Co-Büchi automaton

In automata theory, a co-Büchi automaton is a variant of Büchi automaton. The only difference is the accepting condition: a Co-Büchi automaton accepts an infinite word $w$ if there exists a run, such that all the states occurring infinitely often in the run are in the final state set $F$. In contrast, a Büchi automaton accepts a word $w$ if there exists a run, such that at least one state occurring infinitely often in the final state set $F$.

(Deterministic) Co-Büchi automata are strictly weaker than (nondeterministic) Büchi automata.

Formal definition

Formally, a deterministic co-Büchi automaton is a tuple ${\displaystyle {\mathcal {A}}=(Q,\Sigma ,\delta ,q_{0},F)}$ that consists of the following components:

• $Q$ is a finite set. The elements of $Q$ are called the states of ${\mathcal {A}}$.
• $\Sigma$ is a finite set called the alphabet of ${\mathcal {A}}$.
• ${\displaystyle \delta :Q\times \Sigma \rightarrow Q}$ is the transition function of ${\mathcal {A}}$.
• $q_0$ is an element of $Q$, called the initial state.
• $F\subseteq Q$ is the final state set. ${\mathcal {A}}$ accepts exactly those words $w$ with the run ${\displaystyle \rho (w)}$, in which all of the infinitely often occurring states in ${\displaystyle \rho (w)}$ are in $F$.

In a non-deterministic co-Büchi automaton, the transition function $\delta$ is replaced with a transition relation $\Delta$. The initial state $q_0$ is replaced with an initial state set $Q_{0}$. Generally, the term co-Büchi automaton refers to the non-deterministic co-Büchi Büchi automaton.

For more comprehensive formalism see also ω-automaton.

Acceptance Condition

The acceptance condition of a co-Büchi automaton is formally

${\displaystyle \exists i\forall j:\;j\geq i\quad \rho (w_{j})\in F.}$

The Büchi acceptance condition is the complement of the co-Büchi acceptance condition:

${\displaystyle \forall i\exists j:\;j\geq i\quad \rho (w_{j})\in F.}$

Properties

Co-Büchi automata are closed under union, intersection, projection and determinization.