Non-Deterministic Finite Automata

1. Non-Deterministic Finite Automata

4. Formal Definition of DFA • A DFA consists of: • Alphabet • A set of statesQ • A transition functionδ : Q  Q • One start state q0 • One or more accepting states F  Q • Language accepted by a DFA is the set of strings such that DFA ends at an accepting state • Each string is c1c2…cn with ci   • States are qi = δ(qi-1,ci ) for i=1…n • qn is an accepting state

5. q2 1  q1 q1 q2 q3 1 empty string transition nondeterministic transition NFA and -NFA • Nondeterministic Finite Automata • Same input may produce multiple paths • Allows transition with an empty string or transition from one state to different states given a character

8. The Language Accepted by a NFA

11. EXAMPLE

12. Accept strings containing a 1 in the third position from the back 0,1 0, 1 1 2 3 4 1 0,1 Another Example

13. RECAP: How does NFA work? • Start in start state • Read a symbol, clone a machine for each matching transition • If a symbol is read and there is no way to exit from a state, then that machine dies • At end of input if any machine accepts then accept

14. RECAP Definition: NFA • An NFA is defined by a 5-tuple, with • Alphabet • A set of statesQ • A transition functionδ : Q Σ P(Q) • One start stateq0 • One or more accepting states F  Q • Notation: P(Q) is power set of Q • What is the difference from DFA?

15. RECAP Nondeterministic Transition • The functionδ : Q Σ P(Q)is the key difference! • When reading symbol `a’ while in state q, it may go to one of the states in δ(q,a)Q. • Can δ(q,a) map to empty set?

16. RECAP Languages • Language accepted by a NFA is the set of strings such that NFA ends at an accepting state • Each string of language is c1c2…cn with ciΣ(possibly with ε) • States are qi  δ(qi-1,ci) for i=1…n • qn is an accepting state

17. RECAP DFA = NFA • Theorem: For every language L accepted by an NFA, there is a DFA that accepts L. • In other words, DFA and NFA are equivalent computational models. • Proof idea: When keeping track of nondeterministic computation of NFA N, use many ‘fingers’ to point at the set of states of N that can be reached on a given input string. We can simulate this computation with a DFA M with state space P(Q).

18. 0,1 0, 1 1 2 3 4 1 0,1 Proof of DNA=NFA • More formal proof: Let A be the language recognized by the NFA N = (Q,Σ,δ,q0,F). Define the DFA M = (Q’,Σ,δ’,q’0,F’) by • Q’ = P(Q) • δ’(R,a) = { qQ | qδ(r,a) for an rR } • q’0 = { q0 } • F’ = {RQ’ | R contains an accept state of N}

20. How does NFA work? • Start in start state • If any  transitions, clone a machine for each  transition • Read a symbol, clone a machine for each matching transition • If a symbol is read and there is no way to exit from a state, then that machine dies • At end of input if any machine accepts then accept

21. 0,1 0,1 0,  1 1 2 3 4 1 Example • Accept strings containing either 101 or 11 as a substring

22. How would this NFA work? 0,1 0,1 0,  1 1 2 3 4 1 Example: Read: 010110

23. 1 0 1 1 1 2 3 0 1 3 1 1 2 3 4 1 2 1 3 4 4 0 1 4 4 Read: 010110

24. Definition: εNFA • An NFA is defined by a 5-tuple, with • Alphabet • A set of statesQ • A transition functionδ : Q Σε P(Q) • One start stateq0 • One or more accepting states F  Q • Notation: Σε = Σ{ε}; P(Q) is power set of Q • What is the difference from DFA?

25. Nondeterministic Transition • The functionδ : Q Σε P(Q)is the key difference! • When reading symbol `a’ while in state q, it may go to one of the states in δ(q,a)Q. • ε in Σε allows empty-string transitions • Can δ(q,a) map to empty set?

26. Languages • Language accepted by a NFA is the set of strings such that NFA ends at an accepting state • Each string of language is c1c2…cn with ciΣε (possibly with ε) • States are qi  δ(qi-1,ci) for i=1…n • qn is an accepting state

27. The Golden Chain εNFA  NFA  DFA REGEX

30. EpsilonNFA NFA  DFA

32. Regular Expressions