constraints. As the extensionality rules have an intrinsic and unavoidable difference-reducing character the HO paramodulation approach looses its pure term-rewriting character. On the other hand examples demonstrate that the extensionality rules harmonise rather well with the difference-reducing HO RUE-resolution idea.

Abstract. We provide a complete system of transformation rules for semantic unification with respect to theories defined by convergent rewrite systems. We show that this standard unification procedure, with slight modifications, can be used to solve the satisfiability problem in combinatory logic with a convergent set of algebraic axioms R, thus resulting in a complete higher-order unification procedure for R. Furthermore, we use the system of transformation rules to provide a syntactic characterization for R which results in decidability of semantic unification. 1