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**Formal Models of Computation Part III Computability & Complexity**

Part III-A – Computability Theory

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**Reducibility: Introduction**

Chapter 5 of Sipser’s Book Overview: 2 Examples of proofs using reduction Mapping Reducibility formal models of computation

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**Reducibility: Introduction**

Reducibility: method to prove that problems are computationally unsolvable Reduction: convert problem A into problem B so that B’s solution applies to A OR so that A’s lack of solution applies to B Intuitive example: travelling from ABZ to New York Reduces to: buying a ticket WHICH Reduces to: getting money to buy ticket WHICH Reduces to: getting a Summer job WHICH Reduces to: … Reducibility in maths: Measuring the area of a rectangle reduces to measuring its height and width formal models of computation

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**Reducibility: Introduction (Cont’d)**

Reducibility helps us classifying problems When A is reducible to B Solving A cannot be harder than solving B, because a solution to B gives a solution to A In computability theory: If A is reducible to B and B is decidable, A is also decidable If A is undecidable and reducible to B, B is also undecidable Method to prove that a problem B is undecidable: Show that some other problem A already known to be undecidable reduces to B “If B was decidable then A would also be decidable, and we know that that’s not the case” formal models of computation

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**formal models of computation**

Problem No. 1 Problem: determine if a Turing machine halts (accepting or rejecting) on a given input Let HALTTM = {M, w | M is a TM and M halts on input w } And remember ATM = {M, w | M is a TM that accepts input string w } HALTTM is the real halting problem ATM is called the acceptance problem Theorem: HALTTM is undecidable Proof idea: Reduce ATM to HALTTM then Use undecidability of ATM to prove undecidability of HALTTM formal models of computation

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**formal models of computation**

Problem No. 1 (Cont’d) Proof: Assume (to obtain contradiction) that TM R decides HALTTM We can then construct TM S to decide ATM: S = “On input M, w where M is a TM and w is a string: 1. Run TM R on input M, w . 2. If R rejects, reject. {<M,w> does not halt} 3. If R accepts {<M,w> halts} then simulate M on w until it halts. 4. If M accepts w , accept; If M rejects w , reject.” If R decides HALTTM then S decides ATM Because ATM is undecidable, HALTTM must also be undecidable!! formal models of computation

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**formal models of computation**

Problem No. 1 (Cont’d) ATM consists of two problems: For given <M,w>, Decide whether <M,w> halts. Given R, we can do this! If yes then determine whether M accepts w. (We know a TM exists that recognises ATM , so we can do this too, by constructing a suitable TM.) If no then reject. formal models of computation

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**formal models of computation**

Problem No. 1 (Cont’d) Diagrammatically: R M,w Yes No M halts on w M loops on w S M,w Yes No M accepts w M rejects or loops on w S M,w Yes M accepts w No M rejects or loops on w M accepts w M rejects w simulate M on w R M,w Yes No formal models of computation

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**formal models of computation**

Problem No. 1 (Cont’d) In summary: ATM reduces to HALTTM A TM S would require a TM R if it could be built The added functionality is straightforward Since ATM is undecidable, HALTTM is also undecidable S M,w Yes M accepts w No M rejects or loops on w R M rejects w simulate M on w formal models of computation

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**THE REST OF THIS LECTURE IS OPTIONAL**

formal models of computation

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**formal models of computation**

Problem No. 2 Problem: determine if a Turing machine does not accept any input, that is, its language is empty Let ETM = {M | M is a TM and L(M ) = } Theorem: ETM is undecidable Proof idea: same as before Assume that ETM is decidable and Show that ATM is decidable – a contradiction Let R be a TM that decides ETM We use R to build S that decides ATM formal models of computation

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**formal models of computation**

Problem No. 2 (Cont’d) However, we run R on a modification of M : We modify M to ensure M rejects all strings except w On input w, M works as usual In our notation, the modified machine is The only string M1 may now accept is w M1 language is non-empty if, and only if, it accepts w M1 = “On input x : 1. If x w, reject. 2. If x =w, run M on input w and accept if M does.” formal models of computation

