38.pdf - CMPSCI 250 Introduction to Computation Lecture#38...

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
CMPSCI 250: Introduction to Computation Lecture #38: Turing Machine Semantics David Mix Barrington 8 December 2017
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Turing Machine Semantics Review: A Turing Machine Example Turing Recognizable Languages Turing Decidable Languages The TR/TD Theorem Running Two Machines in Parallel Multitape Turing Machines Nondeterministic Turing Machines
Image of page 2
A Turing Machine Example Here is a machine that solves a problem that a DFA cannot. When started in configuration i w 1 w 2 ...w n , it will halt if and only if w is in the language {a n b n : n 0} -- otherwise it will hang. With input aabb we get i aabb, paabb, ☐☐ qabb, ☐☐ aqbb, ☐☐ abqb, ☐☐ abbq , ☐☐ abrb, ☐☐ asb, ☐☐ sab, s ab, ☐☐ pab, ☐☐☐ qb, ☐☐☐ bq , ☐☐☐ rb, ☐☐ s , ☐☐☐ p , ☐☐☐ h . The string aabb is accepted.
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
A Turing Machine Example In i: Move R and go to p. In p: On , go to h. On b, move L and go to z. On a, print , move R, and go to q. In q: On a or b, move R and stay in q. On , move L and go to r. In r: On a or , move L and go to z. On b, print , move L, and go to s. In s: On a or b, move L and stay in s. On , move R and go to p. In h: Halt (final state). In z: Move left and stay in z.
Image of page 4
The Church-Turing Thesis The Church-Turing Thesis says that any “reasonable” general-purpose model of computation will be able to compute exactly the same functions from strings to strings as Turing machines or the lambda calculus. (More precisely, they compute the same set of partial functions , because a general computation always has the possibility of not returning an output.)
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The Church-Turing Thesis We can’t mathematically prove this thesis, only amass evidence for it. In fact it actually serves as an implicit definition of “reasonable”. Serious people have argued against the thesis -- for example physicist Roger Penrose argues that quantum effects in the brain compute in ways that a Turing machine could not. (He’s wrong.) For more on this see Turing’s article On Minds and Machines or almost anything by Douglas Hofstadter.
Image of page 6
The Church-Turing Thesis You probably believe that we could simulate a Turing machine in Java, given unlimited memory. Could a Turing machine simulate any Java program? We know that Java can be compiled into machine language, so we would have to believe that any machine language program could be simulated by a TM.
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Turing Recognizable Languages We’ve seen that a Turing machine, when started on a string, may or may not ever reach a final state. We define the language of the machine M , called L(M), to be the set of strings on which M eventually halts. If a language X is equal to L(M) for some Turing machine M, we say that X is Turing recognizable . The idea is that the machine “recognizes” strings in the language by halting, but gives no output on strings not in the language.
Image of page 8
Turing Recognizable Languages In the example earlier, the language of our TM is {a n b n : n 0}, a language that we now know is not regular.
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern