Algebraic characterization of logically defined tree languages

We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindström quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization relies on the usage of preclones, an algebraic structure introduced by the authors in a previous paper, and of the block product operation on preclones. Our results generalize analogous results on finite word languages, but it must be noted that, as they stand, they do not yield an algorithm to decide whether a given regular tree language is first-order definable. Classification: ACM: F.4.3, F.4.1. MSC: 03B70, 68Q70, 68Q45 One of Bret Tilson’s lasting contributions is the introduction (with John Rhodes) of the notions of block product and two-sided semidirect product, and their use in the structure theory of finite monoids. This tool was initially introduced to derive iterated decompositions of morphisms and to refine the wreath product-based Krohn-Rhodes decomposition of finite monoids [28]. It quickly found applications in formal language theory (see [29, 30, 38, 2, 35] among others). One of the more fruitful applications of this work has been in the investigation of the logical aspects of automata theory (on finite words). For The first author acknowledges partial support from grant MTM2007 63422 from the Ministry of Education and Science of Spain. The second author acknowledges partial support from the French ANR (projet dots) and the Indo-French project Timed discoveri. Both authors acknowledge support from the European Science Foundation program AutoMathA. Corresponding author. [email protected]. Postal address: LaBRI, 351 cours de la Libération, 33405 Talence Cedex, France.