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**formal models of computation**

Problem No. 2 (Cont’d) Proof: We assume TM R decides ETM We then build TM S (using R ) that decides ATM: N.B.: S must be able to compute M1 from M, w Just add extra states to M to perform x =w test The reasoning: If R were a decider for ETM, S would be a decider for ATM A decider for ATM cannot exist Hence, ETM must be undecidable S = “On input M, w where M is a TM and w is a string: 1. Use the description of M and w to built M1 as explained 2. Run R on input M1. 3. If R accepts, reject; if R rejects, accept.” formal models of computation

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**formal models of computation**

Problem No. 2 (Cont’d) Diagrammatically: S M,w Yes No M accepts w M rejects or loops on w R M L(M ) L(M ) = S M,w Yes M accepts w No M rejects or loops on w R No Yes M1 modify M onto M1 M,w formal models of computation

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**formal models of computation**

Mapping Reducibility Let’s now formalise the notion of reducibility There are various alternatives Ours is called mapping reducibility Also called “many-one reducibility” In a nutshell If we can reduce problem A to problem B and We have a solution to problem B Then we have a solution to problem A Diagrammatically: a Reduction f(a) = b b Solver for B output formal models of computation

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**Mapping Reducibility (Cont’d)**

To reduce problem A to problem B via mapping reducibility: We must find a computable function to convert instances of problem A to instances of problem B If we have such function (called a reduction) we can solve A with a solver for B formal models of computation

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**formal models of computation**

Computable Functions A function f : * * is a computable function if some Turing machine M, on every input w, halts with just f (w ) on its tape Example: A TM that takes input m, n and returns m + n formal models of computation

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**Definition of Mapping Reducibility**

Language A is mapping reducible to language B if there is a computable function f : * * such that for every w, w A f (w ) B This is denoted as A m B Function f is called the reduction of A to B Diagrammatically: B f A formal models of computation

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**Mapping Reducibility (Cont’d)**

If problem A is mapping reducible to problem B If problem B has been previously solved, then We can obtain a solution to problem A Theorem: if A m B and B is decidable, then A is decidable Proof: Let M be the decider for B Let f be the reduction from A to B We describe a decider N for A as N = “On input w : 1. Compute f (w ). 2. Run M on input f (w ) and output whatever M outputs.” – Clearly, if w A then f (w ) B as f is a reduction from A to B. – Thus, M accepts f (w ) whenever w A. – Therefore M works as desired. formal models of computation

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**formal models of computation**

Revisiting an Example Let’s revisit our example and prove that HALTTM = {M, w | M is a TM and M halts on input w } is undecidable. We shall do this using mapping reducibility We give a reduction f from ATM to HALTTM A computable function (i.e., a TM!) that takes as input M, w and returns as output M, w where M, w ATM if and only if M, w HALTTM The following machine F computes a reduction f S = “On input M,w : 1. Construct the following machine M : M = “On input x : 1. Run M on x 2. If M accepts, accept. 3. If M rejects, enter a loop.” 2. Output M,w ” formal models of computation

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**Mapping Reducibility (Cont’d)**

Corollary: if A m B and A is undecidable, then B is undecidable Theorem: if A m B and B is Turing-recognisable, then A is Turing-recognisable if A m B and A is not Turing-recognisable, then B is not Turing-recognisable formal models of computation

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**formal models of computation**

The end ... In the course , this is the end of the material for the CS4026 exam Given more time, we would have shown that the method of reduction is also applicable to proofs concerning the complexity of a problem. A typical reasoning pattern: “If problem A was solvable within time limitations X then B would also be solvable within X. We know (or believe) that B is not solvable within X, therefore A is not either” formal models of computation

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**formal models of computation**

Reading List Introduction to the Theory of Computation. Michael Sipser. PWS Publishing Co., USA, (A 2nd Edition has recently been published). Chapter 5. Algorithmics: The Spirit of Computing. 3rd Edition. David Harel with Yishai Feldman. Addison-Wesley, USA, Chapter 8. formal models of computation

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