Operations on Boolean and Alternating Finite Automata

TL;DR

This paper analyzes the computational complexity of fundamental operations on languages represented by Boolean and alternating finite automata, providing tight bounds and optimal witnesses over small alphabets.

Contribution

It establishes tight upper bounds for regular operations on Boolean and alternating automata, including union, intersection, concatenation, and others, with optimal alphabet witnesses.

Findings

Tight bounds for union, intersection, difference, concatenation, square, and quotients.

Complexity results for complementation, symmetric difference, star, and reversal.

Optimal witnesses over unary or binary alphabets.

Abstract

We examine the complexity of basic regular operations on languages represented by Boolean and alternating finite automata. We get tight upper bounds m+n and m+n+1 for union, intersection, and difference, 2^m+n and 2^m+n+1 for concatenation, 2^n+n and 2^n+n+1 for square, m and m+1 for left quotient, 2^m and 2^m+1 for right quotient. We also show that in both models, the complexity of complementation and symmetric difference is n and m+n, respectively, while the complexity of star and reversal is 2^n. All our witnesses are described over a unary or binary alphabets, and whenever we use a binary alphabet, it is always optimal.