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CSCI 2670 Introduction to Theory of Computing

This text provides announcements, information on DFA design, and details about office hours for the Introduction to Theory of Computing course.

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CSCI 2670 Introduction to Theory of Computing

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  1. CSCI 2670Introduction to Theory of Computing August 31, 2005

  2. Announcements • Quiz tomorrow • Definition of DFA’s and designing DFA’s • Office hours with TA Ryan Foster • Monday 2:30 – 3:30 (Rm. 323) • Friday 10:00 – 11:00 (Rm. 524) • Change in homework policy • I give your homework to the TA at 11:00 the day after it’s due • I will accept it until that time

  3. Agenda • Last class • Discussed non-determinism • Equivalence of DFA’s and NFA’s • Today • Further exploration of equivalence of DFA’s and NFA’s • Tomorrow • Closure of regular languages under regular operators • Another method for describing regular languages

  4. Non-deterministic finite automaton A non-deterministic finite automaton is a 5-tuple (Q,,,q0,F), where • Q is a finite set of states •  is a (finite) alphabet •  : Q × ε  P(Q) is the transition function •  maps to sets of states • q0 is the start state, and • F  Q is the set of accept states

  5. Equivalence of DFAs and NFAs Theorem: Every non-deterministic finite automaton has an equivalent deterministic finite automaton • Both FAs accept the same language • Proof method • Construction • Similar to method used for calculating strings • Follow all paths in parallel where states represent parallel paths

  6. Proof idea • Given NFA M1={Q,,,q0,F} construct DFA M2={Q’,,’,q0’,F’} with L(M1)=L(M2) • Intuition • Recall :Q×εP(Q) • Our DFA’s transition function will generate paths within P(Q) • ’: P(Q)×P(Q)

  7. (r,a) is the set of state reached from r when a is processed R is a set of states from the NFA r is one of the states in R (i.e., a state in the NFA a is a symbol in the alphabet Defining M2 (ignoring  jumps) • Determine Q’, q0’, and F’ • Q’ = P(Q) • q0’ = {q0} • F’ = {R  Q’ | R  F  } • i.e., R contains at least one of M1’s accept states • Defining ’ (for now ignore ε jumps)

  8. Example 1 • Q’={,{q1}, {q2}, {q3}, {q1,q2}, {q1,q3}, {q2,q3}, {q1,q2,q3}} q2 0 q1 1 0 q3

  9. Example 1 • q0’={q1} q2 0 q1 1 0 q3

  10. Example 1 • F’={{q3}, {q1,q3}, {q2,q3}, {q1,q2,q3}} q2 0 q1 1 0 q3

  11. {q2,q3} 0 0 1 1 q2 0 0 q1 {q1} 1 0 q3 1 1  0,1 Example

  12. What about ε jumps? • For each R P(Q), define function E(R) E(R) = {q | q can be reach by 0 or more ε jumps from some r  R} • Redefine ’(R,a) to include E(R) ’(R,a) = {q | q  E((r,a)) for some r  R} • Are we done? No! What if there are  jumps from q0? q0’ = E({q0})

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