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Theorem (Myhill-Nerode) A language L ⊆ Σ is recognizable iff ≡L partitions Σ* into ?nitely many Nerode equivalence classes.
Proof: For the "only if" direction (that every recognizable language has ?nitely many Nerode equivalence classes) observe that L ∈ Recog iff L = L(A) for some DFA A and that if δ(q0,w) = δ(q0, u) (i.e., if the path from the start state labeled w and that labeled u end up at the same state) then w ≡L u. This is a consequence of the fact that the state ˆ δ(q0,w) encodes all the information the automaton remembers about the string w. If v extends w to wv ∈ L(A) then v is the label of a path to an accepting state from δ(q0,w). Since this is the same state as δ(q0, u) the same path witnesses that uv ∈ L. Similarly, if the path leads one to a non-accepting state then it must necessarily lead the other to the same state. The automaton has no way of distinguishing two strings that lead to the same state and, consequently, the language it recognizes cannot distinguish them. Since A is deterministic, every string in Σ* labels a path leading to some state, hence the equivalence classes corresponding to the states partition Σ*. Since the automaton has ?nitely many states, it distinguishes ?nitely many equivalence classes.
dfa for (00)*(11)*
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i have some questions in automata, can you please help me in solving in these questions?
The upper string r ∈ Q+ is the sequence of states visited by the automaton as it scans the lower string w ∈ Σ*. We will refer to this string over Q as the run of A on w. The automa
Lemma 1 A string w ∈ Σ* is accepted by an LTk automaton iff w is the concatenation of the symbols labeling the edges of a path through the LTk transition graph of A from h?, ∅i to
What are codds rule
This close relationship between the SL2 languages and the recognizable languages lets us use some of what we know about SL 2 to discover properties of the recognizable languages.
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