Finite and Infinite Input

If the input word $w$ is a word on $\Sigma^{*}$ then we have a finite input. If the input word $w$ is a word on $\Sigma^{\omega}$ then we have an infinite input. In the latter case is the run of an automaton also infinite. In case of an finite automaton with infinite input we will need instead of the set of final states additionally an acceptance condition.

In the case of a finite input word $w \in \Sigma^{*}$ with $\mid w\mid = n-1$ we can say, that an automaton $A$ accepts word w if there exists a run $\varrho = \langle \varrho(0),w(0),\varrho(1)\rangle, ...,\langle \varrho(n-1),w(n-1),-\rangle$ with $\varrho(n) \in F$.

In case of an infinite input word $w \in \Sigma^{\omega}$ with $\mid w\mid = \omega$ the run must fullfill certain acceptance conditions to be qualified as accepting the word $w$ (see below).