Typed Monoids – An Eilenberg-Like Theorem for Non Regular Languages

Unambiguous Constrained Automata

“…Second, we translated unambiguous CA to a natural model of deterministic register automata; the close inspection of this translation can lead to further advances in our understanding of unambiguity, in particular in the open problems dealing with unambiguous finite automata [4]. Third, we note that the closure properties of L UnCA imply that this class can be described by a natural algebraic object (see [1]). This will certainly help in linking UnCA to a first-order logic framework, and thus to some Boolean circuit classes.…”

A positive extension of Eilenberg’s variety theorem for non-regular languages

“…Another direction is to consider different closure properties for the classes of languages and monoids [1,14,16]. A further development in [2] deals with varieties of non-regular languages and new algebraic structures. More recently, a categorical approach has been used to obtain Eilenberg-like theorems in [22] (for regular languages), and also in [18].…”

“…Later on, Behle, Krebs and Reifferscheid [2] introduce a new algebraic structure, called typed monoid, to describe formal languages. This structure adds additional information to a monoid, which leads to a finer notion of language recognition.…”

“…The strategy of limiting the allowed accepting subset was already used by Sakarovitch [17] for an algebraic study of context-free languages, through the category of pointed monoids. In the new approach proposed in [2] instead of adding a "point" to the monoid, a finite Boolean algebra is attached to the monoid and the accepting subsets are required to be elements of this Boolean algebra. In fact, a preliminary version of the notion of typed monoid, named finitely typed monoid, was first introduced in [11], just controlling the accepting sets but not the morphisms.…”

“…They introduced the notion of C-variety, where C is a category of admissible morphisms between finitely generated free monoids, and the elements of such varieties are not semigroups or monoids, but morphisms from finitely generated free monoids onto finite monoids, called stamps. The proposal in [2] to limit the morphisms is through the use of the so-called units in the definition of a typed monoid. It is to be said that one of the motivations for both developements is that some classes of languages defined through logic or circuit complexity do not form varieties in the usual sense, since they are not closed under arbitrary inverse morphisms.…”