On the Shuffle Automaton Size for Words

TL;DR

This paper studies the state complexity of automata accepting the shuffle of two words, showing conditions for quadratic bounds and cases requiring exponential states, highlighting the impact of small modifications.

Contribution

It provides the first examples of exponential state size for shuffle automata and identifies conditions that guarantee quadratic bounds.

Findings

Exponential DFA size for certain word shuffles

Quadratic upper bounds under specific conditions

Small letter switches can cause exponential growth

Abstract

We investigate the state size of DFAs accepting the shuffle of two words. We provide words u and v, such that the minimal DFA for u shuffled with v requires an exponential number of states. We also show some conditions for the words u and v which ensure a quadratic upper bound on the state size of u shuffled with v. Moreover, switching only two letters within one of u or v is enough to trigger the change from quadratic to exponential.