A Complexity Approach to Tree Algebras: the Bounded Case

In this paper, we initiate a study of the expressive power of tree algebras, and more generally infinitely sorted algebras, based on their asymptotic complexity. We provide a characterization of the expressiveness of tree algebras of bounded complexity. Tree algebras in many of their forms, such as clones, hyperclones, operads, etc, as well as other kind of algebras, are infinitely sorted: the carrier is a multi sorted set indexed by a parameter that can be interpreted as the number of variables or hole types. Finite such algebras - meaning when all sorts are finite - can be classified depending on the asymptotic size of the carrier sets as a function of the parameter, that we call the complexity of the algebra. This naturally defines the notions of algebras of bounded, linear, polynomial, exponential or doubly exponential complexity... We initiate in this work a program of analysis of the complexity of infinitely sorted algebras. Our main result precisely characterizes the tree algebras of bounded complexity based on the languages that they recognize as Boolean closures of simple languages. Along the way, we prove that such algebras that are syntactic (minimal for a language) are exactly those in which, as soon as there are sufficiently many variables, the elements are invariant under permutation of the variables.