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assign3 - w(a(5 Give an example of a DFA that is in S(b(20...

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CS5371 Theory of Computation Homework 3 Due: 2:10 pm, December 4, 2007 (before class) 1. Let k -PDA be a pushdown automaton that has k stacks. Thus a 0-PDA is an NFA and a 1-PDA is a conventional PDA. We already know that 1-PDAs are more powerful than 0-PDAs (since 1-PDAs recognize a larger class of languages). (a) (15%) Show that some language can be recognized by a 2-PDA but not a 1-PDA. Conclude that 2-PDAs are more powerful than 1-PDAs. (b) (Further studies: No marks) Show that if a language L can be recognized by a 3- PDA, L can be recognized by some 2-PDA. Conclude that 2-PDAs are as powerful as 3-PDAs. 2. (20%) Show that a language is decidable if and only if some enumerator enumerates the language in a way that shorter strings are enumerated first, while for equal-length strings, they are enumerated in lexicographic order. 3. Let S = {h M i | M is a DFA that accepts w whenever it accepts the reverse of
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Unformatted text preview: w } . (a) (5%) Give an example of a DFA that is in S . (b) (20%) Show that S is decidable. 4. (20%) Let PAL DFA = {h M i | M is a DFA that accepts some palindrome } . Show that PAL DFA is decidable. (Hint: Prob 2.18 and Prob 4.23 are helpful here.) 5. (20%) Suppose that we have a decider D that decides if the language of a CFG is infinite. That is, D is a decider for the language: INFINITE CFG = {h G i | G is a CFG and L ( G ) is infinite } . By using D or otherwise, show that the following language: C CFG = {h G,k i | G is a CFG and L ( G ) contains exactly k strings where k ≥ 0 or k = ∞} is decidable. 6. (Further studies: No marks) Let C be a language. Prove that C is Turing-recognizable if and only if a decidable language D exists such that C = { x | ∃ y ( h x,y i ∈ D ) } . 1...
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