Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms

Nominal terms extend first-order terms with binding. They lack some properties of first-and higher-order terms: Terms must be reasoned about in a context of 'freshness assumptions'; it is not always possible to 'choose a fresh variable symbol' for a nominal term; it is not always possible to 'α-convert a bound variable symbol' or to 'quotient by α-equivalence'; the notion of unifier is not based just on substitution.Permissive nominal terms closely resemble nominal terms but they recover these properties, and in particular the 'always fresh' and 'always rename' properties. In the permissive world, freshness contexts are elided, equality is fixed, and the notion of unifier is based on substitution alone rather than on nominal terms' notion of unification based on substitution plus extra freshness conditions.We prove that expressivity is not lost moving to the permissive case and provide an injection of nominal terms unification problems and their solutions into permissive nominal terms problems and their solutions.We investigate the relation between permissive nominal unification and higher-order pattern unification. We show how to translate permissive nominal unification problems and solutions in a sound, complete, and optimal manner, in suitable senses which we make formal.

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Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques

Dowek

Gabbay

Mulligan

2010

Logic Journal of the IGPL

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Quantifiers in logic and proof-search using permissive-nominal terms and sets

Gabbay

Wirth

2013

Journal of Logic and Computation

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Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms

Cited by 2 publications

References 29 publications

Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques

Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques

Quantifiers in logic and proof-search using permissive-nominal terms and sets

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