# $\omega$-automata where string is accepted iff a final state is accessible from starting state

I am wondering if $$\omega$$-automata with the following acceptance condition are valid.

An input string is accepted iff one of the final states occurs at least once.

This differs from Buchi automata in that the final state only has to occur once, not infinitely often.

Does this kind of automata have a name? Is it interesting or important?

Your type is called 1-acceptance (if I recall right) and a string is accepted if a state from an accepting set $$D$$ is entered at least once. The dual 1'-acceptance required that the states entered are always within $$D$$. One level up one requires that a state from $$D$$ is entered infinitely often (Büchi), respectively that from some moment on the states are within $$D$$.
These definitions can be considered not only for finite state automata, but also for automata with external storage, like pushdown automata. One common fact for all such automata is that the 1-acceptance (your type) is of low "topological" complexity. Those $$\omega$$-languages are of the form $$L\cdot \Sigma^\omega$$, where $$L$$ is an ordinary language of the type of automata. This means one cannot accept $$(a^*b)^\omega$$.
In general there is a clear link between topological complexity of the $$\omega$$-language and the acceptance type.
L.H. Landweber, Decision problems for $$\omega$$-automata, Math. Systems Theory 3 (1969) 376-384.