Lecture 25

1. Lecture 25 • Myhill-Nerode Theorem • distinguishability • equivalence classes of strings • designing FSA’s • proving a language L is not regular

2. Distinguishability

3. Distinguishable and Indistinguishable • 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 • String x is indistinguishable from string y with respect to language L iff • for all strings z, • xz and yz are both in L OR • xz and yz are both not in L

4. Example • Let EVEN-ODD be the set of strings over {a,b} with an even number of a’s and an odd number of b’s • Is the string aa distinguishable from the string bb with respect to EVEN-ODD? • Is the string aa distinguishable from the string ab with respect to EVEN-ODD?

5. Equivalence classes of strings

6. Definition of equivalence classes • Every language L partitionsS* into equivalence classes via indistinguishability • Two strings x and y belong to the same equivalence class defined by L iff x and y are indistinguishable w.r.t L • Two strings x and y belong to different equivalence classes defined by L iff x and y are distinguishable w.r.t. L

7. Example How does EVEN-ODD partition {a,b}* into equivalence classes? Strings with an EVEN number of a’s and an EVEN number of b’s Strings with an ODD number of a’s and an EVEN number of b’s Strings with an EVEN number of a’s and an ODD number of b’s Strings with an ODD number of a’s and an ODD number of b’s

8. Second Example Let 1MOD3 be the set of strings over {a,b} whose length mod 3 = 1. How does 1MOD3 partition {a,b}* into equivalence classes? Length mod 3 = 0 Length mod 3 = 1 Length mod 3 = 2

9. Designing FSA’s

10. l a a b b b ab a Even Even Odd Even Designing an FSA for EVEN-ODD Even Odd Odd Odd

11. a,b l a aa a,b a,b Length mod 3 = 0 Designing an FSA for 1MOD3 Length mod 3 = 1 Length mod 3 = 2

12. Proving a language is not regular

13. Third Example • Let EQUAL be the set of strings x over {a,b} s.t. the number of a’s in x = the number of b’s in x • How does EQUAL partition {a,b}* into equivalence classes? • Strings with an equal number of a’s and b’s • Strings with one extra a • Strings with one extra b • Strings with two extra a’s • Strings with two extra b’s • … • There are an infinite number of equivalence classes. • Can we construct a finite state automaton for EQUAL? • We shall see that the answer is no.

14. Myhill-Nerode Theorem

15. Theorem Statement • Two part statement • If L is regular, then L partitions S* into a finite number of equivalence classes • If L partitions S* into a finite number of equivalence classes, then L is regular • One part statement • L is regular iff L partitions S* into a finite number of equivalence classes

16. Implication 1 • Method for constructing FSA’s to accept a language L • Identify equivalence classes defined by L • Make a state for each equivalence class • Identify initial and accepting states • Add transitions between the states • You can use a canonical element of each equivalence class to help with building the transition function d

17. Implication 2 • Method for proving a language L is not regular • Identify equivalence classes defined by L • Show there are an infinite number of such equivalence classes • Table format may help, but it is only a way to help illustrate that there are an infinite number of equivalence classes defined by L

18. Proving a language is not regular revisited

19. Proving EQUAL is not regular • Let EQUAL be the set of strings x over {a,b} s.t. the number of a’s in x = the number of b’s in x • We want to show that EQUAL partitions {a,b}* into an infinite number of equivalence classes • We will use a table that is somewhat reminiscent of the table used for diagonalization • Again, you must be able to identify the infinite number of equivalence classes being defined by the table. They ultimately represent the proof that EQUAL or whatever language you are working with is not regular.

20. Table … … … … … … b IN OUT OUT OUT OUT ... bb OUT IN OUT OUT OUT ... bbb OUT OUT IN OUT OUT ... bbbb OUT OUT OUT IN OUT ... bbbbb OUT OUT OUT OUT IN ... a aa aaa aaaa aaaaa ... The strings being distinguished are the rows. The tables entries indicate that the concatenation of the row string with the column string is in or not in EQUAL. Each complete column shows one row string is distinguishable from all the other row strings.

21. Concluding EQUAL is nonregular • We have shown that EQUAL partitions {a,b}* into an infinite number of equivalence classes • In this case, we only identified some of the equivalence classes defined by EQUAL, but that is sufficient • Strings with one extra a • Strings with two extra a’s • Strings with three extra a’s • … • Thus, the Myhill-Nerode Theorem implies that EQUAL is nonregular

22. Summary • Myhill-Nerode Theorem and what it says • It does not say a language L is regular iff L is finite • Many regular languages such as S* are not finite • It says that a language L is regular iff L partitions S* into a finite number of equivalence classes • Provides method for designing FSA’s • Provides method for proving a language L is not regular • Show that L partitions S* into an infinite number of equivalence classes