Myhill-Nerode Theorem

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## Statement

## Proof

## Use and consequences

## Generalization

## See also

## References

## Further reading

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Myhill%E2%80%93Nerode Theorem

In the theory of formal languages, the **Myhill-Nerode theorem** provides a necessary and sufficient condition for a language to be regular. The theorem is named for John Myhill and Anil Nerode, who proved it at the University of Chicago in 1958 (Nerode 1958).

Given a language *L*, and a pair of strings *x* and *y*, define a **distinguishing extension** to be a string *z* such that
exactly one of the two strings *xz* and *yz* belongs to *L*.
Define a relation *R _{L}* on strings by the rule that

The Myhill-Nerode theorem states that *L* is regular if and only if *R _{L}* has a finite number of equivalence classes, and moreover that the number of states in the smallest deterministic finite automaton (DFA) recognizing

If *L* is a regular language, then by definition there is a DFA *A* that recognizes it, with only finitely many states. If there are *n* states, then partition the set of all finite strings into *n* subsets, where subset *S _{i}* is the set of strings that, when given as input to automaton

In the other direction, suppose that *R _{L}* has finitely many equivalence classes. In this case, it is possible to design a deterministic finite automaton that has one state for each equivalence class. The start state of the automaton corresponds to the equivalence class containing the empty string, and the transition function from a state

Thus, the existence of a finite automaton recognizing *L* implies that the Myhill-Nerode relation has a finite number of equivalence classes, at most equal to the number of states of the automaton, and the existence of a finite number of equivalence classes implies the existence of an automaton with that many states.

The Myhill-Nerode theorem may be used to show that a language *L* is regular by proving that the number of equivalence classes of *R _{L}* is finite. This may be done by an exhaustive case analysis in which, beginning from the empty string, distinguishing extensions are used to find additional equivalence classes until no more can be found. For example, the language consisting of binary representations of numbers that can be divided by 3 is regular. Given the empty string,

Another immediate corollary of the theorem is that if a language defines an infinite set of equivalence classes, it is *not* regular. It is this corollary that is frequently used to prove that a language is not regular.

The Myhill-Nerode theorem can be generalized to trees. See tree automaton.

- Pumping lemma for regular languages, an alternative method for proving that a language is not regular. The pumping lemma may not always be able to prove that a language is not regular.

- Hopcroft, John E.; Ullman, Jeffrey D. (1979), "Chapter 3",
*Introduction to Automata Theory, Languages, and Computation*, Reading, Massachusetts: Addison-Wesley Publishing, ISBN 0-201-02988-X. - Nerode, Anil (1958), "Linear Automaton Transformations",
*Proceedings of the AMS*,**9**, JSTOR 2033204. - Regan, Kenneth (2007),
*Notes on the Myhill-Nerode Theorem*(PDF), retrieved .

- Bakhadyr Khoussainov; Anil Nerode (6 December 2012).
*Automata Theory and its Applications*. Springer Science & Business Media. ISBN 978-1-4612-0171-7.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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