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# lect_04 - £ CS 373 Theory of Computation £ Madhusudan...

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Unformatted text preview: £ CS 373: Theory of Computation £ Madhusudan Parthasarathy Lecture 4: The product construction: Closure un- der intersection and union 28 January 2010 This lecture nishes section 1.1 of Sipser and also covers the start of 1.3. 1 Product Construction 1.1 Product Construction: Example Let Σ = { a , b } and L is the set of strings in Σ * that have the form a * b * and have even length. L is the intersection of two regular languages L 1 = a * b * and L 2 = (ΣΣ) * . We can show they are regular by exhibiting DFA s that recognize them. q q 1 drain a b b a a , b r r 1 a , b a , b L 1 L 2 We can run these two DFA s together, by creating states that remember the states of both machines. ( q , r ) ( q 1 , r ) ( drain, r ) ( q , r 1 ) ( q 1 , r 1 ) ( drain, r 1 ) a a b b b b a a a , b a , b Notice that the nal states of the new DFA are the states ( q,r ) where q is nal in the rst DFA and r is nal in the second DFA . To recognize the union of the two languages, rather than the intersection, we mark all the states ( q,r ) such that either q or r are accepting states in the their respective DFA s. State of a DFA after reading a word w . In the following, given a DFA M = ( Q, Σ ,δ,q ,F ) , we will be interested in the state the DFA M is in, after reading the a string...
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lect_04 - £ CS 373 Theory of Computation £ Madhusudan...

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