On Solving Nominal Disunification Constraints

Abstract

This paper proposes an extension of first-order disunification problems by taking into account binding operators according to the nominal approach. In this approach, bindings are implemented through atom abstraction, and renaming of atoms is implemented via atom permutations. In the nominal setting, unification problems consist of equational questions ($s{\approx }_{\alpha }^{?}t$) considered under freshness constraints ($a{\#}^{?}t$) that restrict solutions by forbidding free occurrences of atoms in the instantiation of variables. In addition to equational and freshness constraints, nominal disunification problems include also nominal disunification constraints ($s{\not\approx }_{\alpha }^{?}t$), and their solutions consist of a substitution and additional freshness constraints such that under these constraints the instantiation of the equations, disequations and freshness constraints with the substitution hold. By re-using nominal unification techniques, this paper shows how to decide whether two nominal terms can be made different modulo α-equivalence. This is done by extending previous results on first-order disunification, and defining the notion of solutions with exceptions in the nominal syntax. A discussion on the semantics of disunification constraints is also given.