# From Decidability to Undecidability by Considering Regular Sets of Instances

## TL;DR

This paper explores the transition from decidability to undecidability in classical problems when extended to regular sets of instances, revealing undecidable cases even for problems within L and analyzing factors affecting this boundary.

## Contribution

It introduces the intreg-problem as a generalization of classical problems to regular sets, demonstrating undecidability in cases previously considered decidable and analyzing factors influencing this boundary.

## Key findings

- Undecidability arises for certain problems from L when considering regular sets.
- The paper characterizes the boundary between decidable and undecidable intreg-problems.
- Different encoding schemes and alphabet sizes influence problem decidability.

## Abstract

We are lifting classical problems from single instances to regular sets of instances. The task of finding a positive instance of the combinatorial problem $P$ in a potentially infinite given regular set is equivalent to the so called intreg-problem of $P$, which asks for a given DFA $A$, whether the intersection of $P$ with $L(A)$ is non-empty. The intreg-problem generalizes the idea of considering multiple instances at once and connects classical combinatorial problems with the field of automata theory. While the question of the decidability of the intreg-problem has been answered positively for several NP- and even PSPACE-complete problems, we are presenting natural problems even from L with an undecidable intreg-problem. We also discuss alphabet sizes and different encoding-schemes elaborating the boundary between problem-variants with a decidable respectively undecidable intreg-problem.