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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