Logic and Rational Languages of Scattered and Countable Series-Parallel Posets

Abstract

Let $A$ be an alphabet and $S{P}^{\diamond }(A)$ denote the class of all countable N-free partially ordered sets labeled by $A$, in which chains are scattered linear orderings and antichains are finite. We characterize the rational languages of $S{P}^{\diamond }(A)$ by means of logic. We define an extension of monadic second-order logic by Presburger arithmetic, named P-MSO, such that a language $L$ of $S{P}^{\diamond }(A)$ is rational if and only if $L$ is the language of a sentence of P-MSO, with effective constructions from one formalism to the other. As a corollary, the P-MSO theory of $S{P}^{\diamond }(A)$ is decidable.