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 $$\mathcal{A} = (Q,\Sigma,\delta,q_0,F)$$ 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 $$\mathcal{A}$$.
 * $$\delta : Q \times \Sigma \rightarrow Q$$ 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 $$\rho(w)$$, in which all of the infinitely often occurring states in $$\rho(w)$$ 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 automaton.

For more comprehensive formalism see also ω-automaton.

Acceptance Condition
The acceptance condition of a co-Büchi automaton is formally

$$\exists i \forall j:\; j\geq i \quad \rho(w_j) \in F.$$

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

$$\forall i \exists j:\; j\geq i \quad \rho(w_j) \in F.$$

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

Literature

 * Wolfgang Thomas: Automata on Infinite Objects. In: Jan van Leeuwen (Hrsg.): Handbook of Theoretical Computer Science. Band B: Formal Models and Semantics. Elsevier Science Publishers u. a., Amsterdam u. a. 1990, ISBN 0-444-88074-7, p. 133–164.