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COSC 3340: Introduction to Theory of Computation

COSC 3340: Introduction to Theory of Computation. University of Houston Dr. Verma Lecture 4. Formal definition of NFA acceptance. Define  *(q, w) as a set of states: p ε  *(q, w) if there is a directed path from q to p labeled w Example : consider NFA of Lecture 3

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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 4 UofH - COSC 3340 - Dr. Verma

  2. Formal definition of NFA acceptance • Define *(q, w) as a set of states: p ε*(q, w) if there is a directed path from q to p labeled w • Example: consider NFA of Lecture 3 • *(q0, 1) = ? • Ans: {q0, q1} • *(q0, 11) = ? • Ans: {q0, q1, q2} UofH - COSC 3340 - Dr. Verma

  3. NFA acceptance (contd.) • w is accepted by NFA M iff *(q0, w)  F isnonempty. • L(M) = {w in * | w is accepted by M}. UofH - COSC 3340 - Dr. Verma

  4. NFA vs. DFA • Is NFA more powerful than DFA? • Ans: No. • Theorem: • For every NFA M there is an equivalent DFA M' • Proof Idea: • NFA is in a set of states at any point during reading a string. • DFA will use a lot of states to keep track of this. • Important Assumption: • No transition labeled by epsilon. (Will get rid of this assumption later.) UofH - COSC 3340 - Dr. Verma

  5. Equivalent DFA construction. • NFA M = (Q, , , s, F) • DFA M' = (Q', , , s', F') where: • Q' = 2Q • s' = {s} • F' = {P ε Q' | P  F is nonempty} • ({p1, p2, …, pm}, ) = *(p1, ) *(p2, )  ... *(pm, ) i.e. find all the states that can be reached on  from all the NFA states in a DFA state. UofH - COSC 3340 - Dr. Verma

  6. Example: Equivalent DFA construction NFA UofH - COSC 3340 - Dr. Verma

  7. Equivalent DFA construction (contd.) UofH - COSC 3340 - Dr. Verma

  8. How to handle epsilon transitions? • Define e-closure of state q as *(q, ). • notation: e-closure(q). • Example: UofH - COSC 3340 - Dr. Verma

  9. Handling epsilon transitions (contd.) • Extend e-closure to sets of states by: • e-closure({s1, ... , sm}) = e-closure(s1)  ...  e-closure(sm) Now let s' = e-closure({s}). and, ({p1,..., pm}, ) = e-closure(*(p1, ))  ...  e-closure(*(pm, )) to complete construction of DFA. UofH - COSC 3340 - Dr. Verma

  10. Example: Handling epsilon transitions. UofH - COSC 3340 - Dr. Verma

  11. DFA = ? UofH - COSC 3340 - Dr. Verma

  12. Language Operations • Concatenation. Notation: LL' or just LL' • L  L' = {uv | u in L, v in L'}. • Kleene Star. Notation: L* • L* = { w in * | w = w1...wk for some k >= 0 and each wi in L}. • Examples: if L = {a(2n+1) | n >= 0}. L' = {b(2n) | n > = 0}. • LL' = ? • Ans: LL' = {a(2n+1) b(2m)| n, m > = 0} • L* = ? • Ans: {an | n >= 0} • U, ., * are called regular operations. UofH - COSC 3340 - Dr. Verma

  13. Closure properties of regular languages. • Previously we saw closure under  and . • New: Regular languages are closed under • Concatenation • Kleene star • Complement. UofH - COSC 3340 - Dr. Verma

  14. Examples L = {w in {a,b}* | w has even a’s } UofH - COSC 3340 - Dr. Verma

  15. Examples L' = {w in {a,b}* | w has at least one b} UofH - COSC 3340 - Dr. Verma

  16. Construction for LL' L’’ = (K,,,s,F)K = K1  K2s = s1F = F2 = 1  2  F1 X {e} X {s2} UofH - COSC 3340 - Dr. Verma

  17. L* and L'* L* M = (K, , , s, F)K = {s}  K1F = {s}  F1 = 1  F1 X {e} X {s1}  {(s, e, s1)} Given M1 = (K1, , 1, s1, F1) L’* UofH - COSC 3340 - Dr. Verma

  18. Complement of L and L' Complement of L Complement of L’ UofH - COSC 3340 - Dr. Verma

  19. General Construction for Complement DFA M = (K, , δ, s, F)K = K1s = s1F = K - F1δ = δ1 L(M) = Complement of L(M1) DFA M1 = (K1, , δ1, s1, F1) Exercise: Will this construction work for NFAs? Explain your answer. UofH - COSC 3340 - Dr. Verma

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