Ω-automaton

In automata theory, a branch of theoretical computer science, an ω-automaton (or stream automaton) is a variation of finite automatons 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 latter 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.

Formally, a deterministic ω-automaton is a tuple A = (Q,Σ,δ,Q₀,Acc) that consists of the following components:

An input for A is an infinite string over the alphabet Σ, i.e. it is an infinite sequence α = (a₁,a₂,a₃,...). The run of A on such an input is an infinite sequence ρ = (r₀,r₁,r₂,...) of states, defined as follows:

The main purpose of an ω-automaton is to define a subset of the set of all inputs: The set of accepted inputs. Whereas in the case of an ordinary finite automaton every run ends with a state r_n and the input is accepted if and only if r_n is an accepting state, the definition of the set of accepted inputs is more complicated for ω-automata. Here we must look at the entire run ρ. The input is accepted if the corresponding run is in Acc. The set of accepted input ω-words is called the recognized ω-language by the automaton, which is denoted as L(A).

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