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Unformatted text preview: Exercise 4.2.6 Show that the regular languages are closed under the operations below. For each, well start with L and apply operations under which regular languages are closed (homomorphisms, intersection, set difference) to get the desired language. a) min ( L ) = { w  w is in L , but no proper prefix of w is in L } Describe the strings which are ineligible for min ( L ) and exclude them using set difference. The ineligible strings are L + , since w L + means that w = xy where x L . That is, w has a proper prefix x which is in L . min ( L ) = L L + Comments : many students submitted the (correct) answer L ( L L + ). This probably looks more natural, since L L + is a subset of L , and L + is not (in general). However, recall the definition of set difference: L M = L M . Think of this not as removing M from L , but excluding M from L , since M need not be a subset of L . Also, many students submitted a solution using homomorphisms, which were not necessary. It should be a red flag if you define a pair of homomorphisms that just add and remove hats without otherwise altering the string. This means that your set intersection could be done in the original alphabet . b) max ( L ) = { w  w is in L and for no x other than is wx in L } Again, describe strings which are ineligible for max ( L ), then remove them with set difference. Use homomorphisms to describe all the strings that are prefixes of strings in L . Here is the idea: use an inverse homomorphism h 1 on L to mark arbitrary symbols with hats, apply set intersection to force the...
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This homework help was uploaded on 09/28/2007 for the course COM S 381 taught by Professor Hopcroft during the Fall '05 term at Cornell University (Engineering School).
 Fall '05
 HOPCROFT
 Formal language, Regular expression, Regular language, Nondeterministic finite state machine, regular languages, inverse homomorphism

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