Decidable Expansions of Labelled Linear Orderings

TL;DR

This paper explores conditions under which expanding a linear ordering with additional monadic predicates preserves decidability, focusing on structures with specific infinite intervals.

Contribution

It demonstrates that for certain linear orderings with infinite intervals, non-trivial monadic expansions can still maintain decidability of their theories.

Findings

Decidable expansions exist for linear orderings with  or - type intervals.

Adding a new monadic predicate can preserve decidability under specific conditions.

The results identify when expansions do not compromise the decidability of the structure.

Abstract

Consider a linear ordering equipped with a finite sequence of monadic predicates. If the ordering contains an interval of order type \omega or -\omega, and the monadic second-order theory of the combined structure is decidable, there exists a non-trivial expansion by a further monadic predicate that is still decidable.