The unification problem in first-order predicate calculus is described in general terms as the solution of a system of equations, and a nondeterministic algorithm is given. A new unification algorithm, characterized by having the acyclicity test efficiently embedded into it, is derived from the nondeterministic one, and a PASCAL implementation is given. A comparison with other well-known unification algorithms shows that the algorithm described here performs well in all cases.

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The unification problem and several variants are presented. Various algorithms and data structures are discussed. Research on unification arising in several areas of computer science is surveyed, these areas include theorem proving, logic programming, and natural language processing. Sections of the paper include examples that highlight particular uses

... unification problems. Then, in this framework, we develop a new unification algorithm for a-calculus with dependent function (II) types. This algorithm is especially useful as it provides for mechanization in the very expressive Logical Framework (LF). The development (object-languages). The rich structure of a typed-calculus,asopposedtotraditional,rst- generalideaistousea-calculusasameta-languageforrepresentingvariousotherlanguages thelattercase,thealgorithmisincomplete,thoughstillquiteusefulinpractice. Thelastpartofthedissertationprovidesexamplesoftheusefulnessofthealgorithms.The algorithmrstfordependentproduct()types,andsecondforimplicitpolymorphism.In involvessignicantcomplicationsnotarisingHuet'scorrespondingalgorithmforthesimply orderabstractsyntaxtrees,allowsustoexpressrules,e.g.,programtransformationand typed-calculus,primarilybecauseitmustdealwithill-typedterms.Wethenextendthis Wecanthenuseunicationinthemeta-languagetomechanizeapplicationoftheserules.

This paper presents a connectionist unification strategy for a Prolog-system. The unification strategy utilizes a backprop-net (bp-net) and a distributed connectionist representation. The most-generalunifier is computed by the connectionist heuristic in constant time. To keep the system also efficient in space the distributed representation was choosen. It is possible to show that this representation is assoziative. Moreover this connectionist strategy allows to use semantics and context during the unification. The network is able to learn context sensitive strategies and unification with extensions like E-unification (unification under a certain equational-theory E). So unification by a network becomes not only very cheap cause of the constant propagation-time of the bp-net but it gives also the ability to interpret its output as a mechanism for uncertainty. 1 Introduction Unification in general is a technique which utilizes substitutions of variables with adequate terms to make term...

ion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 80 III A complete example 83 12 ASL: A Small Language 85 12.1 ASL abstract syntax trees : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 85 12.2 Parsing ASL programs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 13 Untyped semantics of ASL programs 91 13.1 Semantic values : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91 13.2 Semantic functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 92 13.3 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93 14 Encoding recursion 95 14.1 Fixpoint combinators : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 95 14.2 Recursion as a primitive construct : : : : : : : : : : : : : : : : : : : : : : : : : : : : CONTENTS 3 15 Static typing, polymorphism and type synthesis 97 15.1 The type syst...

A partial instantiation approach to the solution of the satisfiability problem in the Schoenfinkel-Bernays fragment of 1 st order logic is presented. It is based on a reduction of the problem to a finite sequence of satisfiability problems in the propositional logic and it improves upon the original idea of partial instantiation, as proposed by Jeroslow. In the second part of the paper a new interpretation of the partial instantiation approach in terms of Directed Hypergraphs is proposed and a particular implementation for the Datalog case is described in detail. 1 Introduction. The problem of Logical Inference plays a fundamental role in Decision Sciences and has several applications in fields such as decision support systems, logic circuit design, data bases, and programming languages. Although classical approaches to formalize and solve inference problems have been of symbolic nature, in the last few years many scientists in the Operations Research community have studied ...

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We introduce an connectionist approach to unification using a local and a distributed representation. A Prolog-System using these unification-strategies has been build. Prolog is a Logic Programming Language which utilizes unification. We introduce a uncertainty measurement in unification. This measurement is based on the structure-abilities of the chosen representations. The strategy using a local representation, called `-CUP, utilizes a self-organizing feature-map (FM-net) to determine similarities between terms and induces the representation for a relaxation-network (relax-net). The strategy using a distributed representation, called d-CUP, embeds a similarity measurement by its recurrent representation. It has the advantage that similar terms have a similar representation. The unification itself is done by a backpropagation network (BP-net). We have proven the systems adequacy for unification, its efficient computation, and the ability to do extended unification. 1 Introduction Un...

this paper indicates an unexpected difference between the two relevant properties of sets of equations. As was shown by Paterson and Wegman [6] the property of being unifiable can be tested in linear time. We prove that the property of being STO is NP-hard. Recall that a problem is NP-hard, if its solvability in polynomial time implies that every problem in the class NP is solvable in polynomial time. This shows that, for all practical purposes, the definition of standard Prolog refers to a computationally intractable concept. So the "STO test" apparently cannot be efficiently implemented. A possible remedy could be to identify a more limited property than that of being STO, which could be efficiently implemented and which would still be able to capture the original intention that for a set of equations satisfying the property omission of the occur-check might