By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atoms-abstraction. The results are of interest in their own right, (...) but also, we factor the mathematics so as to maximise the chances that it could be used off-the-shelf for other nominal reasoning systems too. Models with infinite support can be easier to work with, so it is useful to have a semi-automatic theorem to transfer results from classes of infinitely-supported nominal models to the more restricted class of models with finite support. In conclusion, we consider different permissive-nominal syntaxes and nominal models and discuss how they relate to the results proved here. (shrink)

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. (shrink)

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. (shrink)