November 3

November 3 - CSCI 2670 Introduction to Theory of Computing...

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CSCI 2670 Introduction to Theory of Computing November 3, 2005
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November 3, 2005 Agenda •Yester day – Reductions (Section 5.3) •Today – More on reductions – Section 6.3 – We will not be covering section 6.4 • I will discuss some basic issues of this section when covering chapter 7
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November 3, 2005 Announcement • Remember to let me know if you want to come to my pizza party next Wednesday
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November 3, 2005 Mapping reducibility Definition: Language A is mapping reducible to language B, written A m B, if there is a computable function f: * * , where for every w, w A iff f(w) B f f
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November 3, 2005 Why m ? • In some sense, if A m B, then A is “less powerful” than B – For example, we can map from CFG’s to decidable languages that aren’t CF, but not vice versa •Examp le – We can easily map any CFG C to A CFG • f(w) = <C,w> •w C iff <C,w> A CFG • You can use this example to help you remember how to use reductions
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November 3, 2005 Mapping reductions & decidability Theorem: If A m B and B is decidable, then A is decidable. Proof: Let M be a decider for B and let f be a reduction from A to B. Consider the following TM, N: N = “On input w: 1. Compute f(w) 2. Run M on f(w) and report M’s output Then N decides A
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November 3, 2005 Example • Let EV = {<A>| A is a DFA all strings
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November 3 - CSCI 2670 Introduction to Theory of Computing...

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