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Unformatted text preview: CS 181 — Winter 2008 Formal Languages and Automata Theory Problem Set #8 Solutions Problem 8.1. (10 points) Show that P is closed under union, concatenation, and star. Solution Suppose A,B are two languages in P. Then let M A be the polynomial time Turing-decider for A and M B the same for B . The M A has some asymptotic runtime O ( n k ) and M B has runtime O ( n ‘ ) for some integers k and ‘ . Without loss of generality, we assume that k ≥ ‘ . We first show that A ∪ B ∈ P. We can build the following Turing decider M 1 for A ∪ B . M 1 first runs M A on its input, if M A accepts, then M 1 accepts. If M A rejects, then M 1 acts as M B on the same input. Clearly, this machine decides A ∪ B . Let us now analyze the running time of M 1 . Notice that for every input word, in the worst case we run M A then M B . Then this takes O ( n k + n ‘ ) = O ( n max( k,‘ ) ) = O ( n k ). Since k is a constant, it follows that M 1 runs in polynomial time. The existence of M 1 proves that A ∪ B ∈ P. Next, we show that AB ∈ P by building the Turing decider M 2 for it. Notice that given an input string of length n , there are n +1 ways to partition it into two substrings (e.g. 123 can be partitioned as ε · 123 , 1 · 23 , 12 · 3 or 123 · ε ). M 2 simply tries each partitioning one at a time. For each partitioning, it runs M A on the left half of the string then M B on the right half. If both machines accept, then M 2 halts and accepts. If one of them rejects, then M 2 tries the next partitioning. If all partitionings fail, then M 2 rejects. Let us now analyze the running time of M 2 . Notice that for each partitioning, we run M A on a string of length at most n and M B on a string of length at most n . Thus, for each partioning M 2 takes time O ( n max( k,‘ ) ) = O ( n k ). However, there are n + 1 partitionings. Thus, in total, M 2 takes O ( n k +1 ) time. This is still polynomial and so AB ∈ P. Lastly, we show that A * ∈ P by building a Turing decider M 3 for it. M 3 will work by using dynamic programming. Notice that a string s is in A * if s ∈ A or if s can be written s = s 1 s 2 where s 1 ,s 2 ∈ A * and s 1 ,s 2 are non-empty (and so are proper substrings of s ). Then for input string s , we can first calculate which length 1 substrings are in A * (there are n of them). Then using this information, we can calculate which of the length 2 substrings are in A * (all n- 1 of them) by checking to see if they are in A , and otherwise checking to see if we can partition it into substrings in A * . We then continue this for the n- 2 length-3 substrings, the n- 3 length-4 substrings, etc. This continues until we can calculate whether the length n substring (namely, s itself) is in A * ....
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- Spring '08