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COSC 3340: Introduction to Theory of Computation

COSC 3340: Introduction to Theory of Computation. University of Houston Dr. Verma Lecture 2. 1st model -- Deterministic Finite Automaton (DFA). Read only Head. Finite Control. DFA (contd.). The DFA has: a finite set of states 1 special state - initial state

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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 2 UofH - COSC 3340 - Dr. Verma

  2. 1st model -- Deterministic Finite Automaton (DFA) Read only Head Finite Control UofH - COSC 3340 - Dr. Verma

  3. DFA (contd.) • The DFA has: • a finite set of states • 1 special state - initial state • 0 or more special states - final states • input alphabet • transition table containing (state, symbol) -> next state UofH - COSC 3340 - Dr. Verma

  4. Informally -- How does a DFA work? • An input string is placed on the tape (left-justified). • DFA begins in the start state. • Head placed on leftmost cell. • DFA goes into a loop until the entire string is read. • In each step, DFA consults a transition table and changes state based on (s,) where • s - current state •  - symbol scanned by head UofH - COSC 3340 - Dr. Verma

  5. How does a DFA work? (contd.) • After reading input string, • if DFA state final, input accepted • if DFA state notfinal, input rejected • Language of DFA -- set of all strings accepted by DFA. UofH - COSC 3340 - Dr. Verma

  6. Pictorial representation of DFA (q,σ)->q' UofH - COSC 3340 - Dr. Verma

  7. Example: Diagram of DFA L = {a2n + 1 | n >= 0} • Answer: L = {a, aaa, aaaaa, ...} UofH - COSC 3340 - Dr. Verma

  8. JFLAP Step-By-Step aaa UofH - COSC 3340 - Dr. Verma

  9. JFLAP Step-By-Step aaa UofH - COSC 3340 - Dr. Verma

  10. JFLAP Step-By-Step aaa UofH - COSC 3340 - Dr. Verma

  11. JFLAP Step-By-Step aaa UofH - COSC 3340 - Dr. Verma

  12. Formal definition of DFA • DFA M = (Q, , , s, F) • Where, • Q is finite set of states •  is input alphabet • sQ is initial state • FQ is set of final states • : Q X-> Q UofH - COSC 3340 - Dr. Verma

  13. Formal definition of L(M) • L(M) - Language accepted by M • Define * (extends  to strings): • *(q, ) = q • *(q, wσ) = (*(q,w),σ), w - string σ - symbol • Definition: L(M) = { w in *| *(s,w) is in F }. UofH - COSC 3340 - Dr. Verma

  14. Example: L(M) = {w in {a,b}* | w contains even no. of a's} UofH - COSC 3340 - Dr. Verma

  15. JFLAP Step-By-Step aa UofH - COSC 3340 - Dr. Verma

  16. JFLAP Step-By-Step aa UofH - COSC 3340 - Dr. Verma

  17. JFLAP Step-By-Step aa UofH - COSC 3340 - Dr. Verma

  18. JFLAP Step-By-Step ab UofH - COSC 3340 - Dr. Verma

  19. JFLAP Step-By-Step ab UofH - COSC 3340 - Dr. Verma

  20. JFLAP Step-By-Step ab UofH - COSC 3340 - Dr. Verma

  21. Given a language, how to define DFA? • Creative process requiring you to: (i) Put yourself in DFA's shoes (ii) Find finite amount of info, based on which string accepted/rejected (iii) From step (ii), determine number of states and then transitions. UofH - COSC 3340 - Dr. Verma

  22. Regular Languages • Definition: A Language is regular iff there is a DFA that accepts it. • Examples: •  • {} • * • {w in {0,1}* | second symbol of w is a 1} Exercise • {w in {0,1}* | second last symbol of w is a 1} Exercise • {w in {0,1}* | w contains 010 as a substring} - (importance?) UofH - COSC 3340 - Dr. Verma

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