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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.
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]
This string is in exactly when = and thus is not regular by the Myhill–Nerode theorem. 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.
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 ...
Anil Nerode (born 1932) is an American mathematician, known for his work in mathematical logic and for his many-decades tenure as a professor at Cornell University. He received his undergraduate education and a Ph.D. in mathematics from the University of Chicago , the latter under the directions of Saunders Mac Lane .
The study of linear bounded automata led to the Myhill–Nerode theorem, [8] which gives a necessary and sufficient condition for a formal language to be regular, and an exact count of the number of states in a minimal machine for the language.
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
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.