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CSCI 2670 Introduction to Theory of Computing

Explore algorithmic descriptions of Turing machines, understand enumerators, machine equivalence theorems, decidable languages, and depth-first tree algorithms. Learn how Turing machines can simulate various algorithms.

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CSCI 2670 Introduction to Theory of Computing

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  1. CSCI 2670Introduction to Theory of Computing October 11, 2005

  2. Agenda • Last week • Prove equivalence of deterministic and nondeterministic Turing machines • Today • Enumerators • Definition of algorithm • Another class of languages • Decidable October 11, 2005

  3. Announcements • Quiz tomorrow • High-level description of TM • Trace through a TM’s operation • Tape notation & configuration notation • High-level description of equivalences • Homework due next Tuesday (10/18) • 4.4, 4.6, 4.7, 4.12, 4.19 • Old text 4.4, 4.7, 4.8, 4.11, 4.19 October 11, 2005

  4. aaba bbcc aaaaabb Printer M 1 1 2 1 3 ~ ~ ~ Enumerators • Instead of reading an input and processing it, enumerators start with an empty tape and print out strings in Σ* October 11, 2005

  5. Machine equivalence Theorem: A language is Turing-recognizable if and only of some enumerator enumerates it. Proof technique: Construction in each direction October 11, 2005

  6. TM accept enumerator language • TM = “On input w: • Run enumerator E. Every time E prints a string, compare it to w. • If w appears in the output, accept.” October 11, 2005

  7. Enumerator accepts TM language • Let s1, s2, s3, … be all the strings in Σ* • E = “Ignore the input. • For i = 1, 2, 3, … • Run M for i steps on each input s1, s2, …, si • Whenever M accepts a string, print it October 11, 2005

  8. What is an algorithm? • Intuitively, an algorithm is anything that can be simulated by a Turing machine (Church-Turing Thesis) • Many algorithms can be simulated by Turing machines • Inputs can be represented as strings • Graphs • Polynomials • Atomata • Etc. October 11, 2005

  9. Example algorithm • Depth-first walk-through of binary tree • Which nodes do you visit, and in what order, when doing a depth-first search? • Visit each leaf node from left to right • Recursive algorithm October 11, 2005

  10. Binary tree depth-first walk-through • Start at root • Process left subtree (if one exists) • Process right subtree (if one exists) • Process how? • Print the node name • If there is a left subtree then • Process the left subtree • Print the node name again • If there is a right subtree then • Process the right subtree • Print the node name again October 11, 2005

  11. 1 2 3 4 6 5 7 9 8 10 Example 1 2 4 8 4 2 5 9 5 2 1 3 6 10 6 3 7 October 11, 2005

  12. Can a Turing machine do this? • Input must be a string (not a tree) • Can we represent a tree with a string? • Yes. October 11, 2005

  13. 1 2 3 4 6 5 7 9 8 10 1 2 3 4 5 6 7 8 # # 9 10 # # # ~ String representation of a binary tree October 11, 2005

  14. Can a Turing machine do this? • Input must be a string (not a tree) • Can we represent a tree with a string? • Yes • How do we know which node(s) are children of current node • If node is kth node at depth d, it’s position in the string is 2d+k-1 and its children are at position 2d+1+2(k-1) and 2d+1+2k-1 October 11, 2005

  15. What about the output? • Need to write out nodes in a particular order • Can we do this with a TM? • Yes. Add output tape • A TM can move left and right on the input tape writing to the output tape whenever appropriate October 11, 2005

  16. Describing Turing machines • From now on, we can describe Turing machines algorithmically M = “On input w 1. … 2. … …” October 11, 2005

  17. Decidability • A language is decidable if some Turing machine decides it • Not all languages are decidable • We will see examples of both decidable and undecidable languages October 11, 2005

  18. Showing a language is decidable • Write a decider that decides it • Must show the decider • Halts on all inputs • Accepts w iff w is in the language • Can use algorithmic description October 11, 2005

  19. DFA acceptance problem • Consider the language ADFA = {<B,w> | B is a DFA that accepts the string w} Theorem: ADFA is a decidable language Proof: Consider the following TM, M M = “On input string <B,w>, where B is a DFA and w is an input to B • Simulate B on input w • If simulation ends in accept state, accept. Otherwise, reject.” October 11, 2005

  20. NFA acceptance problem • Consider the language ANFA = {<B,w> | B is a NFA that accepts the string w} Theorem: ANFA is a decidable language Proof: Consider the following TM, N N = “On input string <B,w> • Convert B to a DFA C • Run TM M from previous slide on <C,w> • If M accepts, accept. Otherwise, reject.” October 11, 2005

  21. RE acceptance problem • Consider the language AREX = {<R,w> | R is an RE that generates the string w} Theorem: AREX is a decidable language Proof: Consider the following TM, P P = “On input string <R,w> • Convert R to a DFA C using algorithm discussed in class and in text • Run TM M from earlier slide on <C,w> • If M accepts, accept. Otherwise, reject.” October 11, 2005

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