Synchronous Subsequentiality and Approximations to Undecidable Problems

TL;DR

This paper introduces the class of synchronous subsequential relations, exploring their properties and showing how they can approximate undecidable problems, potentially balancing expressivity and decidability for applications.

Contribution

The paper defines the class of synchronous subsequential relations and demonstrates their ability to approximate undecidable problems, offering a new intermediate framework.

Findings

Most decision problems remain undecidable for this class

Relations can be approximated in a meaningful way

Potential applications in balancing expressivity and decidability

Abstract

We introduce the class of synchronous subsequential relations, a subclass of the synchronous relations which embodies some properties of subsequential relations. If we take relations of this class as forming the possible transitions of an infinite automaton, then most decision problems (apart from membership) still remain undecidable (as they are for synchronous and subsequential rational relations), but on the positive side, they can be approximated in a meaningful way we make precise in this paper. This might make the class useful for some applications, and might serve to establish an intermediate position in the trade-off between issues of expressivity and (un)decidability.