Automata Theory in Language Computation

1. Erik Jonsson School of Engineering and Computer Science CS 4384– 501 Automata Theory http://www.utdallas.edu/~pervin and eLearning Tuesday: Start Chapter 2 Look at Ullman’s Lectures 2 & 3 Thursday 1-16-14 FEARLESS Engineering www.utdallas.edu/~pervin

2. SyllabusOfficialS2014.docx SYLLABUS

3. Hopcroft & Ullman, Intro. to Automata Theory, Languages, and Computation • Lewis & Papadimitriou, Elements of the Theory of Computation • Martin, Intro. to Languages and the Theory of Computation (B&N $174) • Sipser, Intro. to the Theory of Computation ($170 for newest edition) • Du & Ko, Problem Solving in Automata, Languages, and Complexity (e-book $160) RESERVE BOOK LIST

4. CHAPTER ONE 1.3 Proof Techniques 1.3.6 Proof by Induction Note: Pigeon Hole CS 3305

7. A palindrome can be defined as a string that reads the same backward and forward, or by the following definition: • 1) e is a palindrome. • 2) If a is any symbol, then the string a is a palindrome. • 3) If a is a symbol and x is a palindrome, then axa is a palindrome. • 4) Nothing else is a palindrome unless it follows from 1) through 3). Palindrome Note: We will prove later that no FA accepts palindrome.

8. CS 4384 14 January 2014 HOMEWORK 1Due: 21 JANUARY 2014 Written carefully on 8.5x11 white paper as for your boss!

9. A = {w : w is a binary string containing an odd number of 1s} • B = {w: w is a binary string containing 101 as a substring} • C = {we {0,1}* : w has a 1 in the third position from the right} M&S Examples 2.2.1 p.26; 2.2.2 p.28; 2.2.3 p.29

10. A = {w : w is a binary string containing an odd number of 1s} What’s wrong with: ….

11. B = {w: w is a binary string containing 101 as a substring}

12. C = {we {0,1}* : w has a 1 in the third position from the right}Just remember the last three symbols

13. Language • A language is a set of strings. For example, {0, 1}, {all English words}, {0, 0, 0, ...} are all languages. • The following are operations on sets and hence also on languages. Union: A U B Intersection: A ∩ B Difference: A \ B (A - B when B A) Complement: A = Σ* - A where Σ* is the set of all strings on alphabet Σ. 0 1 2 _ Du

14. Concatenation: • For example, {0, 1}{1, 2} = {01, 02, 11, 12}. • Especially, we denote A = A, A = AA, ..., and define A = {ε}. 2 1 0 Concatenation of Languages Du

15. Kleene Star

16. The set of regular languages is closed under the union operation Theorem 2.3.1 M&S p. 32

17. Proof:

18. The set of regular languages is closed under the intersection operation Theorem 2.3.1 Addendum M&S p. 32

19. Proof:

20. The set of regular languages is closed under the complement operation Theorem 2.3.1 Addendum M&S p. 32

21. Proof:

22. The set of regular languages is closed under the Kleene star and concatenation operations To be proved later: M&S p. 32

23. This takes quite a bit of thinking, probably.

24. Du p.15

25. Binary expansion of positive integers congruent to zero mod 5

26. Binary expansion of positive integers congruent to zero mod 5

27. Binary expansion of non-negative integers congruent to 0 mod 5

28. What is L(M)?

29. Any number (including zero) of a’s followed by an equal number of b’s. Recursive Definition

30. Any number of a’s (including zero) followed by a less than or equal number of b’s. Another Recursive Definition

31. Problem Recursive Definition

32. Third from the right is a one