Nondeterministic finite automaton: Difference between revisions

Revision as of 19:04, 22 December 2023

Revision as of 17:49, 20 February 2024

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Like DFAs, NFAs only recognize [[regular language]]s.

NFAs were introduced in 1959 by [[Michael O. Rabin]] and [[Dana Scott]],<ref>{{cite journal |doi=10.1147/rd.32.0114 |lastlast1=Rabin |firstfirst1=M. O. |last2=Scott |first2=D. |title=Finite Automata and Their Decision Problems |journal=IBM Journal of Research and Development |volume=3 |issue=2 |pages=114–125 |date=April 1959 }}</ref> who also showed their equivalence to DFAs. NFAs are used in the implementation of [[regular expression]]s: [[Thompson's construction]] is an algorithm for compiling a regular expression to an NFA that can efficiently perform pattern matching on strings. Conversely, [[Kleene's algorithm]] can be used to convert an NFA into a regular expression (whose size is generally exponential in the input automaton).

NFAs have been generalized in multiple ways, e.g., [[#NFA with ε-moves|nondeterministic finite automata with ε-moves]], [[finite-state transducer]]s, [[pushdown automaton|pushdown automata]], [[Alternating finite automaton|alternating automata]], [[ω-automaton|ω-automata]], and [[probabilistic automaton|probabilistic automata]].

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* One can solve in linear time the [[emptiness problem]] for NFA, i.e., check whether the language of a given NFA is empty. To do this, we can simply perform a [[depth-first search]] from the initial state and check if some final state can be reached.

* It is [[PSPACE]]-complete to test, given an NFA, whether it is ''universal'', i.e., if there is a string that it does not accept.<ref>Historically shown in: {{Cite journal |lastlast1=Meyer |firstfirst1=A. R. |last2=Stockmeyer |first2=L. J. |date=1972-10-25 |title=The equivalence problem for regular expressions with squaring requires exponential space |journal=Proceedings of the 13th Annual Symposium on Switching and Automata Theory (SWAT) |___location=USA |publisher=IEEE Computer Society |pages=125–129 |doi=10.1109/SWAT.1972.29}} For a modern presentation, see [http://www.lsv.fr/~jacomme/1718/ca/td4_sol.pdf]</ref> As a consequence, the same is true of the ''inclusion problem'', i.e., given two NFAs, is the language of one a subset of the language of the other.

* Given as input an NFA ''A'' and an integer n, the [[counting problem (complexity)|counting problem]] of determining how many words of length ''n'' are accepted by ''A'' is intractable; it is [[♯P|'''#P'''-hard]]. In fact, this problem is complete (under [[parsimonious reduction]]s) for the complexity class [[SpanL]].<ref>{{Cite journal |lastlast1=Álvarez |firstfirst1=Carme |last2=Jenner |first2=Birgit |date=1993-01-04 |title=A very hard log-space counting class |url=https://wwwdx.sciencedirectdoi.comorg/science10.1016/article/pii/030439759390252O0304-3975%2893%2990252-O |journal=Theoretical Computer Science |volume=107 |issue=1 |pages=3–30 |doi=10.1016/0304-3975(93)90252-O |issn=0304-3975}}</ref>

==Application of NFA==