Myhill-Nerode Theorem

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Lecture 22

• Myhill-Nerode Theorem
• distinguishability
• equivalence classes of strings
• designing FSAs
• 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 as and an odd number of
  bs
• 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(Societies in the
handout)
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Definition of equivalence classes

• Every language L partitions S 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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Example
How does EVEN-ODD partition a,b into
equivalence classes?
Strings with an EVEN number of as and an EVEN
number of bs
Strings with an ODD number of as and an EVEN
number of bs
Strings with an EVEN number of as and an ODD
number of bs
Strings with an ODD number of as and an ODD
number of bs
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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
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Designing FSAs
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Designing an FSA for EVEN-ODD
Even Even
Odd Even
Even Odd
Odd Odd
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Designing an FSA for 1MOD3
Length mod 3 0
Length mod 3 1
Length mod 3 2
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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 as in x the number of bs in x
• How does EQUAL partition a,b into equivalence
  classes?
• Strings with an equal number of as and bs
• Strings with one extra a
• Strings with one extra b
• Strings with two extra as
• Strings with two extra bs
• 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.

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Myhill-Nerode Theorem
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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

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Implication 1

• Method for constructing FSAs 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

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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 as in x the number of bs 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

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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 the 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
• Strings with one extra a
• Strings with two extra as
• Strings with three extra as
• 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 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 FSAs
• Provides method for proving a language L is not
  regular
• Show that L partitions S into an infinite number
  of equivalence classes