On insertion-deletion systems over relational words

TL;DR

This paper introduces relational words with ordered positions and binary relations, generalizes insertion-deletion operations from strings, and studies their decidability and computational power.

Contribution

It defines relational words and extends insertion-deletion operations, analyzing their decidability and showing their capacity to encode any recursively enumerable language.

Findings

Decidable membership for short insertion-deletion rules

Undecidability in the general case due to encoding power

Relational words generalize string operations with new computational properties

Abstract

We introduce a new notion of a relational word as a finite totally ordered set of positions endowed with three binary relations that describe which positions are labeled by equal data, by unequal data and those having an undefined relation between their labels. We define the operations of insertion and deletion on relational words generalizing corresponding operations on strings. We prove that the transitive and reflexive closure of these operations has a decidable membership problem for the case of short insertion-deletion rules (of size two/three and three/two). At the same time, we show that in the general case such systems can produce a coding of any recursively enumerable language leading to undecidabilty of reachability questions.