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Solving equations, also called unification, has made impressive progress during the past decade. Described as existentially quantified formulae over =, unification problems are usually transformed step by step until a solved form is reached from which a most general unifier can be obtained. Kirchner showed how to compute transformation rules for theories having a syntactic presentation, for which a proof of an arbitrary equation uses at most one top equality step. Comon generalized to the case where negation is added, yielding disequations. Both sets of rules may not necessarily terminate. Kirchner gave sufficient proof-theoretic conditions for checking syntacticness, and was improved upon by Rémy, and Nipkov. As an application, a complete type inference algorithm for polymorphic records in CAML was obtained by Rémy. Recently, Kirchner and Klay showed that all finitary unifying theories without collapse axioms were syntactic, a result generalized by Comon, Contejean and Jouannaud to arbitrary finitary unifying and finitary matching theories. In particular, the associative and commutative theory of one operator has a syntactic presentation made of seven axioms. This paper surveys these results which show an intriguing relationship between unification and syntacticness, and suggests some directions for future research.
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