xxx_hd4_8 - 332 CHAPTER 8 SPACE COMPLEXITY 8.28 Show that...

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332 CHAPTER 8 / SPACE COMPLEXITY 8.28 Show that A NFA is NL -complete. 8.29 Show that E DFA is NL -complete. 8.30 Give an example of an NL -complete context-free language. A 8.31 Define CYCLE = { G | G is an directed graph that contains a directed cycle } . Show that CYCLE is NL -complete. SELECTED SOLUTIONS 8.5 Let A 1 and A 2 be languages that are decided by NL -machines N 1 and N 2 . We construct three Turing machines: N deciding A 1 A 2 ; N deciding A 1 A 2 ; and N * deciding A * 1 . Each of these machines receives input w . Machine N nondeterministically branches to simulate N 1 or to simulate N 2 . In either case, N accepts if the simulated machine accepts. Machine N nondeterministically selects a position on the input to divide it into two substrings. Only a pointer to that position is stored on the work tape— insufficient space is available to store the substrings themselves. Then N simulates N 1 on the first substrings, branching nondeterministically to simulate N 1 ’s nonde- terminism. On any branch that reaches
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