Ω-automaton

In automata theory, a branch of theoretical computer science, an ω-automaton (or stream automaton) is a variation of finite automata 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. Formally, a deterministic ω-automaton is a tuple A = (Q,Σ,δ,Q0,Acc) that consists of the following components: Q is a finite set. The elements of Q are called the states of A. Σ is a finite set called the alphabet of A. δ: Q × Σ → Q is a function, called the transition function of A. Q0 is an element of Q, called the initial state. Acc is the acceptance condition, formally a subset of Qω. 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: r0 = q0. r1 = δ(r0,a1). r2 = δ(r1,a2). rn = δ(rn-1,an). The main purpose of an ω-automaton is to define a subset of the set of all inputs: The set of accepted inputs.