Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central University 1 Chapter 5 Reducibility Some slides are in courtesy of Prof. Verma at Univ. of Houston, Max Alekseyev at Univ. of South Carolina Spring 2016 CISG 5115 Theory of Computation

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Overview In Chapter 3 and Chapter 4, established a general purpose computer model, TM, and presented several TM solvable problems and A TM, a computationally unsolvable problem. This chapter introduces additional unsolvable problems. The primary method for proving that problems are computationally unsolvable is called reducibility. 2

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Reducibility Informally speaking, “a problem A is reduced to a problem B ” means that if we can solve B then we can solve A if we cannot solve A then we cannot solve B the problem A is not “harder” than the problem B A at most as “hard” as B and possibly even “simpler” than B B is at least as “hard” as A and possibly even “harder” than A Mostly we will use reduction to prove something is “hard”. Q: What needs to be done to show that problem T is “hard”? To prove that T is “hard” we need to find such S that ( i ) S is known to be hard ( ii ) S is reduced to T. 3

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Reducibility – cont’ In our current language, hardness is measured in terms of decidability: from decidable (“simple”) to undecidable (“hard”). Correspondingly, solution is a decider. We can solve “simple” problems but not “hard” ones. A reduction is a way of converting one problem to another problem in such a way that a solution to the second problem can be used to solve the first problem. To prove a problem is undecidable, we need to show that some other known undecidable problem reduces to it. 4

Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central University 5 Chapter 5.1 Undecidable Problems from Language Theory Some slides are in courtesy of Prof. Verma at Univ. of Houston, Max Alekseyev at Univ. of South Carolina Spring 2016 CISG 5115 Theory of Computation

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Undecidable Problems from Language Theory 6 A TM is undecidable->Halt TM is undecidable  Halt TM is decidable -> A TM is decidable

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Undecidable Problems from Language Theory – cont’ 7 A TM is undecidable->E TM is undecidable  E TM is decidable -> A TM is decidable

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Undecidable Problems from Language Theory – cont’ 8

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Undecidable Problems from Language Theory – cont’ 9

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Reduction via Computation History A computation history of a TM M is a sequence of its configurations C 1, C 2, …, C l such that C l+1 legally follows from C l according to the rules of M. A computation history is accepting/rejecting if C l is accepting or rejecting configuration respectively. In general, for a TM there may exist an infinite number of configurations (because of an infinite tape). However, for TMs that use only limited amount of the tape, the number of configurations is finite. A linear bounded automaton (LBA) is a TM that is permitted to use only portion of the tape containing the input (i.e., cannot move the head beyond the input). 10

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University A LBA is decidable Lemma: Let M be an LBA with q states and g symbols in the tape alphabet. Then there are exactly qng n distinct configurations of M for the tape of length n. Lemma: Let M be an LBA with q states and g symbols in the tape alphabet. If M on input of length n does not produce an output within first qng n steps, it loops. Theorem A LBA = { | M is a LBA and M accepts w } is decidable The algorithm (TM) L that decides A LBA is as follows. On input, where M is an LBA and w is a string, Simulate M on w for qng n steps or until it halts. If M has halted, accept if it has accepted and reject if it has rejected. If it has not halted, reject. 11

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University E LBA is undecidable 12

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University E LBA is undecidable – cont’ 13

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University E LBA is undecidable – cont’ 14

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Computable Functions 15

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Computable Functions – Example 16

Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central University 17 Chapter 5.3 Mapping Reducibility Some slides are in courtesy of Prof. Verma at Univ. of Houston, Max Alekseyev at Univ. of South Carolina Spring 2016 CISG 5115 Theory of Computation

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Mapping Reducibility 18

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Mapping Reducibility – cont’ 19

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Mapping Reducibility – cont’ 20

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University 21

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University 22

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University 23

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University EQ TM = { | M 1, M 2 are TMs and L ( M 1 )= L ( M 2 )} Theorem: EQ TM is neither Turing-recognizable nor co- Turing-recognizable. Q: What would be a high-level outline of the proof? To prove that EQ TM not Turing-recognizable, we need to show that To prove that not Turing-recognizable, we need to show that 24

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University To show that, we need to need to construct a computable to such that Define a TM F that on input does the following: 1.Construct the following TMs M 1 and M 2 : M 1 rejects any input. M 2 ignores its input and run M on w. If it accepts, accept. 2.Output 25

Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central University 26 Chapter 6.3 Turing Reducibility Some slides are in courtesy of Prof. Verma at Univ. of Houston, Max Alekseyev at Univ. of South Carolina Spring 2016 CISG 5115 Theory of Computation

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Oracle Turing Machine An oracle for a language B is an external device that is capable of reporting whether any string w is a member of B. An oracle Turing machine is a modified Turing machine that has the additional capability of querying an oracle. We write M B to describe an oracle Turing machine that has an oracle for language B. 27

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Oracle Turing Machine – cont’ 28

Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Turing Reducibility 29