We consider the rewrite system Rμ with μx.M → μM[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 , 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 as the function type constructor. For general signatures, this rewriting system is widely used to represent and manipulate infinite regular trees.
The main concern of this paper is the convertibility relation for μ-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.