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  1. Erik Jonsson School of Engineering and Computer Science CS 5349 – 001 (CS 4384– HON001) Automata Theory Thursday: Read Chapter 1 Look at Ullman’s Lecture 1 Tuesday 08-26-14 FEARLESS Engineering

  2. Erik Jonsson School of Engineering and Computer Science Dr. William J. Pervin Professor Emeritus of Mathematics, Electrical and Computer Engineering, and Computer Science University of Texas at Dallas Head, Computer Engineering Program (ret.) Master, College of Engineering (ret.) Head, Computer Science Department (ret) Director, UT Regional Comp. Center (ret.) Chair, Mathematics Department (ret.) Office: ECSN 4.626 E-mail: FEARLESS Engineering

  3. Syllabus TEXT “Theory of Computation” by Anil Maheshwari & Michiel Smid of Carleton University (Free Download) Theory of Computation ADDITIONAL MATERIAL “Automata Theory” web-course by J. Ullman on coursera (Free Download) Ullman JFLAP computer program and tutorial (Free Download) JFLAP Syllabus Remember these FREE items at course evaluation time, please.

  4. Hopcroft & Ullman, Intro. to Automata Theory, Languages, and Computation (~$135) • 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 (~$150 for newest edition) < (Other section) [see eLearning for this book material] • Du(UTD) & Ko, Problem Solving in Automata, Languages, and Complexity (e-book $160) RESERVE BOOK LIST

  5. A variety of properties concerning the models, grammars, and languages will be proven. • The existence or non-existence of algorithms for processing languages and language processors will be proven. • These algorithms form the basis of tools for processing languages, e.g., parsers, compilers, assemblers, etc. • Other algorithms will form the basis of tools that automatically construct language processors, e.g., yacc, lex, etc. Note that our perspective will be similar to, yet different from a compiler class. • Additionally, some things will be proven to be non-computable, e.g., the enhanced compiler. Practical Applications

  6. The set of all strings over the alphabet S is denoted by S* M&S, pp. 6-7

  7. CHAPTER ONE 1.3 Proof Techniques 1.3.6 Proof by Induction Note: Pigeon Hole Principle Prerequisite material

  8. Induction

  9. CS 5349/4384 26 August 2014 HOMEWORK 0 Do for 28 August 2014 Keep copies of your homework; they probably will not be returned but instead submitted to ABET/SACS for accreditation

  10. The language associated with a FA: L(M) = {w | w is a string over Σ and M “accepts” w} FINITEAUTOMATAFinite State MachinesDeterministic Finite Automata

  11. Deterministic Finite State Automata (DFA) …….. • One-way, infinite tape, broken into cells • One-way, read-only tape head. • Finite control, I.e., a program, containing the position of the read head, current symbol being scanned, and the current “state.” • A string is placed on the tape, read head is positioned at the left end, and the DFA will read the string one symbol at a time until all symbols have been read. The DFA will then either accept or reject. Finite Control

  12. Formal Definition of a DFA A DFA is a five-tuple: M = (Q, Σ, δ, q0, F) Q A finite set of states Σ A finite input alphabet q0 The initial/starting state, q0 is in Q F A set of final/accepting states, which is a subset of Q δ A transition function, which is a total function from Q x Σ to Q δ: (Q x Σ) –> Q δ is defined for any q in Q and s in Σ, and δ(q, s) = q’ is equal to some state q’ in Q, could be q’=q Intuitively, δ(q, s) is the state entered by M after reading symbol s while in state q.

  13. Strings over {a,b,c} in which all the a’s precede the b’s, which in turn precede the c’s.

  14. Example: Sudkamp 2.6Strings over {a,b,c} in which all the a’s precede the b’s, which in turn precede the c’s. State Diagram JFLAP

  15. Example: Sudkamp 2.6Strings over {a,b,c} in which all the a’s precede the b’s, which in turn precede the c’s. State Diagram The language of this DFA is {aibjck : 0 ≤ i, j, k} JFLAP

  16. M&S p. 26 Extension to Strings

  17. Example: Sudkamp 2.8Strings over {a,b} in which the substring aa occurs exactly once.

  18. Example: Sudkamp 2-22The set of strings over {a,b} with an even number of a’s and an even number of b’s.

  19. Example: Sudkamp 2-13The set of strings over {a,b,c} such that every b is immediately followed by at least one c.

  20. a b b b a q4 q1 q2 q5 q6 q3 q7 a b b a/b a q0 a b a b a Give a DFA M such that: L(M) = {x | x is a string of a’s and b’s such that x contains both aa and bb}

  21. Language • A language is a set of strings. For example, {0, 1}, {all English words}, {ε=00, 01, 02, ...} are all languages. • The following are operations on sets and hence also on languages. Union: A U B Intersection: A ∩ B Difference: A \ B (also A - B) Complement: A = Σ* - A where Σ* is the set of all strings on alphabet Σ. _

  22. A language is regular iff it is accepted by a (deterministic) finite automata. REGULAR LANGUAGES

  23. CS 5349/4384 26 August 2014 HOMEWORK 1Due: 2 September 2014 Written carefully on 8.5x11 white paper as for your boss!

  24. TeX LaTeX Available for UNIX and Windows

  25. 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 Assume concatenation of strings

  26. Kleene closure: A* = A U A U A U ··· • Notation: A = A U A U A U ··· 0 1 2 + 1 2 3 Kleene Closure

  27. What is ? • What is ? • What is ? Where is the empty language.

  28. 0/1 0/1 0/1 q1 q0 q0 q0 q1 0/1 0/1 q0 Let Σ = {0, 1}. Give DFAs for {}, {ε}, Σ*, and Σ+. For {}: For {ε}: For Σ*: For Σ+: 0/1

  29. Regular Languages • The concept of regular languages on an alphabet Σ is defined recursively as follows: (1) The empty language is regular. (2) For every symbol a Σ, {a} is regular. (3) If A and B are regular languages, then A U B, AB, and A* are regular. (4) Nothing else is a regular language. What about epsilon?