What ω-language does a Büchi Automaton recognize?

Question

When I'm reading about Büchi Automatons, it says that x is accepted if there are infinitely many occurrences of states from the set of accepting states in the run.

But for instance, when I'm being asked this:

I would say that both answer A, C and D is the correct answer, even though there is only one correct answer.

My thoughts:

A: When "1" is appearing in all even positions, the run will keep going through the accepting state.

C: If "1" is appearing in infinitely many positions, the run will also keep going through the accepting state.

D: If for all occurrences of the letter "1", there is a subsequent occurence of the letter "2", then the run will keep going from the first state to the accepting state and so forth = the run will also keep going through the accepting state.

What am I missing?

Answer 1

The correct answer is C. Regardless of what state it is currently in, the automaton moves to the initial state on reading a $0$ or $2$ and to the acceptor state on reading a $1$. Thus, it is in the acceptor state infinitely many times while reading the word if and only if the word has infinitely many $1$s.

The sets described in A and D are proper subsets of the language accepted by the automaton: every word in each of those sets belongs to the language, but so do other words that are not in those sets.

Added: As mercio notes in the comments below, that isn’t quite right: the set described in D actually isn’t quite a subset of the language, since it vacuously include all $\omega$-words that contain no $1$ at all.

Answer 2

Let $L$ be the recognized language of the Büchi automaton and $L_X$ the language described in part $X$.

$L_C$ is contained in $L$. Every word in $L$ has to be accepted along a path which contains the finial state infinite many times. The final state can only be assumed by recognizing a $1$ symbol. So $L$ is contained in $L_C$ too and $L = L_C$ follows.

$L_A \subsetneq L_C$ (e.g. $111011^\omega \in L_C$) and $L_B \subsetneq L_C$ ($(10)^\omega \in L_C$).

$D$ is wrong, $L_D$ would not contain $w = 11^\omega$, which is recognized.

That leaves $C$ as answer.