State Complexity of Projection on Languages Recognized by Permutation Automata and Commuting Letters

TL;DR

This paper establishes tight bounds on the state complexity of language projection for permutation automata and extends the class of automata with the property that their projected languages are recognizable with a number of states equal to the original automaton.

Contribution

It derives the tight state complexity bounds for permutation automata and shows that commutative automata also have the property that their projected languages are recognizable with the same number of states.

Findings

Tight bound of 2^{n - rac{m}{2}} - 1 for permutation automata.

Projected languages of automata with the observer property are recognizable with n states.

Commutative automata can also have the property that their projected languages are recognizable with the same number of states.

Abstract

The projected language of a general deterministic automaton with $n$ states is recognizable by a deterministic automaton with $2^{n-1} + 2^{n-m} - 1$ states, where $m$ denotes the number of states incident to unobservable non-loop transitions, and this bound is best possible. Here, we derive the tight bound $2^{n - \lceil \frac{m}{2} \rceil} - 1$ for permutation automata. For a state-partition automaton with $n$ states (also called automata with the observer property) the projected language is recognizable with $n$ states. Up to now, these, and finite languages projected onto unary languages, were the only classes of automata known to possess this property. We show that this is also true for commutative automata and we find commutative automata that are not state-partition automata.