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COSC 3340: Introduction to Theory of Computation

Explore the concept of countability and uncountability in languages. Learn about regular and infinite sets. Discover the Pumping Lemma and its application to identify non-regular languages.

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COSC 3340: Introduction to Theory of Computation

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  1. COSC 3340: Introduction to Theory of Computation University of Houston Dr. Verma Lecture 7 UofH - COSC 3340 - Dr. Verma

  2. Are all languages regular? • Ans: No. • How do we know this? • Ans: Cardinality arguments. • Let C(DFA) = {M | M is a DFA}. • C(DFA) is a countable set. Why? • Let AL = { L | L is a subset of *}. • AL is uncountable. UofH - COSC 3340 - Dr. Verma

  3. Cardinality • Def: A set is countable if it is (i) finite or (ii) countably infinite. • Def: A set S is countable infinite if there is a bijection from N to S. • Note: Instead of N to S we can also say S to N. N = {0, 1, 2, 3, ...} -- the set of natural numbers. • Def: A set that is not countable is uncountable. • Or, any infinite set with no bijection from N to itself. UofH - COSC 3340 - Dr. Verma

  4. Examples 1. The set of chairs in this classroom is finite. 2. The set of even numbers is countably infinite. 3. The set of all subsets of N is uncountable. • Notation: 2N UofH - COSC 3340 - Dr. Verma

  5. Exercises • What is the cardinality of? 1. Z - the set of integers. 2. N X N = {(a,b) | a, b in N}. 3. R - the set of real numbers. UofH - COSC 3340 - Dr. Verma

  6. Pumping Lemma • First technique to show that specific given languages are not regular. • Cardinality arguments show existence of languages that are not regular. • There is a big difference between the two! UofH - COSC 3340 - Dr. Verma

  7. Statement of Pumping Lemma If A is an infinite regular language, then there is a number p (the pumping length) where, if s is any string in Aof length at leastp, then s may be divided into three pieces, s = xyz, satisfying the following conditions: 1. for each i 0, xyiz  A, 2. |y| > 0, and 3. |xy| p. UofH - COSC 3340 - Dr. Verma

  8. Proof of pumping lemma • Idea: If a string w of length m is accepted by a DFA with n states, and n < m, then there is a cycle (repeated state) on the directed path from s to a final state labeled w. • Recall: directed path is denoted by *(s,w). • Uses: Pigeon-hole principle UofH - COSC 3340 - Dr. Verma

  9. Pigeon-hole principle 4 pigeons 3 pigeonholes UofH - COSC 3340 - Dr. Verma

  10. Pigeon-hole principle (contd.) A pigeonhole must have two pigeons UofH - COSC 3340 - Dr. Verma

  11. Pigeon-hole principle (contd.) ........... pigeonholes ........... UofH - COSC 3340 - Dr. Verma

  12. Details of Proof of Pumping Lemma. Consider L - any infinite regular language. 1. L regular  there is a DFA M with L(M) = L. 2. Let DFA have p states (say). 3. Let w in L be of length more than p. • Why does w exist? • Ans: because L is infinite. 4. *(s,w) = f (some final state) must be *(s, w = xyz) = *(q,yz) = *(q,z) = f 5. So xynz in L for n = 0, 1, 2, 3, .... since *(s, xynz) = *(q, ynz) = *(q, y{n-1}z) = ... = *(q,z) = f. UofH - COSC 3340 - Dr. Verma

  13. Describing the pumping lemma Take an infinite regular language DFA that accepts states UofH - COSC 3340 - Dr. Verma

  14. Describing the pumping lemma (contd.) Take string with There is a walk with label: ......... UofH - COSC 3340 - Dr. Verma

  15. Describing the pumping lemma (contd.) number of states If string has length Then, from the pigeonhole principle: A state is repeated in the walk ...... ...... UofH - COSC 3340 - Dr. Verma

  16. Describing the pumping lemma (contd.) Write ...... ...... UofH - COSC 3340 - Dr. Verma

  17. ...... ...... Describing the pumping lemma (contd.) Observations : length number of states length UofH - COSC 3340 - Dr. Verma

  18. ...... ...... Describing the pumping lemma (contd.) Observation: The string is accepted UofH - COSC 3340 - Dr. Verma

  19. ...... ...... Describing the pumping lemma (contd.) The string is accepted Observation: UofH - COSC 3340 - Dr. Verma

  20. ...... ...... Describing the pumping lemma (contd.) The string is accepted Observation: UofH - COSC 3340 - Dr. Verma

  21. ...... ...... Describing the pumping lemma (contd.) In General: The string is accepted UofH - COSC 3340 - Dr. Verma

  22. Some Applications of Pumping Lemma The following languages are not regular. 1. {anbn | n  0 }. 2. {w = wR | w in {a,b}* } (language of palindromes). 3. {wwR | w in {a,b}*}. 4. {a{n2} | n  0}. UofH - COSC 3340 - Dr. Verma

  23. Tips of the trade -- Do not forget! Closure properties can be used effectively for: (1) shortening cumbersome Pumping lemma arguments. • Example: {w in {a, b}* | w has equal a's and b's}. (2) for showing that certain languages are regular. • Example: {w in {a, b}* | w begins with a and w contains a b}. UofH - COSC 3340 - Dr. Verma

  24. References • Pigeonhole Principle & Pumping Lemma Description • www.cs.rpi.edu/courses/fall00/modcomp3/class8.ppt • Author: Costas Busch, Rensselaer Polytechnic Institute, NY UofH - COSC 3340 - Dr. Verma

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