On the Logic of Unification

Unification, or solving equations on finite trees, is a P-complete problem central to symbolic manipulation, especially in Resolution, Type Inference and Rewriting. We present a natural logic dedicated to unification, which includes a constructive version of equational logic. This logic enjoys the classical proof-theoretic properties: atomicity; strong normalization; Church-Rosserness; left right, introduction elimination and positive negative symmetries. Motivated by the Type Inference problem, we introduce, besides a model-theoretic semantics and its completeness, a geometrical interpretation of deductions describing their operational content. This allows the design of a normalization process. This unification logic provides significant tools in investigations of higher-order unification, especially for the Type Inference problem, via fixed-point equations deducible from the given equations in unification logic. We also present some results on the classification problem of these fixed-point equations. To this end, we introduce the notion of elementary cyclic sets, that essentially possess a single associated fixed-point equation. The finite set of elementary cyclic sets embedded in some unification problem is obtained by a linearization process of the input equations. Finally, up to permutation, there exists a minimum equational deduction associated to an elementary cyclic set. We give a deterministic algorithm computing this deduction.