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Understanding Computable Functions in Computer Science

Learn about computable functions in computer science and the concept of Turing Machines. Discover the Church-Turing Thesis and the distinction between computable and unknown functions. Delve into the Traveling Salesman Problem and the Halting Problem in this informative lecture.

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Understanding Computable Functions in Computer Science

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  1. CS1001 Lecture 23

  2. Overview • Incompleteness and the Halting Problem • Methods in Artificial Intelligence

  3. Reading • Brookshear – Chapters 10 and 11

  4. Functions • Computer Science is concerned with implementing and evaluating mathematical functions • There is now a practical concern – how to actually produce an output given an input • Functions can be classified as “computable” or “non-computable”

  5. Turing Machines • A theoretical device • Tape (memory) • A head (for reading/writing the tape) • Instructions for moving/reading/writing based on a current state • As powerful as any modern machine (in other words, it can compute everything a modern computer can compute) • http://wap03.informatik.fh-wiesbaden.de/weber1/turing/tm.html

  6. Example • The “successor” function • A function that takes an input and returns its value plus 1 • Succ(x) = x + 1 • Succ(5) is equal to 6

  7. Turing Computable • Succ is Turing Computable because it can be computed using a turing machine • How? • Start with the number (input) in binary on the tape • Go to the rightmost bit, if 0, change to 1 and halt • Otherwise, set to 0 and enter the “carry” state. Then, move backwards trying to find a 0…

  8. Church-Turing Thesis • “Turing-Computable” means the same thing as Algorithmically computable. • In other words, if you can prove a function computable on a turing machine, it must also be computable by a definite algorithm • The converse also holds

  9. Computable vs Unknown • Key Point: A non computable function is one where there is a definite correct answer and we have enough information to find it. However, actually getting the answer from the inputs is not feasible due to the complexity of the problem. • This is not like asking “will the human race exist in 5,000 years” – we do not have enough information to answer that

  10. Traveling Salesman Problem • http://www.math.princeton.edu/tsp/cpapp/us41.html

  11. The Halting Problem • Given a program and inputs, is it possible to write another program to predict whether the initial program will terminate or loop for ever? • No!

  12. Public Key Cryptography

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