# Higher Order Unification via Explicit Substitutions

@article{Dowek2000HigherOU, title={Higher Order Unification via Explicit Substitutions}, author={Gilles Dowek and Th{\'e}r{\`e}se Hardin and Claude Kirchner}, journal={Inf. Comput.}, year={2000}, volume={157}, pages={183-235} }

Higher order unification is equational unification for βη-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus, the methods for equational unification (such as narrowing) built upon grafting (i.e., substitution without renaming) cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory. This is achieved by…

## 24 Citations

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The unification procedure was developed for the calculi with ES that belong to the paradigm known as “act at a distance”, and explicit substitutions are not propagated to the level of variables, as usual.

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A restricted version of Dougherty's algorithm that is incomplete, terminating and does not require polymorphism is presented, including a novel use of a substitution tree as a filtering index for higher-order unification.

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A characterization of the class of higher-order rewriting systems which can be encoded by first- order rewriting modulo an empty theory (that is, Ɛ = θ), which includes of course the λ-calculus.

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The undecidability of the expansion problem for the λσ calculus is proved by using a version of Rice's theorem, which is more straightforward and general than the one based on Scott’s theorem.

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The main technical novelty of this work is that it is compatible with the axiom-of-choice (unlike earlier nominal logic work by Pitts et al); thus it was able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for Church-Rosser and strong-normalisation.

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This work uses Martelli-Montanari style multi-equation reduction to generate name management problems from arbitrary unification terms that preserve names while using special representations of de Bruijn numbers to enable efficient name management.

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