2. 1 Theory of Computation 計算理論

3. 2 Instructor: 顏嗣鈞 E-mail: yen@cc.ee.ntu.edu.tw Web: http://www.ee.ntu.edu.tw/~yen Time: 9:10-12:10 PM, Monday Place: BL 103 Office hours: by appointment Class web page: http://www.ee.ntu.edu.tw/~yen/courses/TOC-2009.html

4. 3 TEXTBOOK Introduction to Automata Theory, Languages, and Computation 3rd Edition John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman, (Addison-Wesley, 2006)

5. 4 Introduction to Automata Theory, Languages, and Computation John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman, (2nd Ed. Addison-Wesley, 2001) 2 nd Edition

6. 5 Introduction to Automata Theory, Languages, and Computation John E. Hopcroft, Jeffrey D. Ullman, (Addison-Wesley, 1979) 1 st Edition

7. 6 Grading HW/Project : 20% Midterm exam.: 40% Final exam.: 40%

8. 7 A simple computer BATTERY SWITCH input: switch output: light bulb actions: flip switch states: on, off

9. 8 A simple “ computer ” BATTERY SWITCH off on start f f input: switch output: light bulb actions: f for “flip switch” states: on, off bulb is on if and only if there was an odd number of flips

10. 9 Another “ computer ” BATTERY off start 1 inputs: switches 1 and 2 actions: 1 for “flip switch 1” actions: 2 for “flip switch 2” states: on, off bulb is on if and only if both switches were flipped an odd number of times 1 2 1 off on 1 1 2 2 2 2

11. 10 A design problem Can you design a circuit where the light is on if and only if all the switches were flipped exactly the same number of times? 4 BATTERY 1 2 3 5 ?

12. 11 A design problem Such devices are difficult to reason about, because they can be designed in an infinite number of ways By representing them as abstract computational devices, or automata, we will learn how to answer such questions

13. 12 These devices can model many things They can describe the operation of any “ small computer ”, like the control component of an alarm clock or a microwave They are also used in lexical analyzers to recognize well formed expressions in programming languages: ab1 is a legal name of a variable in C 5u= is not

14. 13 Different kinds of automata This was only one example of a computational device, and there are others We will look at different devices, and look at the following questions: –What can a given type of device compute, and what are its limitations? –Is one type of device more powerful than another?

15. 14 Some devices we will see finite automataDevices with a finite amount of memory. Used to model “ small ” computers. push-down automata Devices with infinite memory that can be accessed in a restricted way. Used to model parsers, etc. Turing MachinesDevices with infinite memory. Used to model any computer. time-bounded Turing Machines Infinite memory, but bounded running time. Used to model any computer program that runs in a “ reasonable ” amount of time.

16. 15 Why Study Automata Theory? Finite automata are a useful model for important kinds of hardware and software: Software for designing and checking digital circuits. Lexical analyzer of compilers. Finding words and patterns in large bodies of text, e.g. in web pages. Verification of systems with finite number of states, e.g. communication protocols.

17. 16 Why Study Automata Theory? (2) The study of Finite Automata and Formal Languages are intimately connected. Methods for specifying formal languages are very important in many areas of CS, e.g.: Context Free Grammars are very useful when designing software that processes data with recursive structure, like the parser in a compiler. Regular Expressions are very useful for specifying lexical aspects of programming languages and search patterns.

18. 17 Why Study Automata Theory? (3) Automata are essential for the study of the limits of computation. Two issues: What can a computer do at all? (Decidability) What can a computer do efficiently? (Intractability)

19. 18 Applications Theoretical Computer Science Automata Theory, Formal Languages, Computability, Complexity … Compiler Prog. languages Comm. protocols circuits Pattern recognition Supervisory control Quantum computing Computer-AidedVerification...

