General Unifier Research Articles

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.

Embedding boolean expressions into logic programming

Boolean expressions play a major role in computer science. They can be used to represent sets, formulas of propositional logic and digital circuits. In this paper we present an algorithm to compute the most general unifier (mgu) of boolean expressions, which is unique, if it exists. The fact that there is always at most one mgu, i.e. the theory of boolean expressions is unitary, is of particular importance for the embedding of the data type ‘boolean expression’ into logic programming. First experiemes with an integration into a PROLOG system are shown.

The Regular Expression Descriptions of Unifier Set in the Typed λ-Calculus

λ-language over simple type structure is considered. We investigate unification problem such that number of elementary substitutions in a potentially infinite matching tree introduced by Huet is finite (compare [8]). The substitutions are generated using term grammar technique. Such a grammar consists of productions which are on the form; nonterminal variable of certain type produces a term of the same type. The most interesting is the case when matching tree is infinite and includes an infinite number of most general unifiers. We will show an interesting property of the infinite unification trees from examined class: for every infinite branch there is a node which occurs earlier in this tree. It means that there are some fragments which are repeated in this tree. Therefore the set of most general unifiers which are represented by terminal nodes can be described by regular expression over finite alphabet. Regular expression are constructed as a solution of linear language equalities. For every problem from this class the set of unifiers is described by some language. This language can be approximated by regular languages.

On unification: Equational theories are not bounded

We are interested in first-order unification problems and, more specifically, in the hierarchy of equational theories based on the cardinality of the set of most general unifiers. The following result is established in this paper: if T is a suitable first-order equational theory that is not unitary, then T is not bounded; that is, there is no integer n > 1 such that for every unification problem ( s = t ) T , the cardinality of the set of most general unifiers for ( s = t ) T is at most n . Hence, the class of (non-unitary) finitary theories cannot be decomposed into a hierarchy obtained by uniformly bounding the eardinalities of the sets of most general unifiers.

Properties of substitutions and unifications

In this paper the set of first order substitutions with a partial ordering “more general than” is investigated. It is proved that the set of equivalence classes of idempotent substitutions together with an added greatest element is a complete lattice. A simultaneous unification of finitely many finite sets of terms can be reduced to unifying each of the sets of terms separately and to building the supremum of the most general unifiers in this lattice afterwards. This saves time in an automatic proof procedure when combined with the concept of weak unification also introduced in this paper.

Deduction Plans: A Basis for Intelligent Backtracking

A proof procedure is described that relies on the construction of certain directed graphs called ``deduction plans.'' Plans represent the structure of proofs in such a way that problem reduction may be used without imposing any ordering on the solution of subproblems, as required by other systems. The structure also allows access to all clauses deduced in the course of a proof, which may then be used as lemmas. Economy of representation is the maximum attainable, consistent with this unrestricted availability of lemmas. Restricted versions of this deduction system correspond to existing linear deduction procedures, but do not suffer from some of their shortcomings, such as redundant representation, strict ordering of subproblems, and explicit substitution. One of the rules for constructing plans, however, allows a subproblem to be factored to a previously solved one: this has no equivalent in existing system. A further economy is obtained by making it unnecessary to perform substitutions and calculate most general unifiers. The source of unification failure can be located when a subproblem is found to be unsolvable, so that exact backtracking can be performed rather than the blind backtracking performed by existing systems. Therefore, a deduction system based on the construction of plans can avoid the wasteful search of irrelevant areas of the search space that results from the usual backtracking methods.

A unification algorithm for typed [formula omitted]-calculus

A semi-decision algorithm is presented, to search for unification of formulas in typed ω-order λ-calculus, and its correctness is proved. It is shown that the search space is significantly smaller than the one for finding the most general unifiers. In particular, our search is not redundant. This allows our algorithm to have good directionality and convergence properties.