Menu. cs3102: Theory of Computation. Class 17: Undecidable Languages. Alternate Proof Input. Alternate Proof Input. Contradiction! Contradiction!

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1 cs3102: Theory of Computation Class 17: Undecidable Languages Spring 2010 University of Virginia David Evans Menu Another SELF-REJECTINGargument: diagonalization A language that is Turing-recognizable but not Turing-decidable Monday, March 29 3:30pm in MEC 205 Ed Clarke, 2007 Turing Award Thursday, April 1 2:00pm in Chemistry Barbara Liskov, 2008 Turing Award Yes? Contradiction! No? Contradiction! The assumption leads to a contradiction: thus, M SR must not exist! Machine Alternate Proof Input ε M(ε) M(0) M(1) M(00) M(01) M(10) M(11) M(000) M(w) Which of the machines are in SELF-REJECTING? Machine Alternate Proof Input ε M(ε) M(0) M(1) M(00) M(01) M(10) M(11) M(000) M(w) Where is w SR? Languages that can be recognized by any mechanical computing machine SELF-REJECTING All Languages

2 Recognizing vs. Deciding Turing-Recognizable Turing-Decidable Context-Free SELF-REJECTING All Languages Turing-recognizable: A language L is Turingrecognizable if there exists a TM Msuch that for all strings w: If w L: eventually Menters q accept. If w L: either Menters q reject or Mnever 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 Menters q accept. If w L: eventually Menters q reject. Detour: Exam Revisions Proof that SF is not CFL Contradiction means one of the two assumptions must be false, but we don t know which! Is SF Context-Free? Squarefree Sequences in {a, b, c}* There areinfinitely long squarefreesequences with at least 3 alphabet symbols Some interesting applications and lots of interesting efficient ways to generate them Ron Rivest s paper If you solved PS4 question 2 do you know an inefficient way?

3 Proving Recognizability Accepted by TM How do we prove a language is Turing-recognizable? How do we prove a language is Turing-decidable? Is this language Turing-recognizable? How do we prove a language is not Turing-decidable? Accepted by TM Universal Turing Machine Can we really do this? Is this language Turing-recognizable? <M> w Universal Turing Machine Output of running M starting on tape w Universal Turing Machine: a TM that can simulate every other TM. 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) Manchester Illuminated Universal Turing Machine, #9 from

4 2-state, 3-symbol Universal TM Sequence of configurations Alex Smith, University of Birmingham 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. Rough Sketch of Proof Accepted by TM None of these steps involve universal computation themselves 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. Is this language Turing-decidable? See for the 40-page version with all the details Proof that A TM is Undecidable Proof that A TM is Undecidable

5 Proof that ATM is Undecidable Proof that ATM is Undecidable Both are contractions! So, D must not exist. But, if H exists, we can make D. So, H must not exist! But, if ATM is decidable, H must exist. Thus, ATM must not be decidable. Halting Problem Turing-Recognizable Turing-Decidable Context-Free ATM SELF-REJECTING All Languages Halting Problem is Undecidable Halting Problem is Undecidable

6 HALTSANY Crashes Any equivalent to a TM enters some bad state Model Checking in Theory Model Checking is Undecidable. Impossible to write a program that answers this correctly for all inputs. Edmund M. Clarke, The Birth of Model Checking Model Checking in Practice Monday s Talk MEC 205, 3:30pm (cookies after talk) Model Checking: My 27 year Quest to Overcome the State Explosion Problem Edmund Clarke 2007 Turing Award Winner (with Allen Emerson, Joseph Sifakis)

7 Return PS4 and Exam Revisions