Giorgi Japaridze Theory of Computability. Reducibility. Chapter 5. 5.1.a. Giorgi Japaridze Theory of Computability. The undecidability of the halting problem. Let HALT TM = {<M,w> | M is a TM and M halts on input w} HALT TM is called the halting problem . .

ByOverview of the theory of computation. Episode 3. Turing machines The traditional concepts of computability, decidability and recursive enumerability The limitations of the power of Turing machines The Church-Turing thesis Mapping reducibilty Turing reducibility

ByHalting Problem. Introduction to Computing Science and Programming I. Alan Turing. Alan Turing 1912-1954 “Father” of modern computing science 1936 Turing Machine Church-Turing thesis Halting Problem 1950: Turing Test. Turing Machine.

By15-453. FORMAL LANGUAGES, AUTOMATA, AND COMPUTABILITY. * Read chapter 4 of the book for next time * . Lecture9x.ppt. REVIEW. A Turing Machine is represented by a 7-tuple T = (Q, Σ , Γ , , q 0 , q accept , q reject ): . Q is a finite set of states.

ByCSE115/ENGR160 Discrete Mathematics 03/10/11. Ming-Hsuan Yang UC Merced. 3.3 Complexity of algorithms. Algorithm Produce correct answer Efficient Efficiency Execution time (time complexity) Memory (space complexity) Space complexity is related to data structure . Time complexity.

ByUndecidable Problems (unsolvable problems). Decidable Languages. Recall that: A language is decidable , if there is a Turing machine ( decider ) that accepts the language and halts on every input string. Decision On Halt:. Turing Machine. YES. Accept.

ByUndecidable Problems (unsolvable problems). Decidable Languages. Recall that: A language is decidable , if there is a Turing machine ( decider ) that accepts the language and halts on every input string. Decision On Halt:. Turing Machine. YES. Accept.

ByReducibility. Sipser 5.1 (pages 187-198). Reducibility. Driving directions. Boston. Cambridge. Western Mass. If you can’t drive to London…. If something’s impossible…. Theorem 4.11: A TM = { < M,w > | M is a TM and M accepts w } is undecidable . Define: HALT TM =

ByProjects Schedule. By Tuesday, October 18: Project “abstract” due By Thursday October 20: Feedback from me Week of October 24: Present project abstract to class Month of November: Time in class for help on projects December 9: Final paper due. Brainstorming on projects. Computation.

ByDecidability in One Day – or maybe two (I can’t decide). CS 331, Tandy Warnow Some slides by Luay Nakhleh , Rice University. Outline. Countability and uncountability What is decidability? There has to be a language that is not decidable An example of a language that is not decidable

ByNP – HARD . JAYASRI JETTI CHINMAYA KRISHNA SURYADEVARA. P and NP. P – The set of all problems solvable in polynomial time by a deterministic Turing Machine (DTM). Example: Sorting and searching. P and NP.

ByNP-Hard. Nattee Niparnan. Easy & Hard Problem. What is “difficulty” of problem? Difficult for computer scientist to derive algorithm for the problem? Difficult for computer to solve (run the derived algorithm) the problem?. Basic Intuition.

ByThe halting problem - proof. Review. What makes a problem decidable ? 3 properties of an efficient algorithm? What is the meaning of “ complete ”, “ mechanistic ”, and “ deterministic ”? Is the Halting Problem decidable/undecidable? How would you define the Universal Turing Machine ?.

ByThe Halting Problem. Sipser 4.2 (pages 173-182). Taking stock. All languages. Turing-recognizable. ?. D. Turing-decidable. Context-free languages. a n b n c n. Regular languages. 0 n 1 n. 0 * 1 *. Are there problems a computer can’t solve?!. But they seem so powerful…

By600.103 Solutions to Homework #1. Kenneth.Church@jhu.edu. Sqrt (4) . Newton’s Method. x1 > 0 2 x1 < 0 -2 x1 = 0 NaN. R & Fib. F(15)=610. Halting Problem. What does this have to do with computability? The Halting Problem demonstrates that you cannot compute everything

ByProgram Slicing for Refactoring. Advanced SW Tools Seminar. Jan 2005 Yossi Peery. Agenda. Slicing Overview Slicing Algorithms Slicing with Inference Rules Refactoring Overview Slice Extraction Refactoring Example NATE – Slicing Based Refactoring Tool. Starter.

ByCircuit Complexity meets the Theory of Randomness. SUNY Buffalo, November 11, 2010. Today’s Goal:. To raise awareness of the tight connection between two fields: Circuit Complexity Kolmogorov Complexity (the theory of randomness) And to show that this is useful.

ByChapter 12 Theory of Computation. Introduction to CS 1 st Semester, 2014 Sanghyun Park. Outline. Functions and Their Computation Turing Machines Universal Programming Languages A Noncomputable Function Complexity of Problems. Functions and Their Computation.

ByReducibility. Problem is reduced to problem. If we can solve problem then we can solve problem. Problem is reduced to problem. If is decidable then is decidable. If is undecidable then is undecidable. Example:. the halting problem. is reduced to.

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