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1 Module 20 NFA’s with -transitions –NFA- ’s Formal definition Simplifies construction –LNFA- –Showing LNFA is a subset of LNFA (extra credit) and therefore a subset of LFSA

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2 Defining NFA- ’s

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3 Change: -transitions We now allow an NFA M to change state without reading input That is, we add the following categories of transitions to – (q is allowed

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4 Example * a,b a aba b aab

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5 Defining L(M) and LNFA- * M accepts string x if one of the configurations reached is an accepting configuration –(q 0, x) |- * (f, ),f e A M rejects string x if all configurations reached are either not halting configurations or are rejecting configurations L(M) or Y(M) N(M) LNFA- –Language L is in language class LNFA- iff

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6 LNFA- subset LFSA Recap of what we already know –Let M be any NFA –There exists an algorithm A 1 which constructs an FSA M’ such that L(M’) = L(M) New goal –Let M be any NFA- –There exists an algorithm A 2 which constructs an FSA M’ such that L(M’) = L(M)

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7 Visualization Goal –Let M be any NFA- –There exists an algorithm A 2 which constructs an FSA M’ such that L(M’) = L(M) NFA- M FSA M’ A2A2

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8 Modified Goal Question –Can we use any existing algorithms to simplify the task of developing algorithm A 2 ? Yes, we can use algorithm A 1 which converts an NFA M 1 into an FSA M’ such that L(M’) = L(M 1 ) NFA- M FSA M’A2A2 NFA- M FSA M’ Algorithm A 2 NFA M 1 A1A1 A 2’

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9 New Goal (extra credit) Difficulty –NFA- M can make transitions on –How can the NFA M 1 simulate these -transitions? NFA- M NFA M 1 A 2’ a bb 234561

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10 Basic Idea For each state q of M and each character of , figure out which states are reachable from q taking any number of -transitions and exactly one transition on that character . In the NFA- M 1, directly connect q to each of these states using an arc labeled with . NFA- M NFA M 1 A 2’ a bb 234561 234561 b b b

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11 Process State 2 NFA- M NFA- M 1 A 2’ a bb 234561 234561 b b b b b b

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12 Process State 3 NFA- M NFA- M 1 A 2’ a bb 234561 234561 b b b b b b a a b

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13 Final Picture NFA- M NFA- M 1 A 2’ a bb 234561 234561 b b b b b b a a b b b a a

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14 Construction Input NFA- M = (Q, , q 0, , A) Output NFA M 1 = (Q 1, 1, q 1, 1, A 1 ) –What is Q 1 ? Q 1 = Q In this case, Q 1 = {1,2,3,4,5,6} –What is 1 ? 1 = In this case, 1 = = {a,b} –What is q 1 ? We always make q 1 = q 0 In this case q 1 = 1 a bb 234561

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15 Construction Input NFA- M = (Q, , q 0, , A) Output NFA M 1 = (Q 1, 1, q 1, 1, A 1 ) –What is 1 ? 1 (q,a) = the set of states reachable from state q in M taking any number of -transitions and exactly one transition on the character a –More on this later In this case – 1 (1,a) = {} – 1 (1,b) = {3,4,5} –What is A 1 ? A 1 = A with one minor change –If an accepting state is reachable from q 0 using only -transitions, then we make q 1 an element of A 1 In this case, using only -transitions, no accepting state is reachable from q 0, so A 1 = A a bb 234561

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16 Computing 1 (q,a) 1 (q,a) = the set of states reachable from state q in M taking 0 or more -transitions and exactly one transition on the character a –Break this down into three steps First compute all states reachable from q using 0 or more -transitions –We call this set of states (q) Next, compute all states reachable from any element of (q) using the character a –We can denote these states as ( (q),a) Finally, compute all states reachable from states in ( (q),a) using 0 or more -transitions –We denote these states as ( ( (q),a)) –This is the desired answer a bb 234561

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17 Example 1 (1,b) = {3,4,5} –Compute (1), all states reachable from state 1 using 0 or more - transitions (1) = {1,2} –Compute ( (q),b), all states reachable from any element (1) of using the character b: ( (q),b) = ({1,2},b) = (1,b) U (2,b) = {} U {3} = {3} –Compute ( ( (q),a)), all states reachable from states in ( (q),a) using 0 or more -transitions ( ( (q),a)) = (3) = {3,4,5} a bb 234561

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18 Comments For extra credit, you should be able to execute this algorithm –Convert any NFA- into an equivalent NFA. For extra credit, you should understand the idea behind this algorithm –Why the transition function is computed the way it is –Why A 1 may need to include q 1 in some cases You should understand the importance of this algorithm –Compiler role again –Use in combination with previous algorithm for converting any NFA into an equivalent FSA to create a new algorithm for converting any NFA- into an equivalent FSA

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19 LNFA- = LFSA Implications –Let us primarily use the term LFSA to refer to this language class –Given a language L is in LFSA We know there exists an FSA M s.t. L(M) = L We know there exists an NFA M s.t. L(M) = L –To show a language L is in LFSA Show there exists an FSA M s.t. L(M) = L Show there exists an NFA- M s.t. L(M) = L

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1 Lecture 20 LNFA subset of LFSA –Theorem 4.1 on page 105 of Martin textbook –Compare with set closure proofs Main idea –A state in FSA represents a set.

1 Lecture 20 LNFA subset of LFSA –Theorem 4.1 on page 105 of Martin textbook –Compare with set closure proofs Main idea –A state in FSA represents a set.

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