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Nominal terms extend first-order terms with binding. They lack someproperties of first- and higher-order terms: Terms must be reasoned about in acontext of 'freshness assumptions'; it is not always possible to 'choose afresh variable symbol' for a nominal term; it is not always possible to'alpha-convert a bound variable symbol' or to 'quotient by alpha-equivalence';the notion of unifier is not based just on substitution.  Permissive nominal terms closely resemble nominal terms but they recoverthese properties, and in particular the 'always fresh' and 'always rename'properties. In the permissive world, freshness contexts are elided, equality isfixed, and the notion of unifier is based on substitution alone rather than onnominal terms' notion of unification based on substitution plus extra freshnessconditions.  We prove that expressivity is not lost moving to the permissive case andprovide an injection of nominal terms unification problems and their solutionsinto permissive nominal terms problems and their solutions. We investigate therelation between permissive nominal unification and higher-order patternunification. We show how to translate permissive nominal unification problemsand solutions in a sound, complete, and optimal manner, in suitable senseswhich we make formal.

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