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The Myhill–Nerode theorem may be used to show that a language is regular by proving that the number of equivalence classes of 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.
The Myhill–Nerode theorem provides a test that exactly characterizes regular languages. The typical method for proving that a language is regular is to construct either a finite-state machine or a regular expression for the language.
By the Myhill–Nerode theorem, A/ ≈ is a deterministic automaton that recognizes the same language as A. [1]: 65–66 As a consequence, the quotient of A by every refinement of ≈ also recognizes the same language as A.
To prove that a language is not regular, one often uses the Myhill–Nerode theorem and the pumping lemma. Other approaches include using the closure properties of regular languages [28] or quantifying Kolmogorov complexity. [29] Important subclasses of regular languages include Finite languages, those containing only a finite number of words. [30]
The state of a deterministic finite automaton = (,,,,) is unreachable if no string in exists for which = (,).In this definition, is the set of states, is the set of input symbols, is the transition function (mapping a state and an input symbol to a set of states), is its extension to strings (also known as extended transition function), is the initial state, and is the set of accepting (also ...
The Myhill–Nerode theorem states: a language is regular if and only if the family of quotients {|} is finite, or equivalently, the left syntactic equivalence has finite index (meaning it partitions into finitely many equivalence classes).
With John Myhill, Nerode proved the Myhill–Nerode theorem specifying necessary and sufficient conditions for a formal language to be regular. [3] [4] [5] With Bakhadyr Khoussainov, Nerode founded the theory of automatic structures, an extension of the theory of automatic groups.
The Myhill–Nerode theorem for tree automata states that the following three statements are equivalent: [14] L is a recognizable tree language; L is the union of some equivalence classes of a congruence of finite index; the relation ≡ L is a congruence of finite index