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Unification in Monoidal Theories is Solving Linear Equations over Semirings
 Intelligenz, DFKI GmbH, Stuhlsatzenhausweg 3, D66123 Saarbrucken
, 1992
"... Although unification algorithms have been developed for numerous equational theories there is still a lack of general methods. In this paper we apply algebraic techniques to the study of a whole class of theories, which we call monoidal. Our approach leads to general results on the structure of unif ..."
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Although unification algorithms have been developed for numerous equational theories there is still a lack of general methods. In this paper we apply algebraic techniques to the study of a whole class of theories, which we call monoidal. Our approach leads to general results on the structure of unification algorithms and the unification type of such theories. An equational theory is monoidal if it contains a binary operation which is associative and commutative, an identity for the binary operation, and an arbitrary number of unary symbols which are homomorphisms for the binary operation and the identity. Monoidal theories axiomatize varieties of abelian monoids. Examples are the theories of abelian monoids (AC), idempotent abelian monoids (ACI), and abelian groups. To every monoidal theory we associate a semiring. Intuitively, semirings are rings without subtraction. We show that every unification problem in a monoidal theory can be translated into a system of linear equations over t...
Unification algebras: an axiomatic approach to unification, equation solving and constraint solving
 Universitat Kaiserslautern
, 1988
"... Abstract. Traditionally unification is viewed as solving an equation in an algebra given an explicit construction method for terms and substitutions. We abstract from this explicit term construction methods and give a set of axioms describing unification algebras that consist of objects and mappings ..."
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Abstract. Traditionally unification is viewed as solving an equation in an algebra given an explicit construction method for terms and substitutions. We abstract from this explicit term construction methods and give a set of axioms describing unification algebras that consist of objects and mappings, where objects abstract terms and mappings abstract substitutions. A unification problem in a given unification algebra is the problem to find mappings for a system of equations 〈si = tii ∈ I〉, where si and ti are objects, such that si and ti are mapped onto the same object. Typical instances of unification algebras and unification problems are: Term unification with respect to equational theories and sorts, standard equation solving in mathematics, unification in the λcalculus, constraint solving, disunification, and unification of rational terms. Within this framework we give general purpose unification rules that can be used in every unification algorithm in unification algebras. Furthermore we demonstrate the use of this framework by investigating the analogue of syntactic unification and unification of rational terms.