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Theorem The class of recognizable languages is closed under Boolean operations.
The construction of the proof of Lemma 3 gives us a DFA that keeps track of whether or not a given string is in either or both of any pair of recognizable languages. We can modify the construction for other Boolean operations simply by selecting the appropriate set of accepting states:
• Union: Let F′
= {(q, p) | q ∈ F1 or p ∈ F2}. Then L(A′ ) = L1 ∪ L2.
• Relative complement: Let F′ = F1 × (Q2 - F2). Then L(A′ ) = L1 -L2.
• Complement: Let L1 = Σ* and use the construction for relative complement.
(c) Can you say that B is decidable? (d) If you somehow know that A is decidable, what can you say about B?
The fact that the Recognition Problem is decidable gives us another algorithm for deciding Emptiness. The pumping lemma tells us that if every string x ∈ L(A) which has length grea
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Construct a Moore machine to convert a binary string of radix 4.
let G=(V,T,S,P) where V={a,b,A,B,S}, T={a,b},S the start symbol and P={S->Aba, A->BB, B->ab,AB->b} 1.show the derivation sentence for the string ababba 2. find a sentential form
Find the Regular Grammar for the following Regular Expression: a(a+b)*(ab*+ba*)b.
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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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