# fall 2004comp 3351 reducibility. fall 2004comp 3352 problem is reduced to problem if we can solve...

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• Slide 1
• Fall 2004COMP 3351 Reducibility
• Slide 2
• Fall 2004COMP 3352 Problem is reduced to problem If we can solve problem then we can solve problem
• Slide 3
• Fall 2004COMP 3353 If is undecidable then is undecidable If is decidable then is decidable Problem is reduced to problem
• Slide 4
• Fall 2004COMP 3354 Example: the halting problem is reduced to the state-entry problem
• Slide 5
• Fall 2004COMP 3355 The state-entry problem Inputs: Turing Machine State Question: Does String enter state on input ?
• Slide 6
• Fall 2004COMP 3356 Theorem: The state-entry problem is undecidable Proof: Reduce the halting problem to the state-entry problem
• Slide 7
• Fall 2004COMP 3357 state-entry problem decider YES NO enters doesnt enter Suppose we have a Decider for the state-entry algorithm:
• Slide 8
• Fall 2004COMP 3358 Halting problem decider YES NO halts on doesnt halt on We want to build a decider for the halting problem:
• Slide 9
• Fall 2004COMP 3359 State-entry problem decider Halting problem decider YES NO YES NO We want to reduce the halting problem to the state-entry problem:
• Slide 10
• Fall 2004COMP 33510 Halting problem decider YES NO YES NO Convert Inputs ? We need to convert one problem instance to the other problem instance State-entry problem decider
• Slide 11
• Fall 2004COMP 33511 Convert to : Add new state From any halting state of add transitions to halting states Single halt state
• Slide 12
• Fall 2004COMP 33512 halts on input halts on state on input if and only if
• Slide 13
• Fall 2004COMP 33513 Generate Halting problem decider YES NO YES NO State-entry problem decider
• Slide 14
• Fall 2004COMP 33514 Since the halting problem is undecidable, the state-entry problem is undecidable END OF PROOF We reduced the halting problem to the state-entry problem
• Slide 15
• Fall 2004COMP 33515 Another example: the halting problem is reduced to the blank-tape halting problem
• Slide 16
• Fall 2004COMP 33516 The blank-tape halting problem Input:Turing Machine Question:Doeshalt when started with a blank tape?
• Slide 17
• Fall 2004COMP 33517 Theorem: Proof: Reduce the halting problem to the blank-tape halting problem The blank-tape halting problem is undecidable
• Slide 18
• Fall 2004COMP 33518 blank-tape halting problem decider YES NO halts on blank tape doesnt halt on blank tape Suppose we have a decider for the blank-tape halting problem:
• Slide 19
• Fall 2004COMP 33519 halting problem decider YES NO halts on doesnt halt on We want to build a decider for the halting problem:
• Slide 20
• Fall 2004COMP 33520 Blank-tape halting problem decider Halting problem decider YES NO YES NO We want to reduce the halting problem to the blank-tape halting problem:
• Slide 21
• Fall 2004COMP 33521 Blank-tape halting problem decider Halting problem decider YES NO YES NO We need to convert one problem instance to the other problem instance Convert Inputs ?
• Slide 22
• Fall 2004COMP 33522 Construct a new machine When started on blank tape, writes Then continues execution like then write step 1step2 if blank tapeexecute with input
• Slide 23
• Fall 2004COMP 33523 halts on input string halts when started with blank tape if and only if
• Slide 24
• Fall 2004COMP 33524 Generate blank-tape halting problem decider Halting problem decider YES NO YES NO
• Slide 25
• Fall 2004COMP 33525 Since the halting problem is undecidable, the blank-tape halting problem is undecidable END OF PROOF We reduced the halting problem to the blank-tape halting problem
• Slide 26
• Fall 2004COMP 33526 Does machine halt on input ? Summary of Undecidable Problems Halting Problem: Membership problem: Does machine accept string ?
• Slide 27
• Fall 2004COMP 33527 Does machine enter state on input ? Does machine halt when starting on blank tape? Blank-tape halting problem: State-entry Problem:
• Slide 28
• Fall 2004COMP 33528 Uncomputable Functions
• Slide 29
• Fall 2004COMP 33529 Uncomputable Functions A function is uncomputable if it cannot be computed for all of its domain Domain Values region
• Slide 30
• Fall 2004COMP 33530 An uncomputable function: maximum number of moves until any Turing machine with states halts when started with the blank tape
• Slide 31
• Fall 2004COMP 33531 Theorem:Function is uncomputable Proof: Then the blank-tape halting problem is decidable Assume for contradiction that is computable
• Slide 32
• Fall 2004COMP 33532 Decider for blank-tape halting problem: Input: machine 1. Count states of : 2. Compute 3. Simulate for steps starting with empty tape If halts then return YES otherwise return NO
• Slide 33
• Fall 2004COMP 33533 Therefore, the blank-tape halting problem is decidable However, the blank-tape halting problem is undecidable Contradiction!!!
• Slide 34
• Fall 2004COMP 33534 Therefore, function in uncomputable END OF PROOF
• Slide 35
• Fall 2004COMP 33535 Undecidable Problems for Recursively Enumerable Languages
• Slide 36
• Fall 2004COMP 33536 is empty? is finite? contains two different strings of the same length? Take a recursively enumerable language Decision problems: All these problems are undecidable
• Slide 37
• Fall 2004COMP 33537 Theorem: For any recursively enumerable language it is undecidable to determine whether is empty Proof: We will reduce the membership problem to this problem
• Slide 38
• Fall 2004COMP 33538 empty language problem decider YES NO Suppose we have a decider for the empty language problem: Let be the TM with empty not empty
• Slide 39
• Fall 2004COMP 33539 membership problem decider YES NO accepts rejects We will build the decider for the membership problem:
• Slide 40
• Fall 2004COMP 33540 YES NO YES Membership problem decider empty language problem decider We want to reduce the membership problem to the empty language problem:
• Slide 41
• Fall 2004COMP 33541 YES NO YES Membership problem decider empty language problem decider We need to convert one problem instance to the other problem instance Convert inputs ?
• Slide 42
• Fall 2004COMP 33542 Construct machine : When enters a final state, compare with Accept only if executes the same as with On arbitrary input string
• Slide 43
• Fall 2004COMP 33543 is not empty if and only if
• Slide 44
• Fall 2004COMP 33544 construct YES NO YES Membership problem decider END OF PROOF empty language problem decider

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