20. 19 Aims of the Course To familiarize you with key Computer Science concepts in central areas like - Automata Theory - Formal Languages - Models of Computation - Complexity Theory To equip you with tools with wide applicability in the fields of CS and EE, e.g. for - Complier Construction - Text Processing - XML

21. 20 Fundamental Theme What are the capabilities and limitations of computers and computer programs? –What can we do with computers/programs? –Are there things we cannot do with computers/programs?

22. 21 Studying the Theme How do we prove something CAN be done by SOME program? How do we prove something CANNOT be done by ANY program?

23. 22 Example: The Halting Problem (1) Consider the following program. Does it terminate for all values of n  1? while (n > 1) { if even(n) { n = n / 2; } else { n = n * 3 + 1; }

24. 23 Example: The Halting Problem (2) Not as easy to answer as it might first seem. Say we start with n = 7, for example: 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 In fact, for all numbers that have been tried (a lot!), it does terminate...... but in general?

25. 24 Example: The Halting Problem (3) Then the following important undecidability result should perhaps not come as a total surprise: It is impossible to write a program that decides if another, arbitrary, program terminates (halts) or not. What might be surprising is that it is possible to prove such a result. This was first done by the British mathematician Alan Turing.

26. 25 Our focus Automata Computability Complexity

27. 26 Topics 1. Finite automata, Regular languages, Regular grammars: deterministic vs. nondeterministic, one-way vs. two-way finite automata, minimization, pumping lemma for regular sets, closure properties. 2. Pushdown automata, Context-free languages, Context-free grammars: deterministic vs. nondeterministic, one-way vs. two-way PDAs, reversal bounded PDAs, linear grammars, counter machines, pumping lemma for CFLs, Chomsky normal form, Greibach normal form, closure properties. 3.

28. 27 Topics (cont’d) 3. Linear bounded automata, Context- sensitive languages, Context-sensitive grammars. 4. Turing machines, Recursively enumerable sets, Type 0 grammars: variants of Turing machines, halting problem, undecidability, Post correspondence problem, valid and invalid computations of TMs.

29. 28 Topics (cont’d) 5. Basic recursive function theory 6. Basic complexity theory: Various resource bounded complexity classes, including NLOGSPACE, P, NP, PSPACE, EXPTIME, and many more. reducibility, completeness. 7. Advanced topics: Tree Automata, quantum automata, probabilistic automata, interactive proof systems, oracle computations, cryptography.

30. 29 Who should take this course? YOU

31. 30 Languages The terms language and word are used in a strict technical sense in this course: A language is a set of words. A word is a sequence (or string) of symbols.  (or ) denotes the empty word, the sequence of zero symbols.

32. 31 Symbols and Alphabets What is a symbol, then? Anything, but it has to come from an alphabet which is a finite set. A common (and important) instance is  = {0, 1}. , the empty word, is never an symbol of an alphabet.

33. 32 Computation CPU memory

34. 33 CPU input memory output memory Program memory temporary memory

35. 34 CPU input memory output memory Program memory temporary memory compute Example:

36. 35 CPU input memory output memory Program memory temporary memory compute

37. 36 CPU input memory output memory Program memory temporary memory compute

38. 37 CPU input memory output memory Program memory temporary memory compute

39. 38 Automaton CPU input memory output memory Program memory temporary memory Automaton

40. 39 Different Kinds of Automata Automata are distinguished by the temporary memory Finite Automata: no temporary memory Pushdown Automata: stack Turing Machines: random access memory

41. 40 input memory output memory temporary memory Finite Automaton Finite Automaton Example: Vending Machines (small computing power)

42. 41 input memory output memory Stack Pushdown Automaton Pushdown Automaton Example: Compilers for Programming Languages (medium computing power) Push, Pop

43. 42 input memory output memory Random Access Memory Turing Machine Turing Machine Examples: Any Algorithm (highest computing power)

44. 43 Finite Automata Pushdown Automata Turing Machine Power of Automata Less powerMore power Solve more computational problems