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The instances of the DFA minimization problem that cause the worst-case behavior are the same as for Hopcroft's algorithm. The number of steps that the algorithm performs can be much smaller than n , so on average (for constant s ) its performance is O ( n log n ) or even O ( n log log n ) depending on the random distribution on automata chosen ...
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.
A Myhill graph over an alphabet A is a directed graph with vertex set A and subsets of vertices labelled "start" and "finish". The language accepted by a Myhill graph is the set of directed paths from a start vertex to a finish vertex: the graph thus acts as an automaton. [6] The class of languages accepted by Myhill graphs is the class of ...
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.
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
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.
Brzozowski's algorithm for DFA minimization uses the powerset construction, twice. It converts the input DFA into an NFA for the reverse language, by reversing all its arrows and exchanging the roles of initial and accepting states, converts the NFA back into a DFA using the powerset construction, and then repeats its process.
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.