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Infinite-state automata are a new invention: they are automata that have an infinite number of states represented by words, transitions defined us- ing rewriting, and with sets of initial and final states.
(An infinite language is a language with infinitely many strings in it. {an | n 0}, {ambn | m, n 0}, and {a, b} are all infinite regular languages.)
A formal semantics should give, for each program, an abstract model that represents just the relevant features of all possible executions of that program. Then two programs are regarded as semantically equivalent when their models are the same (up to isomorphism).
Is a given regular language infinite? Start with a DFA for the language. Key idea: if the DFA has n states, and the language contains any string of length n or more, then the language is infinite. Otherwise, the language is surely finite.
Omega language. In formal language theory within theoretical computer science, an infinite word is an infinite-length sequence (specifically, an -length sequence) of symbols, and an -language is a set of infinite words.
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A finite language can be described by listing its strings which is not the case for infinite languages; even for some finite languages the listing could be quite cumbersome. Moreover, in general, for a language to be interesting mathematically, it needs to have an infinite set of strings.

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