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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.
The path function δ : Q × Σ*→ P(Q) is the extension of δ to strings: Again, this just says that to ?nd the set of states reachable by a path labeled w from a state q in an
Computer has a single LIFO stack containing ?xed precision unsigned integers (so each integer is subject to over?ow problems) but which has unbounded depth (so the stack itself nev
When we study computability we are studying problems in an abstract sense. For example, addition is the problem of, having been given two numbers, returning a third number that is
s-> AACD A-> aAb/e C->aC/a D-> aDa/bDb/e
Give the Myhill graph of your automaton. (You may use a single node to represent the entire set of symbols of the English alphabet, another to represent the entire set of decima
The computation of an SL 2 automaton A = ( Σ, T) on a string w is the maximal sequence of IDs in which each sequential pair of IDs is related by |- A and which starts with the in
A context free grammar G = (N, Σ, P, S) is in binary form if for all productions A we have |α| ≤ 2. In addition we say that G is in Chomsky Normaml Form (CNF) if it is in bi
We can then specify any language in the class of languages by specifying a particular automaton in the class of automata. We do that by specifying values for the parameters of the
The k-local Myhill graphs provide an easy means to generalize the suffix substitution closure property for the strictly k-local languages. Lemma (k-Local Suffix Substitution Clo
(c) Can you say that B is decidable? (d) If you somehow know that A is decidable, what can you say about B?
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