Unification Of Terms Research Articles

Contraction algebras and unification of (infinite) terms

We introduce the class of contraction algebras in which the algebra of infinite and finite terms is free over the set of variables. We develop a general theory of systems of equations at the level of categories in close connection with the Banach Principle of Contraction. Two applications of this theory are given. The first is the case of regular systems of equations with arbitrary terms. The second is the case of systems of equations attached to a context-free grammar (sometimes called ALGOL-like systems). Systems of equations with terms are used to extend the original unification algorithm of Robinson to the case of infinite terms.

The undecidability of the second order predicate unification problem

We prove that the second order predicate unification problem is undecidable by reducing the second order term unification problem to it.

On the relationship of congruence closureand unification

Congruence closure is a fundamental operation for symbolic computation. Unification closureis defined as its directional dual, i.e., on the same inputs but top-down as opposed to bottom-up. Unifying terms is another fundamental operation for symbolic computation and is commonly computed using unification closure. We clarify the directional duality by reducing unification closure to a special form of congruence closure. This reduction reveals a correspondence between repeated variables in terms to be unified and equalities of monadic ground terms. We then show that: (1) single equality congruence closure on a directed acyclic graph, and (2) acyclic congruence closure of a fixed number of equalities, are in the parallel complexity class NC. The directional dual unification closures in these two cases are known to be log-space complete for PTIME. As a consequence of our reductions we show that if the number of repeated variables in the input terms is fixed, then term unification can be performed in NC; this extends the known parallelizable cases of term unification. Using parallel complexity we also clarify the relationship of unification closure and the testing of deterministic finite automata for equivalence.

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.

Basic narrowing revisited

In this paper we study basic narrowing as a method for solving equations in theinitial algebra specified by a ground confluent and terminating term rewriting system. Since we are interested in equation solving, we don't study basic narrowing as a reduction relation on terms but consider immediately its reformulation as an equation solving rule. This reformulation leads to a technically simpler presentation and reveals that the essence of basic narrowing can be captured without recourse to term unification. We present an equation solving calculus that features three classes of rules. Resolution rules, whose application is don't know nondeterministic, are the basic rules and suffice for a complete solution procedure. Failure rules detect inconsistent parts of the search space. Simplification rules, whose application is don't care nondeterministic, enhance the power of the failure rules and reduce the number of necessary don't know steps. Three of the presented simplification rules are new. The rewriting ruleallows for don't care nondeterministic rewriting and thus yields a marriage of basic and normalizing narrowing. The safe blocking rule is specific to basic narrowing and is particularly useful in conjunction with the rewriting rule. Finally, the unfolding rule allows for a variety of search strategies that reduce the number of don't know alternatives that need to be explored.

Delaying unification algorithms for lambda calculi

Huet proposed delaying some unifications of typed lambda terms, as part of a higher-order resolution method. This paper abstracts that idea from the resolution context. Unification delays are formalised independently of any context, by defining a lambda calculus where a unification constraint forms an integral part of each term. This calculus is shown to support an unusually simple unification theory, where most general unifiers trivially always exist. Attention shifts from the existence of unifiers to the simplification of expressions for most general unifiers. The approach is convenient for discussing the unification of untyped lambda terms and has promise for discussing schematic unification of term schemes. It may also be a convenient format for unification algorithms in rewriting systems which use lazy evaluation to compute with notations for infinite structures.

On the sequential nature of unification

The problem of unification of terms is log-space complete for P. In deriving this lower bound no use is made of the potentially concise representation of terms by directed acyclic graphs. In addition, the problem remains complete even if infinite substitutions are allowed. A consequence of this result is that parallelism cannot significantly improve on the best sequential solutions for unification. However, we show that for the problem of term matching, an important subcase of unification, there is a good parallel algorithm using O( log 2 n) time and n O(1) processors on a PRAM. For the O( log 2 n) parallel time upper bound we assume that the terms are represented by directed acyclic graphs; if the longer string representation is used we obtain an O( log n) parallel time bound.

Zur terminologischen arbeit für die theorie der maschinen und mechanismen (TMM)

This paper deals with the methodological problems associated with terminology activities in the field of TMM, conducted by ITFoMM-Commission A “Standards for Terminology” and national groups. It is outlined that unification of terms on the international level becomes more and more important not only for teaching and communication but also for research work associated with the development of new fields, especially those that are disciplinary in nature (for instance robotics, automatic controls, and computer aided design). The basic problems of classification, articulation and definition of terms are solved using certain rules which are explained and applied to several examples. Future work should pay more attention to stricter consideration of logic, to the growing complexity of systems of motion and to the increased need for co-operation between national and international standardization groups.

Proposals for unification of principal terms in research on atmospheric ozone

Proposals for unification of principal terms in research on atmospheric ozone

The Physical Status of “Space”

IT does not appear from Dr. Jeffreys's letter in NATURE of May 26, p. 394, that we are at variance about anything really vital. What I do contend is this: that, thanks to the searching character of the theory of relativity, the time has come when it is profitable to attempt a much-needed unification of fundamental terms and conceptions, particularly in face of the curious indifference to such matters shown by some of those physicists who, with consummate skill, have developed the differential equations representing the natural forces. As the space of NATURE is limited, may I briefly, in a series of categories, amplify my previous letters (April 7 and 21 and May 5), stating the case for the extension theory suggestively, but in no way dogmatically?