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 of states in original NFA

2 LNFA subset LFSA Let L be an arbitrary language in LNFA Let M be the NFA such that L(M) = L –M exists by definition of L in LNFA Construct an FSA M’ such that L(M’) = L Argue L(M’) = L There exists an FSA M’ such that L(M’) = L L is in LFSA –By definition of L in LFSA

3 Visualization LNFA LFSA NFA’s FSA’s L L M M’ Let L be an arbitrary language in LNFA Let M be an NFA such that L(M) = L M exists by definition of L in LNFA Construct FSA M’ from NFA M Argue L(M’) = L There exists an FSA M’ such that L(M’) = L L is in LFSA

4 Comparison L 1 intersect L 2 L1L1 L2L2 LFSA M3M3 M1M1 M2M2 FSA’s LNFA LFSA NFA’s FSA’s L L M M’

5 Construction Specification We need an algorithm which does the following –Input: NFA M –Output: FSA M’ such that L(M’) = L(M)

6 An NFA can be in several states after processing an input string x –Example The given NFA can be in states {1,2,3} after processing string aa Difficulty aaab a,b Input string aaaaba (1, aaaaba) (1, aaaba)(2, aaaba) (1, aaba)(2, aaba)(3, aaba) (1, aba)(2, aba)(3, aba)crash (1, ba)(2, ba)(3, ba) (1, a)(4, a) (1, )(2, )(5, )

7 All strings which end up in the set of states {1,2,3} are indistinguishable with respect to L(M) –Example Strings aa, aaa, aaaa all end up in the same set of states {1,2,3} For all strings z, if aaz is in L(M), then aaaz and aaaaz are also in L(M) Observation aaab a,b Input string aaaaba (1, aaaaba) (1, aaaba)(2, aaaba) (1, aaba)(2, aaba)(3, aaba) (1, aba)(2, aba)(3, aba)crash (1, ba)(2, ba)(3, ba) (1, a)(4, a) (1, )(2, )(5, )

8 Given an NFA M = (Q, ,q 0, ,A), the equivalent FSA M’ should have one state for each subset of Q Example –In this case there are 5 states in Q –There are 2 5 subsets of Q including {} and Q –The FSA M’ will have 2 5 states What strings end up in state {1,2,3} of M’? –The strings which end up in states 1, 2, and 3 of NFA M. –In this case, strings which do not contain aaba and end with aa such as aa, aaa, and aaaa. Idea aaab a,b Input string aaaaba

9 Idea Illustrated aaab a,b Input string aaaaba (1,aaaaba)({1}, aaaaba) (1, aaaba)(2, aaaba)({1,2}, aaaba) (1, aaba)(2, aaba)(3, aaba) ({1,2,3}, aaba) (1, aba)(2, aba)(3, aba)({1,2,3}, aba) (1, ba)(2, ba)(3, ba)({1,2,3}, ba) (1, a)(4, a) ({1,4}, a) ({1,2,5}, ) (1, )(2, )(5, )

10 Construction Input NFA M = (Q, , q 0, , A) Output FSA M’ = (Q’,  ’, q’,  ’, A’) –What is Q’? all subsets of Q including Q and {} In this case, Q’ = {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} –What is  ’? We always make  ’ =  In this case,  ’ =  = {a,b} –What is q’? We always make q’ = {q 0 } In this case q’ = {1} a,b aa NFA M 1 23

11 Construction Input NFA M = (Q, , q 0, , A) Output FSA M’ = (Q’,  ’, q’,  ’, A’) –What is A’? Suppose a string x ends up in states 1 and 2 of the NFA M above. –Is x accepted by M? »No, neither 1 nor 2 are in A. –Should {1,2} be an accepting state in FSA M’? »No, {1,2} should not be in A’ Suppose a string x ends up in states 1 and 2 and 3 of the NFA M above. –Is x accepted by M? »Yes, because state 3 is in A. –Should {1,2,3} be an accepting state in FSA M’? »Yes, {1,2,3} should be in A’. Suppose p = {q 1, q 2, …, q k } where q 1, q 2, …, q k are in Q p is in A’ iff at least one of the states q 1, q 2, …, q k is in A In this case, A’ = {{3},{1,3},{2,3},{1,2,3}} a,b aa NFA M 1 23

