Algebraic Solutions to Recursion Schemes

Abstract The main problem in recursive scheme theory is determining how to solve a scheme and express its solution. Up to now this was always achieved by adding restrictive hypotheses either on the schemes themselves, or on the domains where they take their values, e.g., assuming the domains have a metric or an order structure and are complete with respect to this structure, or are iterative. Here we develop a strictly algebraic theory of recursion schemes with second-order substitutions. As it is strictly algebraic, the theory applies not only to all recursion schemes on trees, but also to recursion schemes on arbitrary algebras presented in the usual way by generators and relations. In particular, this gives a semantics for nondeterminism and for process algebras.