On equal μ-terms

We consider the rewrite system Rμ with μx.M →μ M[x := μx.M] as its single rewrite rule, where the signature consists of the variable binding operator commonly denoted by μ, first-order symbols, which in this paper are restricted to a binary function symbol F, and possibly some constant symbols. This kernel system denoting recursively defined objects occurs in several contexts, e.g. it is the framework of recursive types, with F as the function type constructor. For general signatures, this rewriting system is widely used to represent and manipulate infinite regular trees. Themain concern of this paper is the convertibility relation forμ-terms as given by the μ-rule, in particular, its decidability. This relation is sometimes called weak μ-equality, in contrast with strong μ-equality, which is given by the equality of the possibly infinite tree unwinding of μ-terms. While strong equality has received much attention, the opposite is the case for weak μ-equality. We present three alternative proofs for decidability of weak μ-equality. The first two proofs build upon an ingenious proof method of Cardone and Coppo. Prior to that, we prepare the way by an analysis of α-conversion. We then give a decidability proof in an ‘α-free’ manner, essentially treating μ-terms as first-order terms, and next, a proof in higher-order style, employing α-equivalence classes and viewing Rμ as a higher-order rewriting system. The third decidability proof is also derived in an α-free manner, exploiting the regular nature of the set of μ-reducts, enabling an appeal to the theory of tree automata. We conclude with additional results concerning decidability of reachability, and upward-joinability of μ-reduction, and of convertibility by α-free μ-reduction. © 2011 Elsevier B.V. All rights reserved.