On the sequential nature of unification
Journal of Logic Programming
Logic for computer science: foundations of automatic theorem proving
Logic for computer science: foundations of automatic theorem proving
New Classes for Parallel Complexity: A Study of Unification and Other Complete Problems for P
IEEE Transactions on Computers
Theoretical Computer Science
Information and Computation - Semantics of Data Types
A Machine-Oriented Logic Based on the Resolution Principle
Journal of the ACM (JACM)
An Efficient Unification Algorithm
ACM Transactions on Programming Languages and Systems (TOPLAS)
Information and Computation
Information and Computation
The Functional Interpretation of Direct Computations
Electronic Notes in Theoretical Computer Science (ENTCS)
Polymorphic type reconstruction using type equations
IFL'03 Proceedings of the 15th international conference on Implementation of Functional Languages
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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.