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Module 24

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

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Distinguishability

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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

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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?

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Equivalence classes of strings

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Definition of equivalence classes

• Every language L partitions * 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

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ExampleHow does EVEN-ODD partition {a,b}* into equivalence classes?

Strings with anEVEN number of a’s

and anEVEN number of b’s

Strings with anEVEN number of a’s

and anODD number of b’s

Strings with anODD number of a’s

and anEVEN number of b’s

Strings with anODD number of a’s

and anODD number of b’s

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Second ExampleLet 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

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Designing FSA’s

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Designing an FSA for EVEN-ODD

EvenEven

EvenOdd

OddOdd

OddEven

b ab

aa

b b

a

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Designing an FSA for 1MOD3

Length mod 3 = 0

Length mod 3 = 1

Length mod 3 = 2

a aaa,b a,b

a,b

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Proving a language is not regular

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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?• How many equivalence classes are there?• Can we construct a finite state automaton for

EQUAL?

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Myhill-Nerode Theorem

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Theorem Statement

• Two part statement– If L is regular, then L partitions * into a finite

number of equivalence classes– If L partitions * into a finite number of

equivalence classes, then L is regular

• One part statement– L is regular iff L partitions * into a finite

number of equivalence classes

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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

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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

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Proving a language is not regular revisited

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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.

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Table *

aaa

aaaaaaa

aaaaa...

bINOUTOUTOUTOUT...

bbOUTINOUTOUTOUT...

bbbOUTOUT1

INOUTOUT...

bbbbOUTOUTOUTINOUT...

bbbbbOUTOUTOUTOUTIN...

………………

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.

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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

• Thus, the Myhill-Nerode Theorem implies that EQUAL is nonregular

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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 * are not finite

– It says that a language L is regular iff L partitions * 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 * into an infinite number of

equivalence classes

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Three Types of Problems

• Create a table that helps prove that a specific language L is not regular– You get to choose the “row” and “column”

strings– I choose the “row” strings

• Identify the equivalence classes defined by L as highlighted by a given table