Weak Büchi automaton

In computer science and automata theory, a Weak Büchi automaton is a formalism which represents a set of infinite words. A Weak Büchi automaton is a modification of Büchi automaton such that for all pair of states ${\displaystyle q}$ and ${\displaystyle q'}$ belonging to the same strongly connected component, ${\displaystyle q}$ is accepting if and only if ${\displaystyle q'}$ is accepting.

A Büchi automaton accepts a word ${\displaystyle w}$ if there exists a run, such that at least one state occurring infinitely often in the final state set ${\displaystyle F}$. For Weak Büchi automata, this condition is equivalent to the existence of a run which ultimately stays in the set of accepting states.

Weak Büchi automata are strictly less-expressive than Büchi automata and than Co-Büchi automata.

Properties

The deterministic Weak Büchi automata can be minimized in time ${\displaystyle O(n\log(n))}$.^[1]

The languages accepted by Weak Büchi automata are closed under union and intersection but not under complementation. For example, ${\displaystyle (a+b)^{*}b^{\omega }}$ can be recognised by a Weak Büchi automaton but its complement ${\displaystyle (b^{*}a)^{\omega }}$ cannot.

Non-deterministic Weak Büchi automata are more expressive than Weak Büchi automata. As an example, the language ${\displaystyle (a+b)^{*}b^{\omega }}$ can be decided by a Weak Büchi automaton but by no deterministic Büchi automaton.