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)