Outfix-guided insertion

TL;DR

This paper introduces an outfix-guided insertion operation inspired by bio-operations on DNA, demonstrating that the closure of certain languages under this operation can be non-regular and analyzing related decision problems.

Contribution

It generalizes the overlap assembly operation, constructs a finite language with non-regular closure, and studies closure properties and decision complexities for various language classes.

Findings

Outfix-guided insertion closure of a finite language can be non-regular.

Deciding closure under outfix-guided insertion for DFA languages is polynomial.

The complexity for NFA languages remains unresolved.

Abstract

Motivated by work on bio-operations on DNA strings, we consider an outfix-guided insertion operation that can be viewed as a generalization of the overlap assembly operation on strings studied previously. As the main result we construct a finite language $L$ such that the outfix-guided insertion closure of $L$ is non-regular. We consider also the closure properties of regular and (deterministic) context-free languages under the outfix-guided insertion operation and decision problems related to outfix-guided insertion. Deciding whether a language recognized by a deterministic finite automaton is closed under outfix-guided insertion can be done in polynomial time. The complexity of the corresponding question for nondeterministic finite automata remains open.