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

TL;DR

This paper characterizes rational languages of countable N-free posets with scattered chains and finite antichains using an extended logic called P-MSO, establishing a logical and decidable framework for these structures.

Contribution

It introduces P-MSO, an extension of monadic second-order logic with Presburger arithmetic, to characterize rational languages of specific scattered posets, and proves its effectiveness and decidability.

Findings

Rational languages of $SP^ullet(A)$ are characterized by P-MSO.

The P-MSO theory of $SP^ullet(A)$ is decidable.

Effective translations exist between formalism and logic.

Abstract

Let $A$ be an alphabet and $SP^\diamond(A)$ 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 $SP^\diamond(A)$ 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 $SP^\diamond(A)$ 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 $SP^\diamond(A)$ is decidable.