What's the true meaning of $(a + b)^\omega$ in regular expression

Question

I'm starting to dabble in language theory, regular expression & infinite words.

I'm not quite sure I completely get the meaning of this expression:

$(a + b)^\omega$

$^w$ meaning infinite repetition, I'm not positive it's standard syntaxe or just our class.

My main question is - do I have to "settle" on either a or b after some position in the infinite word? $abbbbaab^\omega$ would be fine but not $abbbbaa[ a | b]^\omega$ (e.g. I keep having either a or b infinitely)? My understanding is that an English translation would be "you must either have a, or b, at any given position indefinitely". So the + would be like a or condition, essentially.

Expending on that, then $((a+b)(c+d))^\omega$ would accept any alternating patter of (a or b)(c or d), e.g. things like acbcbcadad... but again wouldn't have to "settle" on either $(ac)^\omega$ or $(ad)^\omega$ or $(cc)^\omega$ etc...

Answer 1

The notation is standard but it's an omega ($\omega$, the first infinite ordinal; basically, the infinity in "There are infinitely many natural numbers"), not a $w$ (the 23rd letter of the Latin alphabet).

Both as a closure operator applied to languages and as an operator in $\omega$-regular expressions, $^\omega$ is directly analogous to ${}^*$, except it denotes the concatenation of infinitely many things, rather than any finite number of them.

Answer 2

For a language $L$ not containing $\epsilon$, $L^\omega$ is obtained by concatenating infinitely many words from $L$: $$ L^\omega = \{ w_1 w_2 \ldots : w_1,w_2, \ldots \in L \}. $$ The words can be arbitrary words from $L$. In particular, $\Sigma^\omega$ consists of all $\omega$-words over $\Sigma$.

When $\epsilon \in L$, then the definition is the same, but we only accept a resulting word if it is infinite. This is equivalent to removing $\epsilon$ from $L$, or to require that at most finitely many of the words are $\epsilon$.

Compare this to the definition of $L^*$: $$ L^* = \{ w_1 w_2 \ldots w_n : n \in \mathbb{N}, w_1,w_2,\ldots,w_n \in L \}. $$ Here also, the words need not be the same.