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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.
Theorem The class of ?nite languages is a proper subclass of SL. Note that the class of ?nite languages is closed under union and concatenation but SL is not closed under either. N
Differentiate between DFA and NFA. Convert the following Regular Expression into DFA. (0+1)*(01*+10*)*(0+1)*. Also write a regular grammar for this DFA.
Exercise Show, using Suffix Substitution Closure, that L 3 . L 3 ∈ SL 2 . Explain how it can be the case that L 3 . L 3 ∈ SL 2 , while L 3 . L 3 ⊆ L + 3 and L + 3 ∈ SL
i have some questions in automata, can you please help me in solving in these questions?
can you plz help with some project ideas relatede to DFA or NFA or anything
Normal forms are important because they give us a 'standard' way of rewriting and allow us to compare two apparently different grammars G1 and G2. The two grammars can be shown to
write grammer to produce all mathematical expressions in c.
Suppose A = (Σ, T) is an SL 2 automaton. Sketch an algorithm for recognizing L(A) by, in essence, implementing the automaton. Your algorithm should work with the particular automa
Exercise: Give a construction that converts a strictly 2-local automaton for a language L into one that recognizes the language L r . Justify the correctness of your construction.
how to understand DFA ?
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