1 Module 16 Distinguishability –Definition –Help in designing/debugging FSA’s

2 Distinguishability

3 Questions * Let L be the set of strings over {a,b} which end with aaba. Let M be an FSA such that L(M) = L. Questions –Can aaba and aab end up in the same state of M? Why or why not? –How about aa and aab? –How about or a? –How about b or bb? –How about  or bbab?

4 Definition * String x is distinguishable from string y with respect to language L iff there exists a string z such that –xz is in L and yz is not in L OR –xz is not in L and yz is in L When reviewing, identify the z for pair of strings on the previous slide

5 Questions * Let L be the set of strings over {a,b} that have length 2 mod 5 or 4 mod 5. Let M be an FSA such that L(M) = L. Questions –Are aa and aab distinguishable with respect to L? Can they end up in the same state of M? –How about aa and aaba? –How about and a? –How about b and aabbaa?

6 One design method –Is in L? Implication? –Is a distinguishable from  wrt L? Implication? –Is b distinguishable from  wrt L? Implication? –Is b distinguishable from  a wrt L? Implication? L = set of strings x over {a,b} such that length of x is 2 or 4 mod 5 Design an FSA to accept L *

7 Design continued –Is aa distinguishable from  wrt L? Implication? –Is aa distinguishable from  a wrt L? Implication? L = set of strings x over {a,b} such that length of x is 2 or 4 mod 5 Design an FSA to accept L

8 Design continued –What strings would we compare ab to? –What results do we get? –Implications? –How about ba? –How about bb? L = set of strings x over {a,b} such that length of x is 2 or 4 mod 5 Design an FSA to accept L *

9 Design continued –We can continue in this vein, but it could go on forever –Now lets try something different –Consider string. What set of strings are indistinguishable from it wrt L? Implications? L = set of strings x over {a,b} such that length of x is 2 or 4 mod 5 Design an FSA to accept L *

10 Design continued –Consider string a. What set of strings are indistinguishable from it wrt L? Implications? –Consider string aa. What set of strings are indistinguishable from it wrt L? Implications? L = set of strings x over {a,b} such that length of x is 2 or 4 mod 5 Design an FSA to accept L *

11 Debugging an FSA Do essentially the same thing –Identify some strings which end up in each state –Try and generalize each state to describe the language of strings which end up at that state.

12 Example 1 aaa a,b b b b b a IIIIIIIVV VI

13 Example 2 IIIIIIIV V aaa a,b b b b b a

14 Example 3 I II III IV a b a a b b a,b