49+Slides--Complexity%2C+the+Class+P

49+Slides--Complexity%2C+the+Class+P - CS103 HO#49...

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CS103 HO#49 Slides--Complexity, the class P 5/20/11 1 maps to A B ¬ Decidable(A) ¬ Decidable(B) A language known not to be decidable A language whose decidability we are trying to determine If B decidable Then A decidable But A not decidable So B not decidable A m B Using Mapping Reductions: The Most Common Scenario Using Mapping Reductions: A Less Common Scenario maps to A B A language known to be decidable A language whose decidability we are trying to determine A m B Decidable(B) Decidable(A) If B decidable Then A decidable Left TM = { M, w | M on w tries to move its head left from the leftmost tape cell at some point in its computation } f M, w Decider for Left TM Accept Decider for A TM M', w' Reject If Left TM is decidable and we can produce a mapping function f that shows A TM m Left TM , we could build a decider for A TM . So, we can show Left TM is undecidable by defining a mapping that would work. The mapping f uses M, w to build M' and w' as follows: M' : M' first moves its input over one square to the right and writes a new symbol $ on the leftmost square. Then M' simulates M on w'. If M' sees $, it moves right and stays in the same state. If M accepts, M' moves all the way to the left and then tries to move left. w' = w For improper inputs, f outputs M', w'  Left TM . So M accepts w if and only if M' tries to move its head left from the leftmost cell. Left TM = { M, w | M on w tries to move its head left from the leftmost tape cell at some point in its computation } f M, w Decider for Left TM Accept Decider for A TM M', w' Reject Prove that A is decidable if and only if A m 0*1*. Proof. ( direction) Suppose that A is decidable. Then we can reduce A to 0*1* with the function computed by the following Turing machine F: F = "On input w: 1. Run a decider for A on w to determine whether w A. 2. If w A, output 01. 3. If w A, output 10." F w Decider for 0*1* Accept Decider for A 01 or 10 Reject Prove that A is decidable if and only if A m 0*1*. Proof. ( direction) Suppose that A is decidable. Then we can reduce A to 0*1* with the function computed by the following Turing machine F: F = "On input w: 1. Run a decider for A on w to determine whether w A. 2. If w A, output 01. 3. If w A, output 10." w A if and only if f(w) 0*1* A m 0*1* means there exists a computable function f such that
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CS103 HO#49 Slides--Complexity, the class P 5/20/11 2 Complexity What makes some problems computationally hard and others easy? We do not have an answer to this question.
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49+Slides--Complexity%2C+the+Class+P - CS103 HO#49...

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