Data Monoids

We develop an algebraic theory for languages of data words. We prove that, under certain conditions, a language of data words is definable in first-order logic if and only if its syntactic monoid is aperiodic. 1 Introduction This paper is an attempt to combine two fields. The first field is the algebraic theory of regular languages. In this theory, a regular language is represented by its syntactic monoid, which is a finite monoid. It turns out that many important properties of the language are reflected in the structure of its syntactic monoid. One particularly beautiful result is the Schützenberger-McNaughton-Papert theorem, which describes the expressive power of first-order logic. Let L ⊆ A * be a regular language. Then L is definable in first-order logic if and only if its syntactic monoid M L is aperiodic. For instance, the language " words where there exists a position with label a " is defined by the first-order logic formula (this example does not even use the order on positions <, which is also allowed in general) ∃x. a(x). The syntactic monoid of this language is isomorphic to {0, 1} with multiplication, where 0 corresponds to the words that satisfy the formula, and 1 to the words that do not. Clearly, this monoid does not contain any non-trivial group. There are many results similar to theorem above, each one providing a connection between seemingly unrelated concepts of logic and algebra, see e.g. the book [8]. The second field is the study of languages over infinite alphabets, commonly called languages of data words. Regular languages are usually considered for finite alphabets. Many current motivations, including XML and verification, require the use of infinite alphabets. In this paper, we use the following definition of data words. We fix for the rest of the paper a single infinite alphabet D, and study words and languages over D. A typical language, which we use as our running example, is " some two consecutive positions have the same letter " , i.e. 2 Data Monoids A number of automata models have been developed for such languages. The general theme is that there is a tradeoff between the following three properties: expressivity, good closure properties, and decidable (or even efficiently decidable) emptiness. The existing models strike different balances in this tradeoff, and it is not clear which balance, if any, should deserve the name of " regular language ". …