Positive and Negative Results for Higher-Order Disunification

Abstract

This paper is devoted to higher-order disunification which is the process of solving quantified formulae built on simply-typed lambda-terms, the equality induced by the η and the β reductions, boolean connectives and the negation. This problem is motivated by tests of completeness of definitions in algebraic higher-order specification languages which combine the advantages of algebraic specification languages and higher-order programming languages. We show that higher-order disunification is not semi-decidable and we prove the undecidability of second-order complement problems which are the formulae expressing the completeness of some scheme, by encoding Minsky machines. On the other hand, we propose a set of transformation rules for simplifying such formulae, and we show how to extend this set of rules into a quantifier elimination procedure when we assume some restrictions on the formulae that we consider. We prove that second-order complement problems are decidable when some conditions are imposed on second-order variables and bound variables, and we are able to prove the decidability of any formula when all the terms occurring in the formula are patterns, i.e. terms s.t. the arguments of free variables are distinct bound variables. The quantifier elimination process involves classical rules for unification and their dual through negation, elimination rules for universal variables which are more complex than their first-order counterparts since variables may have arguments, and rules for solving dependence constraints which state that a function depends only on some of its arguments.