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

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

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

Alternate Proof  … 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 ?

Alternate Proof  … 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 ?

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

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

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.

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 are infinitely long squarefree sequences 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?

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?

Accepted by TM Is this language Turing-recognizable?

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

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

Manchester Illuminated Universal Turing Machine, #9 from

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)

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

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

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 for the 40-page version with all the details… See for the 40-page version with all the details… None of these steps involve universal computation themselves

Accepted by TM Is this language Turing-decidable?

Proof that A TM 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 A TM is decidable, H must exist. Thus, A TM must not be decidable.

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

Halting Problem

Halting Problem is Undecidable

HALTSANY

Crashes Any equivalent to a TM enters some bad state

Edmund M. Clarke, The Birth of Model Checking

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

Model Checking in Practice

Monday’s 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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