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CSCI 2670 Introduction to Theory of Computing

CSCI 2670 Introduction to Theory of Computing. August 24, 2005. Announcements. Please send me an email by midnight tonight Office hours changed Tues & Wed 11:00 – 12:30. Agenda. Last class Introduced deterministic finite automata Formally defined DFA’s Informally described DFA’s

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CSCI 2670 Introduction to Theory of Computing

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  1. CSCI 2670Introduction to Theory of Computing August 24, 2005

  2. Announcements • Please send me an email by midnight tonight • Office hours changed • Tues & Wed 11:00 – 12:30

  3. Agenda • Last class • Introduced deterministic finite automata • Formally defined DFA’s • Informally described DFA’s • This class • Quiz • Continue Chapter 1

  4. Group 1 0,1 0 q2 1 q3 0 q4 q1 0 1 1 q5 0,1 Hint: What string doesn’t this DFA accept?

  5. Group 2 0,1 q2 0,1 q3 0 q1 0,1 1 q4 0,1 q5 Hint: String length counts.

  6. Group 3 0,1 1 q1 q2 0 q3 0,1 Hint: Symbol position counts.

  7. Group 4 0 q1 q2 1 0,1 0,1 0,1 q3 q4 Hint: Can you simplify this DFA?

  8. 0,1 Group 5 q7 1 1 q1 q3 q5 1 1 0 0 0 0 0 0 q4 q6 q2 1 1 Hint: For each state, what do you know about how many times each symbol has appeared?

  9. Group 6 0 1 q1 q2 0 1 q3 0, 1 Hint: What happens when you get to q3?

  10. Remainder of class • Formal definition of computation • Definition of a regular language • Designing finite automata

  11. Formalizing computation • Let M = (Q,,,q0,F) be a finite automaton and let w = w1w2…wn be any string over . Macceptsw if there is a sequence of states r0, r1, …, rn, of Q such that • r0 = q0 • start in the start state • ri=(ri-1, wi) • the transition function determines each step • rn F • the last state is one of the final states

  12. Regular languages • A deterministic finite automaton Mrecognizes the language A if A = { w | M accepts w } • Alternatively, we say the language of M is A L(M) = { w | M accepts w } • Any language recognized by a deterministic finite automaton is called a regular language

  13. Designing finite automata • Select states specifically to reflect some important concept • even number of 0’s • odd number of occurrences of the string 010 • Ensure this meaning is relevant to the language you are trying to define • Try to get “in the head” of the automaton

  14. Example • Design a DFA accepting all strings over {0,1,2,3} such that the sum of the symbols in the string is equivalent to 2 modulo 4 or 3 modulo 4

  15. Step 1 • What states do we need? • One state for each value modulo 4 • q1 represents 1 modulo 4 • q2 represents 2 modulo 4 • q3 represents 3 modulo 4 • q4 represents 0 modulo 4

  16. Step 2 • Create the state transition table

  17. Step 3 • What elements of the 5-tuple do we know? • Q, , and  • So we still need q0 and F • q0 = q4 • F = {q2, q3}

  18. 3 1 0 q1 q2 0 2 2 3 1 1 3 2 2 q4 q3 0 0 1 3 Draw DFA

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