Quantitative Semantics for Jumping Automata

TL;DR

This paper introduces four quantitative semantics for jumping automata, measuring jump costs in different ways, and studies the decidability and complexity of the boundedness problem for these automata.

Contribution

It proposes four new cost-based semantics for jumping automata and analyzes the decidability and complexity of the boundedness problem under these semantics.

Findings

Decidability results vary across different semantics.

Complexity bounds are established for the boundedness problem.

The semantics provide a nuanced way to measure jumps in automata.

Abstract

Jumping automata are finite automata that read their input in a non-sequential manner, by allowing a reading head to ``jump'' between positions on the input, consuming a permutation of the input word. We argue that allowing the head to jump should incur some cost. To this end, we propose four quantitative semantics for jumping automata, whereby the jumps of the head in an accepting run define the cost of the run. The four semantics correspond to different interpretations of jumps: the \emph{absolute distance} semantics counts the distance the head jumps, the \emph{reversal} semantics counts the number of times the head changes direction, the \emph{Hamming distance} measures the number of letter-swaps the run makes, and the \emph{maximum jump} semantics counts the maximal distance the head jumps in a single step, We study these measures, with the main focus being the \emph{boundedness problem}: given a jumping automaton, decide whether its (quantitative) language is bounded by some given number $k$. We establish the decidability and complexity for this problem under several variants.