Myhill–Nerode theorem

Myhill–Nerode theorem This article needs additional citations for verification. Aiutaci a migliorare questo articolo aggiungendo citazioni a fonti affidabili. Il materiale non fornito può essere contestato e rimosso. Trova fonti: "Myhill–Nerode theorem" – news · newspapers · books · scholar · JSTOR (settembre 2020) (Scopri come e quando rimuovere questo messaggio modello) 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).

Contenuti 1 Dichiarazione 2 Prova 3 Use and consequences 4 generalizzazioni 5 Guarda anche 6 Riferimenti 7 Further reading Statement Given a language {stile di visualizzazione L} , and a pair of strings {stile di visualizzazione x} e {stile di visualizzazione y} , define a distinguishing extension to be a string {stile di visualizzazione con} such that exactly one of the two strings {displaystyle xz} e {displaystyle yz} belongs to {stile di visualizzazione L} . Define a relation {stile di visualizzazione {}_{l}{sim }} on strings as {stile di visualizzazione x;{}_{l}{sim } y} iff there is no distinguishing extension for {stile di visualizzazione x} e {stile di visualizzazione y} . It is easy to show that {stile di visualizzazione {}_{l}{sim }} is an equivalence relation on strings, and thus it divides the set of all strings into equivalence classes.

The Myhill–Nerode theorem states that a language {stile di visualizzazione L} is regular if and only if {stile di visualizzazione {}_{l}{sim }} has a finite number of equivalence classes, and moreover, that this number is equal to the number of states in the minimal deterministic finite automaton (DFA) recognizing {stile di visualizzazione L} . In particolare, this implies that there is a unique minimal DFA for each regular language (Hopcroft & Ullman 1979).

Some authors refer to the {stile di visualizzazione {}_{l}{sim }} relation as Nerode congruence,[1][2] in honor of Anil Nerode.

Proof If {stile di visualizzazione L} is a regular language, then by definition there is a DFA {stile di visualizzazione A} that recognizes it, with only finitely many states. If there are {stile di visualizzazione n} states, then partition the set of all finite strings into {stile di visualizzazione n} subsets, where subset {stile di visualizzazione S_{io}} is the set of strings that, when given as input to automaton {stile di visualizzazione A} , cause it to end in state {stile di visualizzazione i} . For every two strings {stile di visualizzazione x} e {stile di visualizzazione y} that belong to the same subset, and for every choice of a third string {stile di visualizzazione con} , automaton {stile di visualizzazione A} reaches the same state on input {displaystyle xz} as it reaches on input {displaystyle yz} , and therefore must either accept both of the inputs {displaystyle xz} e {displaystyle yz} or reject both of them. Perciò, no string {stile di visualizzazione con} can be a distinguishing extension for {stile di visualizzazione x} e {stile di visualizzazione y} , so they must be related by {stile di visualizzazione {}_{l}{sim }} . così, {stile di visualizzazione S_{io}} is a subset of an equivalence class of {stile di visualizzazione {}_{l}{sim }} . Combining this fact with the fact that every member of one of these equivalence classes belongs to one of the sets {stile di visualizzazione S_{io}} , this gives a many-to-one relation from states of {stile di visualizzazione A} to equivalence classes, implying that the number of equivalence classes is finite and at most {stile di visualizzazione n} .

In the other direction, supporre che {stile di visualizzazione {}_{l}{sim }} has finitely many equivalence classes. In questo caso, 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 {stile di visualizzazione X} on input symbol {stile di visualizzazione y} takes the automaton to a new state, the state corresponding to the equivalence class containing string {displaystyle xy} , dove {stile di visualizzazione x} is an arbitrarily chosen string in the equivalence class for {stile di visualizzazione X} . The definition of the Myhill–Nerode relation implies that the transition function is well-defined: no matter which representative string {stile di visualizzazione x} is chosen for state {stile di visualizzazione X} , the same transition function value will result. A state of this automaton is accepting if the corresponding equivalence class contains a string in {stile di visualizzazione L} ; in questo caso, ancora, the definition of the relation implies that all strings in the same equivalence class must also belong to {stile di visualizzazione L} , for otherwise the empty string would be a distinguishing string for some pairs of strings in the class.

così, the existence of a finite automaton recognizing {stile di visualizzazione 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.

Use and consequences The Myhill–Nerode theorem may be used to show that a language {stile di visualizzazione L} is regular by proving that the number of equivalence classes of {stile di visualizzazione {}_{l}{sim }} è finito. 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. Per esempio, the language consisting of binary representations of numbers that can be divided by 3 is regular. Given the empty string, {stile di visualizzazione 00} (o {stile di visualizzazione 11} ), {stile di visualizzazione 01} e {stile di visualizzazione 10} are distinguishing extensions resulting in the three classes (corresponding to numbers that give remainders 0, 1 e 2 when divided by 3), but after this step there is no distinguishing extension anymore. The minimal automaton accepting our language would have three states corresponding to these three equivalence classes.

Another immediate corollary of the theorem is that if for a language {stile di visualizzazione L} the relation {stile di visualizzazione {}_{l}{sim }} has infinitely many equivalence classes, it is not regular. It is this corollary that is frequently used to prove that a language is not regular.

Generalizations The Myhill–Nerode theorem can be generalized to tree automata.

See also 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. Syntactic monoid References ^ Brzozowski, Janusz; Szykuła, Marek; Ye, Yuli (2018), "Syntactic Complexity of Regular Ideals", Theory of Computing Systems, 62 (5): 1175–1202, doi:10.1007/s00224-017-9803-8, hdl:10012/12499, S2CID 2238325 ^ Crochemore, Maxime; et al. (2009), "From Nerode's congruence to suffix automata with mismatches", Theoretical Computer Science, 410 (37): 3471–3480, doi:10.1016/j.tcs.2009.03.011 Hopcroft, John E.; Ullman, Jeffrey D. (1979), "Capitolo 3", Introduction to Automata Theory, Languages, and Computation, Lettura, Massachusetts: Addison-Wesley Publishing, ISBN 0-201-02988-X. Nerode, Anil (1958), "Linear Automaton Transformations", Proceedings of the AMS, 9 (4): 541–544, doi:10.1090/S0002-9939-1958-0135681-9, JSTOR 2033204. Regan, Kenneth (2007), Notes on the Myhill-Nerode Theorem (PDF), recuperato 2016-03-22. Further reading Bakhadyr Khoussainov; Anil Nerode (6 Dicembre 2012). Automata Theory and its Applications. Springer Science & Business Media. ISBN 978-1-4612-0171-7. Categorie: Formal languagesTheorems in discrete mathematicsFinite automata