Author: Saeed Salehi (Saeed @ Math.Net) Title: Varieties of Tree Languages

Trees are among the most fundamental and ubiquitous structures in mathematics. Tree languages and automata on trees have been studied extensively since the 1960s from both a purely mathematical and application point of view. When trees are defined as terms, universal algebra becomes directly applicable to tree automata and, on the other hand, the theory of tree automata suggests new notions and problems to universal algebra. Different syntactic invariants have been proposed as bases for classifications of regular tree languages: syntactic algebras (Steinby 1979, 1992; Almeida 1990), syntactic monoids and syntactic semigroups (Thomas 1983; Nivat and Podelski 1989), tree algebras (Wilke 1996) and syntactic theories (Esik 1999). However, so far variety theorems comparable with Eilenberg’s classical theorems for regular string languages were known for syntactic algebras and syntactic theories only. In this thesis we consider several aspects of varieties of tree languages and settle some open questions concerning the various formalisms. In Chapter 2 we extend the variety theorem for general recognizable subsets of free algebras (Steinby 1979) to the many-sorted case. In Chapter 3 we formulate Pin’s (1996) theory of positive varieties for tree languages and prove a variety theorem that establishes a correspondence between positive varieties of tree languages and varieties of finite ordered algebras. It has been known already for quite a long time that not all varieties of tree languages can be defined by syntactic monoids or semigroups, and the question about the exact defining power of these syntactic invariants has been raised by several authors. In Chapter 4 we answer this question by characterizing the varieties of tree languages that correspond to some variety of finite monoids or semigroups. In Chapter 5 we characterize the families of tree languages definable by ordered monoids and study some special instances of the above mentioned variety theorems. Chapter 6 is devoted to Wilke’s tree algebras. We introduce a convergent term rewriting system that yields an efficient method to decide the word problem of tree algebras. By using the notions introduced in Chapter 2 for many-sorted algebras and languages, we obtain a variety theorem for families of tree languages defined by tree algebras. Moreover, we prove that, for any sufficiently rich alphabet, all congruence-preserving functions of the tree term algebra are obtained as compositions of the basic tree-constructing operations.