SOME COMMENTS ON A RECENT NOTE BY RAVIKUMA R Ernst L . Leis s Department of Computer Scienc e University of Houston, Houston Texas 77204-347 5 In [L], Ravikumar draws attention to a technique by Sakoda and Sipser [2] for provin g lower bounds on the size of the smallest deterministic finite automaton (dfa) equivalen t to a finite automaton with some type of nondeterminism . There is no question that thei r method is a very useful technique . However, several comments on Ravikuma r ' s note are i n order . 1. Ravikurrmr ' s Problem 5 does not require any complicated proof since it is a direct con sequence of his Problem 1, by virtue of the following fact . Let R=(A,Q,T,g0,F) be a reduced dfa with n states accepting the language L . Define D=(A,Q,T,g0,Q-F) . Then D has n states, is reduced, and accepts the complement of the language L . 2. Ravikumar does not solve his Problem 4, he merely gives a lower bound . It is well known that the deterministic complexity of the intersection of two regular languages is bounde d from above by the product of their deterministic complexities . Therefore, an obvious uppe r bound is nn . (I am not aware of any construction for arbitrary n that attains this bound . ) 3. Ravikumar states ' that for every integer n, there is an nfa with n states such that th e shortest string not accepted by it is of length 2 n-l . ' This statement is false . A sketc h of the proof follows . Let N=(A,Q,T,g0,F) be an nfa with card(Q)=n . Assume that the statement is true, i .e ., the shortest string w rejected by N is of length 2 n-1 . This implie s that there is a simple path of length 2 n -1 in the state graph that corresponds to the df a equivalent to N . Therefore, every one of the 2 n subsets of Q must be visited by w . This , however, is impossible since not both Q and the empty set can be visited by the same word . [l] B . Ravikumar : Some Applications of a Technique of Sakoda and Sipser, Sigact News, Vol . 21, No . 4, Fall 1990, 73-77 . [2] W . J . Sakoda, M . Sipser : Nondeterminism and the Size of Two-Way Finite Automata , Proceed . Tenth Ann . Symp . on Theory of Computing, 1978, 275-286 .
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Some comments on a recent note by Ravikumar
Leiss, Ernst L.
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, Volume 22 (1)
Association for Computing Machinery – Mar 1, 1991
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