381-hw5-solns - Exercise 4.2.6 Show that the regular...

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Exercise 4.2.6 Show that the regular languages are closed under the operations below. For each, we’ll 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 Σ + , since w L Σ + means that w = xy where x L . That is, w has a proper prefix x which is in L . min ( L ) = L - L Σ + Comments : many students submitted the (correct) answer L - ( L L Σ + ). This probably looks more natural, since L L Σ + is a subset of L , and L Σ + 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 Σ. 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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  • Fall '05
  • HOPCROFT
  • Formal language, Regular expression, Regular language, Nondeterministic finite state machine, regular languages, inverse homomorphism

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