cs373 fa10 hw2sol - Solutions for Problem Set 2 CS 373:...

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Unformatted text preview: Solutions for Problem Set 2 CS 373: Theory of Computation Assigned: September 7, 2010 Due on: September 14, 2010 at 10am Homework Problems Problem 1 . [Category: Design] Design an NFA for the language D given in Problem 1.48. You need not formally prove the correctness of your construction, but your construction should be clear and understand- able. Solution: Language D is the collection of all binary strings that have an equal number of 01 and 10 occurrences. Observe that we have one occurrence of 01 when there is switch from a sequence of 0s to a sequence of 1s. Similarly there is an occurrence of 10 when there is a switch from a sequence of 1s to a sequence of 0s. Thus, since the runs of 0s and 1s alternate in a binary string, a string has equal number of 0s and 1s iff the strings begins and ends in the same symbol. This yields the following NFA. Formally, the q q 1 q 2 q 3 q 4 0,1 1 1 0,1 0,1 NFA N = ( { q ,q 1 ,q 2 ,q 3 ,q 4 } , { , 1 } ,δ,q , { q ,q 2 ,q 4 } ) where δ is given by δ ( q , 0) = { q 1 ,q 2 } δ ( q , 1) = { q 2 ,q 3 } δ ( q 1 , 0) = { q 1 ,q 4 } δ ( q 1 , 1) = { q 1 } δ ( q 3 , 0) = { q 3 } δ ( q 3 , 1) = { q 3 ,q 4 } For all other values of q and a ∈ { , 1 } ∪ { } , δ ( q,a ) = ∅ . Problem 2 . [Category: Design] Problem 1.31. Hint: Show that if M is a DFA recognizing A then there is an NFA recognizing A R . Solution: Given a DFA M = ( Q, Σ ,δ,q ,F ) recognizing A , the NFA N recognizing A R will be obtained by “reversing” the transitions and switching the initial and final states. Formally,“reversing” the transitions and switching the initial and final states....
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This note was uploaded on 10/04/2011 for the course CS 373 taught by Professor Viswanathan,m during the Fall '08 term at University of Illinois, Urbana Champaign.

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cs373 fa10 hw2sol - Solutions for Problem Set 2 CS 373:...

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