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# Lec16 - S 1 and S 2 S 1 ∩ S 2 = Ø S 1 ∪ S 2 = S such...

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Section 5.3 Mapping Reducibility computable function : f there exists a Turing Machine M , that is given w as input, it can produce f ( w ) on the tape, and halt for all the strings w in the domain of f A ≤ m B (mapping reduction) iff for w Є A, f ( w ) Є B and there exists a f that is a computable function 1. Assume we have a solution to B 2. Goal to solve A Solve A Given input w, 1. Compute f ( w ) – this can be done because f is a computable function 2. Run the solution to B on f ( w ) as input and output whatever it does subset sum – give a set, S, of integers, and a target, T, determine if there exists a subset of values S that sums to T partition – given a set, S , of integers, is there a way to partition that set into two sets
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Unformatted text preview: S 1 and S 2 ( S 1 ∩ S 2 = Ø, S 1 ∪ S 2 = S ) such that the sum of elements in S 1 equals the sum of elements in S 2 f ( S : set of integers) (f is a function that does computation) sum = sum of the values in S If sum is odd, output: S = {}, T = 1 sum = sum/2 output S as your set T = sum/2 partition( S ) S = S’ { compute f ( S ) output = S’ , T return SubsetSum( S’ , T ) } If A ≤ m B and B is decidable, then A is decidable If A ≤ m B and A is undecidable, then B is undecidable If A ≤ m B and B is Turing-recognizable, then so is A If A ≤ m B and A is not Turing-recognizable, then B isn’t either...
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