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2110711 THEORY OF COMPUTATION

2110711 THEORY OF COMPUTATION. 07 NON-DETERMINISTIC FINITE AUTOMATA. IMPORTANT. We introduce a conceptual machine that occurs in practice more frequently than the transition graph. TRANSITION GRAPH. Definition A transition graph (abbreviated TG) is a 5-tuple (Q,  , q 0 ,  , A) where

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2110711 THEORY OF COMPUTATION

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  1. 2110711THEORY OF COMPUTATION 07 NON-DETERMINISTIC FINITE AUTOMATA

  2. IMPORTANT • We introduce a conceptual machine that occurs in practice more frequently than the transition graph.

  3. TRANSITION GRAPH • Definition • A transition graph (abbreviated TG) is a 5-tuple (Q, , q0, , A) where • Q means a finite set of states. •  is a finite input alphabet. • q0  Q named Initial state. • A  Q , A is the set of all accepted states. •  is a function from Q* to P(Q), called transition function.

  4. TRANSITION GRAPH 1 1 q1 01 q0 0 1 001 010 q3 q2 1

  5. EXAMPLE • (0+1)*00(0+1)* 0,1 0,1 0 0 q0 q1 q2

  6. EXAMPLE • (010+101)*0(0+1)* q3 0 q2 1 0,1 1 0 q0 q1 0 0 q4 q5 1

  7. NON-DETERMINISTIC MACHINE • A nondeterministic finite state machine (NFA) is a transition graph with a unique start state and each of its edge labels is a single alphabet.

  8. NON-DETERMINISTIC MACHINE • Definition • A nondeterministic finite automaton (abbreviated NFA) is a 5-tuple (Q,,q0,A,). where • Q means a finite set of states. •  is a finite input alphabet. • q0 is in Q named Initial state. • A is a subset of Q, A is the set of all accepted states. •  is a function from Q   to P(Q) • where P(Q) is the power set of Q.

  9. NON-DETERMINISTIC MACHINE 0,1 0,1 0,1 0 A C D B *(A,00) *(A,00) = (r,0) : r *(A,0) : r  { A , B } = { A , B }  { C } = { A , B , C } How to define the transition function?

  10. NON-DETERMINISTIC MACHINE 0,1 0,1 0,1 0 A C D B *(A,010) =  (r,0) : r  *(A,01) *(A,01) =  (s,1) : s  *(A,0) : s  { A , B } = (A,1)  (B,1) = { A }  { C } : r  { A , C } *(A,010) = (A,0)  (C,0) = { A , B }  { D } = { A , B , D } *(A,010)

  11. NON-DETERMINISTIC MACHINE • For a NFA, M = (Q, , q0, , A) and any pQ, *(p,) = {p}. • For any pQ and x = a1a2a3…an  * (with n  1) • *(p, x) is the set of all states q for which • there is a sequence of states p=p0, p1p2…pn-1, pn = q satisfying : pi(pi-1,ai) for each i with 1  i  n. • String x in * is accepted by a NFA if *(q0,x)  A  . *(p,) = {p} *(p,yan) = all r*(p,y)(r,an)

  12. NON-DETERMINISTIC MACHINE • Definition • A nondeterministic finite automaton with -transition (NFA-) is a 5-tuple (Q, , q0, , A) where • Q means a finite set of states. •  is a finite input alphabet. • q0  Q named Initial state. • A  Q , A is the set of all accepted states. •  is a function from Q({}) to P(Q) • where P(Q) is the power set of Q.

  13. NON-DETERMINISTIC MACHINE • For an NFA-, M=(Q, , q0, , A) states p and q  Q and a string • x = a1a2a3…an *, we will say M moves from p to q by a sequence of transitions corresponding to x if • there exist an integer m  n, a sequence b1b2b3…bm{} • satisfying x = b1b2b3…bm and a sequence of states • p = p0, p1, p2, …, pm = q • so that for each i, 1  i  m, pi (pi-1,bi). • For x * and p  Q, *(p,x) is the set of all states q  Q such that there is a sequence of transitions corresponding to x by which M moves from p to q.

  14. -CLOSURE • Definition • Let M=(Q, , q0, , A) be a NFA-. • Let S be any subset of Q. • The -closure of S is the set (S) defined as follows: • Every element of S is an element of (S). • For any q (S), every element of (q,) is in (S). • No other elements of Q are in (S).

  15. -CLOSURE • For a NFA- M = (Q, , q0, , A). • The extended transition function *: Q *  P(Q) is defined as follows: • For any q  Q, *(q,) = ({q}) • For any q  Q, y * and a  , • *(q,ya) = (  r*(q,y)(r,a) ) • A string x is accepted by M if *(q0,x)  A  . • The language recognized by M is the set L(M) of all strings accepted by M.

  16. NON-DETERMINISTIC MACHINE • Theorem • For every nondeterministic finite state machine, there is a finite state machine accepts exactly the same language. PROOF

  17. NON-DETERMINISTIC MACHINE • Theorem • For every nondeterministic finite state machinewith -transition, there is a finite state machine accepts exactly the same language. PROOF

  18. EXERCISE • จงยกตัวอย่าง ภาษาที่บรรยายด้วยการบรรยายแบบสม่ำเสมอ ที่มี non-deterministic automaton with -transition ที่ accept ทุกสมาชิกในภาษาได้ แต่ไม่สามารถถูกยอมรับได้ด้วย non-deterministic finite automaton ที่มี accepted state เพียงสถานะเดียว

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