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HW5ans - Farley Lai,00764474 TheoryofComputation,Homework5...

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Farley Lai, 00764474 Theory of Computation, Homework 5 7.6 i.) union Without loss of generality, let and be the TMs that decide two languages and in M 1 M 2 L 1 L 2 P. Then we construct a TM that decides the union of and in polynomial time. M L 1 L 2 ”on input M = w 1. Run on . Accept if it accepts. M 1 w 2. Run on . Accept if it accepts. M 2 w 3. Otherwise, reject.” Since stage1 and stage2 both run in polynomial time, TM can decide the union of and M L 1 L 2 in polynomial time. Hence, is closed under union. P ii.) concatenation Without loss of generality, let and be the TMs that decide two languages and in M 1 M 2 L 1 L 2 P. Then we construct a TM that decides the concatenation of and in polynomial time. M L 1 L 2 ”on input M = w 1. For each possible split of w = w w 1 2 a. Run on and run on . Accept if both accept. M 1 w 1 M 2 w 2 2. Otherwise, reject.” Assume the . There are possible splits of at most. Besides, the sum of the time w j j = n ( n ) O w complexities of and is also polynomial. Thus, step1 run in polynomial time and TM M 1 M 2 M can decide the concatenation of and in polynomial time. Hence, is closed under L 1 L 2 P concatenation. iii.) complement Without loss of generality, let be the TM that decides a language in P. Then we construct M L a TM that decides the complement of in polynomial time. M 0 L ”on input M 0 = w 1. Run on M w 2. Accept if it rejects. Otherwise, reject if it accepts.” Since step1 run in polynomial time, TM can decide the complement of in polynomial time. M 0 L Hence, is closed under complement. P
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7.7 i.) union Without loss of generality, let and be the NTMs that decide two languages and N 1 N 2 L 1 L 2 in NP. Then we construct an NTM that decides the union of and in polynomial time. N L 1 L 2 ”on input N = w 1. Run on nondeterministically. Accept if it accepts. N 1 w 2. Run on nondeterministically. Accept if it accepts. N 2 w 3. Otherwise, reject.” Since stage1 and stage2 both run in polynomial time nondeterministically, NTM can decide N the union of and in polynomial time. Hence, is closed under union. L 1 L 2 P N ii.) concatenation Without loss of generality, let and be the NTMs that decide two languages and N 1 N 2 L 1 L 2 in NP. Then we construct an NTM that decides the concatenation of and in polynomial N L 1 L 2 time.
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