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cs3102: Theory of Computation Class 17: Undecidable Languages Spring 2010 University of Virginia David Evans

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Menu Another S ELF -R EJECTING argument: diagonalization A language that is Turing-recognizable but not Turing-decidable Ed Clarke, 2007 Turing Award Barbara Liskov, 2008 Turing Award Monday, March 29 3:30pm in MEC 205 Thursday, April 1 2:00pm in Chemistry

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Yes? Contradiction! No? Contradiction! The assumption leads to a contradiction: thus, M SR must not exist!

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Alternate Proof 0100011011000001010 M()M() M(0) M(1) M(00) M(01) M(10) M(11) M(000) M(w)M(w) Input Machine Which of the machines are in S ELF -R EJECTING ?

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Alternate Proof 0100011011000001010 M()M() M(0) M(1) M(00) M(01) M(10) M(11) M(000) M(w)M(w) Input Machine Where is w SR ?

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s Languages that can be recognized by any mechanical computing machine All Languages S ELF -R EJECTING

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s Turing-Recognizable All Languages S ELF -R EJECTING Turing-Decidable Context-Free

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Recognizing vs. Deciding Turing-recognizable: A language L is Turing- recognizable if there exists a TM M such that for all strings w : If w L : eventually M enters q accept. If w L : either M enters q reject or M never terminates. Turing-decidable: A language L is Turing-decidable if there exists a TM M such that for all strings w : If w L : eventually M enters q accept. If w L : eventually M enters q reject.

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Detour: Exam Revisions

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Proof that SF is not CFL Contradiction means one of the two assumptions must be false, but we dont know which!

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Is SF Context-Free?

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Squarefree Sequences in {a, b, c}* There are infinitely long squarefree sequences with at least 3 alphabet symbols Some interesting applications and lots of interesting efficient ways to generate them Ron Rivests paper If you solved PS4 question 2 do you know an inefficient way?

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Proving Recognizability How do we prove a language is Turing-recognizable? How do we prove a language is Turing-decidable? How do we prove a language is not Turing-decidable?

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Accepted by TM Is this language Turing-recognizable?

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Accepted by TM Is this language Turing-recognizable? Can we really do this? Universal Turing Machine: a TM that can simulate every other TM.

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Universal Turing Machine Universal Turing Machine w Output of running M starting on tape w

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Manchester Illuminated Universal Turing Machine, #9 from http://www.verostko.com/manchester/manchester.html

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Universal Turing Machines Universal Turing Machines designed with: 4 symbols, 7 states (Marvin Minsky) 4 symbols, 5 states 2 symbols, 22 states 18 symbols, 2 states 2 states, 5 symbols (Stephen Wolfram)

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2-state, 3-symbol Universal TM Sequence of configurations

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Of course, simplicity is in the eye of the beholder. The 2,3 Turing machine described in the dense new 40-page proof chews up a lot of tape to perform even a simple job, Smith says. Programming it to calculate 2 + 2, he notes, would take up more memory than any known computer contains. And image processing? It probably wouldn't finish before the end of the universe, he says. Alex Smith, University of Birmingham

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Rough Sketch of Proof System 0 (the claimed UTM) can simulate System 1 which can simulate System 2 which can simulate System 3 which can simulate System 4 which can simulate System 5 which can simulate any 2-color cyclic tag system which can simulate any TM. See http://www.wolframscience.com/prizes/tm23/TM23Proof.pdfhttp://www.wolframscience.com/prizes/tm23/TM23Proof.pdf for the 40-page version with all the details See http://www.wolframscience.com/prizes/tm23/TM23Proof.pdfhttp://www.wolframscience.com/prizes/tm23/TM23Proof.pdf for the 40-page version with all the details None of these steps involve universal computation themselves

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Accepted by TM Is this language Turing-decidable?

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Proof that A TM is Undecidable

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Both are contractions! So, D must not exist. But, if H exists, we can make D. So, H must not exist! But, if A TM is decidable, H must exist. Thus, A TM must not be decidable.

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s Turing-Recognizable All Languages S ELF -R EJECTING Turing-Decidable Context-Free A TM

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Halting Problem

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Halting Problem is Undecidable

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HALTSANY

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Crashes Any equivalent to a TM enters some bad state

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Edmund M. Clarke, The Birth of Model Checking

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Model Checking in Theory Model Checking is Undecidable. Impossible to write a program that answers this correctly for all inputs.

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Model Checking in Practice

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Mondays Talk Model Checking: My 27 year Quest to Overcome the State Explosion Problem MEC 205, 3:30pm (cookies after talk) Edmund Clarke 2007 Turing Award Winner (with Allen Emerson, Joseph Sifakis)

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Return PS4 and Exam Revisions