12 Construction Input NFA M = (Q, , q 0, , A) Output FSA M’ = (Q’,  ’, q’,  ’, A’) –What is  ’? If string x ends up in states 1 and 2 after being processed by the NFA above, where does string xa end up after being processed by the NFA above? –From state 1, the a will take the NFA to states 1 and 2 –From state 2, the a will take the NFA to state 3 Figuring out  ’(p,a) in general –Suppose p = {q 1, q 2, …, q k } where q 1, q 2, …, q k are in Q –Then  ’(p,a) =  (q 1,a) union  (q 2,a) union … union  (q k,a) »Similar to 2 FSA to 1 FSA construction –In this case »  ’({1,2},a) =  (1,a) union  (2,a) » = {1,2} union {3} » = {1,2,3} a,b aa NFA M 1 23

13 Construction Summary Input NFA M = (Q, , q 0, , A) Output FSA M’ = (Q’,  ’, q’,  ’, A’) –Q’ = all subsets of Q including Q and {} In this case, Q’ = {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} –  ’ =  In this case,  ’ =  = {a,b} –q’ ={q 0 } In this case, q’ = {1} –A’ Suppose p = {q 1, q 2, …, q k } where q 1, q 2, …, q k are in Q p is in A’ iff at least one of the states q 1, q 2, …, q k is in A –  ’ Suppose p = {q 1, q 2, …, q k } where q 1, q 2, …, q k are in Q Then  ’(p,a) =  (q 1,a) union  (q 2,a) union … union  (q k,a) a,b aa NFA M 1 23

14 Example Summary a,b aa NFA M 1 23 {1}{1,2}{1,2,3}{1,3} {}{2}{3}{2,3} a,b a a a a a b b b b b FSA M’

15 Example Summary Continued a,b aa NFA M 1 23 {1}{1,2}{1,2,3}{1,3} {}{2}{3}{2,3} a,b a a a a a b b b b b FSA M’ These states cannot be reached from initial state and are unnecessary.

16 Example Summary Continued a,b aa NFA M 1 23 {1}{1,2}{1,2,3}{1,3} a a a a b b b b Smaller FSA M’ By examination, we can see that state {1,3} is unnecessary. However, this is a case by case optimization. It is not a general technique or algorithm.

17 Example 2 a,b ab Step 1: name the three states of NFA M ABC NFA M

18 Step 2: transition table a,b ab ABC NFA M {A}{B}{} ab {B}{B,C}{B} {} {B,C}{B}{B,C}  ’({B,C},a) =  (B,a) U  (C,a) = {B} U {} = {B}  ’({B,C},b) =  (B,b) U  (C,b) = {B,C} U {} = {B,C}

19 Step 3: accepting states a,b ab ABC NFA M {A}{B}{} ab {B} {B,C} {} {B,C}{B}{B,C} Which states should be accepting? Any state which includes an accepting state of M, in this case, C. A’ = {{B,C}}

20 Step 4: Answer a,b ab ABC NFA M Initial state is {A} Set of final states A’ = {{B,C}} {A}{B}{} ab {B} {B,C} {} {B,C}{B}{B,C} This is sufficient. You do NOT need to turn this into a diagram.

21 Step 5: Optional a,b ab ABC NFA M a ab {A}{B}{B,C} b a a,b {} b FSA M’

22 Comments You should be able to execute this algorithm –You should be able to convert any NFA into an equivalent FSA. You should understand the idea behind this algorithm –For an FSA M’, strings which end up in the same state of M’ are indistinguishable wrt L(M’) –For an NFA M, strings which end up in the same set of states of M are indistinguishable wrt L(M)

23 Comments You should understand the importance of this algorithm –Design tool We can design using NFA’s A computer will convert this NFA into an equivalent FSA –FSA’s can be executed by computers whereas NFA’s cannot (or at least cannot easily be run by computers) –Chaining together algorithms Perhaps it is easy to build NFA’s to accept L 1 and L 2 Use this algorithm to turn these NFA’s to FSA’s Use previous algorithm to build FSA to accept L 1 intersect L 2 You should be able to construct new algorithms for new closure property proofs