Abstract
We define nominal equational problems of the form exists overline{W} forall overline{Y} : P, where P consists of conjunctions and disjunctions of equations sapprox _alpha t, freshness constraints a#t and their negations: s not approx _alpha t and , where a is an atom and s, t nominal terms. We give a general definition of solution and a set of simplification rules to compute solutions in the nominal ground term algebra. For the latter, we define notions of solved form from which solutions can be easily extracted and show that the simplification rules are sound, preserving, and complete. With a particular strategy for rule application, the simplification process terminates and thus specifies an algorithm to solve nominal equational problems. These results generalise previous results obtained by Comon and Lescanne for first-order languages to languages with binding operators. In particular, we show that the problem of deciding the validity of a first-order equational formula in a language with binding operators (i.e., validity modulo alpha -equality) is decidable.
Highlights
Nominal unification [23] is the problem of solving equations modulo α-equivalence
This paper introduces nominal equational problems (NEPs) and presents simplification rules to find solutions in the ground nominal algebra
If the NEP consists only of existentially quantified conjunctions of freshness and α-equality constraints, we obtain solved forms consisting of a substitution and a freshness context, as in the standard nominal unification algorithm [23]
Summary
Nominal unification [23] is the problem of solving equations modulo α-equivalence. A solution consists of a substitution and a freshness context ∇, i.e., a set of primitive constraints of the form a#X (read: “a is fresh for X”), which intuitively means that a cannot occur free in the instances of X. Nominal disunification is the problem of solving disequations i.e., negated equations of the form s ≈α t. If the NEP consists only of existentially quantified conjunctions of freshness and α-equality constraints, we obtain solved forms consisting of a substitution and a freshness context, as in the standard nominal unification algorithm [23]. A nominal approach to disunification problems was proposed by Ayala et al [1], including only conjunctions of equations and disequations and freshness constraints, without quantified variables. We generalise this previous work to deal with general formulas including disjunction, conjunction and negation of equations and freshness constraints, as well as existential and universal quantification over variables.
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