Nominal Syntax with Atom Substitutions

Abstract Nominal syntax is a generalisation of first-order syntax that includes names, a notion of name binding and an elegant axiomatisation of alpha-equivalence, based on nominal set theory. However, it does not take into account non-capturing atom substitution, which is not a primitive notion in nominal syntax. We consider an extension of nominal syntax with non-capturing atom substitutions and show that matching is decidable and finitary but unification is undecidable in general. The proof of undecidability of unification is obtained by reducing Hilbert's tenth problem to unification of extended nominal terms. We provide a general matching algorithm and characterise a class of problems for which matching is unitary, giving rise to expressive and efficient rewriting systems.