Ω-automaton
variation of finite automata that runs on infinite, rather than finite, strings as input
In automata theory, a branch of theoretical computer science, an ω-automaton (or stream automaton) is a variation of a finite automaton that runs on infinite, rather than finite, strings as input. Since ω-automata do not stop, they have a variety of acceptance conditions rather than simply a set of accepting states.
ω-automata are useful for specifying behavior of systems that are not expected to terminate, such as hardware, operating systems and control systems. For such systems, one may want to specify a property such as "for every request, an acknowledge eventually follows", or its negation "there is a request that is not followed by an acknowledge". The former is a property of infinite words: one cannot say of a finite sequence that it satisfies this property.
Classes of ω-automata include the Büchi automata, Rabin automata, Streett automata, parity automata and Muller automata, each deterministic or non-deterministic. These classes of ω-automata differ only in terms of acceptance condition. They all recognize precisely the regular ω-languages except for the deterministic Büchi automata, which is strictly weaker than all the others. Although all these types of automata recognize the same set of ω-languages, they nonetheless differ in succinctness of representation for a given ω-language.
Deterministic ω-automata
Formally, a deterministic ω-automaton is a tuple
A
=
(
Q
,
Σ
,
δ
,
Q
0
,
A
a
c
c
)
{\textstyle A=(Q,\Sigma ,\delta ,Q_{0},A_{acc})}
, that consists of the following components:
Q
{\textstyle Q}
, is a finite set.
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