Theory of Computation - Lecture Slides | CS 1010, Study notes of Computer Science

Download Theory of Computation - Lecture Slides | CS 1010 and more Study notes Computer Science in PDF only on Docsity! Introduction to Computer Science • Robert Sedgewick and Kevin Wayne • http://www.cs.Princeton.EDU/IntroCS 7: Theory of Computation 2 Introduction to Theoretical CS Two fundamental questions.  What can a computer do?  What can a computer do with limited resources? General approach.  Don't talk about specific machines or problems.  Consider minimal abstract machines.  Consider general classes of problems. Pentium IV running Linux kernel 2.4.22 3 Why Learn Theory In theory . . .  Deeper understanding of what is a computer and computing.  Foundation of all modern computers.  Pure science.  Philosophical implications. In practice . . .  Web search: theory of pattern matching.  Sequential circuits: theory of finite state automata.  Compilers: theory of context free grammars.  Cryptography: theory of computational complexity.  Data compression: theory of information. "In theory there is no difference between theory and practice. In practice there is." -Yogi Berra Introduction to Computer Science • Robert Sedgewick and Kevin Wayne • http://www.cs.Princeton.EDU/IntroCS Regular Expressions and DFAs a* | (a*ba*ba*ba*)* 0 21 b b a a a b 5 Pattern Matching Applications Test if a string matches some pattern.  Process natural language.  Scan for virus signatures.  Search for information using Google.  Access information in digital libraries.  Retrieve information from Lexis/Nexis.  Search-and-replace in a word processors.  Filter text (spam, NetNanny, Carnivore, malware).  Validate data-entry fields (dates, email, URL, credit card).  Search for markers in human genome using PROSITE patterns. Parse text files.  Compile a Java program.  Crawl and index the Web.  Read in data stored in TOY input file format.  Automatically create Java documentation from Javadoc comments. 6 Regular Expressions: Basic Operations Regular expression. Notation to specify a set of strings. every other stringaabaabaabaabConcatenation every other stringaaaababaab a(a|b)aab Parentheses (ab)*a ab*a aa | baab .u.u.u. Regular Expression ε abbbaa a ababababa ab ababa aa abbba Closure Union Wildcard Operation every other stringaabaab succubus tumultuous cumulus jugulum NoYes 7 Regular Expressions: Examples Regular expression. Notation is surprisingly expressive. b bb baabbbaa bbb aaa bbbaababbaa a* | (a*ba*ba*ba*)* multiple of three b’s 111111111 403982772 1000234 98701234 .*0.... fifth to last digits is 0 subspace subspecies raspberry crispbread .* spb .* contains the trigraph spb gcgcgg cggcggcggctg gcgcaggctg gcgctg gcgcggctg gcgcggaggctg gcg (cgg|agg)* ctg fragile X syndrome indicator Regular Expression NoYes 8 Generalized Regular Expressions Regular expressions are a standard programmer's tool.  Built in to Java, Perl, Unix, Python, . . . .  Additional operations typically added for convenience.  Ex: [a-e]+ is shorthand for (a|b|c|d|e)(a|b|c|d|e)*. 111111111 166-54-111 08540-1321 19072-5541 [0-9]{5}-[0-9]{4}Exactly k decaderhythm[^aeiou]{6}Negations camelCase 4illegal capitalized Word [A-Za-z][a-z]*Character classes ade bcde abcde abcbcde a(bc)+deOne or more Regular ExpressionOperation NoYes 17 Fundamental Questions Which languages CANNOT be described by any RE?  Bit strings with equal number of 0s and 1s.  Decimal strings that represent prime numbers.  Genomic strings that are Watson-Crick complemented palindromes.  Many more. . . . How can we extend REs to describe richer sets of strings?  Context free grammar (e.g., Java). Q. How can we make simple machines more powerful? Q. Are there any limits on what kinds of problems machines can solve? Reference: http://java.sun.com/docs/books/jls/second_edition/html/syntax.doc.html 18 Summary Programmer.  Regular expressions are a powerful pattern matching tool.  Implement regular expressions with finite state machines. Theoretician.  Regular expression is a compact description of a set of strings.  DFA is an abstract machine that solves pattern match problem for regular expressions.  DFAs and regular expressions have limitations. You. Practical application of core CS principles. Variations.  Terminology: DFA, FSA, FSM.  DFAs with output: Moore machines, Mealy machines. Introduction to Computer Science • Robert Sedgewick and Kevin Wayne • http://www.cs.Princeton.EDU/IntroCS 7.1: Extra Slides Y 0 mod 3 2 mod 3 0 1 mod 3 1 0 1 1 20 Pattern Matching in Google Google. Supports * for full word wildcard and | for union. 21 Pattern Matching in TiVo TiVo. WishList has very limited pattern matching. Reference: page 76, Hughes DirectTV TiVo manual 22 Order 3 Markov Model: bbbabbabbbbaba bbb bab bba abb aba 23 Fundamental Questions Which languages CANNOT be described by any RE?  Set of all bit strings with equal number of 0s and 1s.  Set of all bit strings of the form ww where w is some string.  Many more. . . . Are Java regular expressions more expressive than REs?  No: extra rules can be expressed in terms of minimal core.  One important exception: back references. – \1 matches whatever string that (.*) matches – words of the form ww where w is some word (beriberi, couscous) (.*)\1 not-so-regular expression 24 DFA Applications Software applications.  Implementing regular expressions.  Lexical analysis in compilers.  Parsing html in a web browser.  Controlling graphical user interfaces. Hardware applications.  Bounce filter.  Traffic lights.  Dishwashers.  Remote controls.  Computer microprocessors.  . . . 25 Limitations of FSA FSA are simple machines.  N states ⇒ can't "remember" more than N things.  Some languages require "remembering" more than N things. Theorem. No FSA can recognize the language of all bit strings with an equal number of 0's and 1's. A warmup exercise: 0 1 0 If 01xyz accepted then so is 00001xyz 0 26 Limitations of FSA Theorem. No FSA can recognize the language of all bit strings with an equal number of 0's and 1's.  Suppose an N-state FSA can recognize this language.  Consider following input: 0000000011111111  FSA must accept this string.  Some state x is revisited during first N+1 0's since only N states. 0000000011111111 x x  Machine would accept same string without intervening 0's. 000011111111  This string doesn't have an equal number of 0's and 1's. N+1 0's N+1 1's 27 Text Searching Build an FSA that accepts all strings that contain 'acat' as a substring.  tatgacatg  acacatg Start building: N N N N Y a c a t acatacaaca φ State name represents longest prefix of acat that input currently matches. 28 Text Searching Build an FSA that accepts all strings that contain 'acat' as a substring.  tatgacatg  acacatg Continue building: N N N N Y a c a t acgtcgt acatacaaca φ cgt