State complexity: Difference between revisions

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* DFA: <math>mn</math> states, see Maslov<ref name="Maslov" /> and Yu, Zhuang and Salomaa.<ref name="YuZhuang1994" />

* NFA: <math>m+n+1</math> states, see Holzer and Kutrib.<ref name="HolzerKutrib2003" />

* UFA: at least <math>\min(n,m)^{\Omega(\log(\min(n,m)))}</math>;<ref name="GoosKieferYuan2022" /> between <math>mn+m+n</math> and <math>m + nm 2^{0.79m}</math> states, see Jirásek, Jirásková and Šebej.<ref name="JirásekJirásková2016">{{Cite book|last1=Jirásek|first1=Jozef|title=Operations on Unambiguous Finite Automata|last2=Jirásková|first2=Galina|last3=Šebej|first3=Juraj|volume=9840|year=2016|pages=243–255|issn=0302-9743|doi=10.1007/978-3-662-53132-7_20|series=Lecture Notes in Computer Science|isbn=978-3-662-53131-0}}</ref>

* SVFA: <math>mn</math> states, see Jirásek, Jirásková and Szabari.<ref name="JirásekJirásková2015">{{Cite book|last1=Jirásek|first1=Jozef Štefan|title=Computer Science -- Theory and Applications|last2=Jirásková|first2=Galina|last3=Szabari|first3=Alexander|volume=9139|year=2015|pages=231–261|issn=0302-9743|doi=10.1007/978-3-319-20297-6_16|series=Lecture Notes in Computer Science|isbn=978-3-319-20296-9}}</ref>

* 2DFA: between <math>m+n</math> and <math>4m+n+4</math> states, see Kunc and Okhotin.<ref name="KuncOkhotin2012">{{cite journal|last1=Kunc|first1=Michal|last2=Okhotin|first2=Alexander|title=State complexity of operations on two-way finite automata over a unary alphabet|journal=Theoretical Computer Science|volume=449|year=2012|pages=106–118|issn=0304-3975|doi=10.1016/j.tcs.2012.04.010|doi-access=free}}</ref>

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If language L requires n states

then how many states does its ''[[complementation of automata|complement]]'' require?

* DFA: <math>n</math> states, by exchanging accepting and rejecting states.

* NFA: <math>2^n</math> states, see Birget.<ref name="Birget1993b">{{cite journal|last1=Birget|first1=Jean-Camille|title=Partial orders on words, minimal elements of regular languages, and state complexity|journal=Theoretical Computer Science|volume=119|issue=2|year=1993|pages=267–291|issn=0304-3975|doi=10.1016/0304-3975(93)90160-U}}</ref> or Jirásková<ref>{{Cite journal |last=Jirásková |first=Galina |date=2005 |title=State complexity of some operations on binary regular languages |url=https://linkinghub.elsevier.com/retrieve/pii/S0304397504006577 |journal=Theoretical Computer Science |language=en |volume=330 |issue=2 |pages=287–298 |doi=10.1016/j.tcs.2004.04.011}}, Theorem 5</ref>

* UFA: at least <math>n^{\tilde{\Omega}(\log n)}</math> states, see Göös, Kiefer and Yuan,<ref name="GoosKieferYuan2022">{{cite arXiv |last1=Göös |first1=Mika |last2=Kiefer |first2=Stefan |last3=Yuan |first3=Weiqiang |title=Lower Bounds for Unambiguous Automata via Communication Complexity |date=12 February 2022 |class=cs.FL |eprint=2109.09155}}</ref> (this follows an earlier bound by Raskin<ref>{{Cite journal |last=Raskin |first=Mikhail |date=2018 |title=A Superpolynomial Lower Bound for the Size of Non-Deterministic Complement of an Unambiguous Automaton |journal=DROPS-IDN/V2/Document/10.4230/LIPIcs.ICALP.2018.138 |language=en |publisher=Schloss-Dagstuhl - Leibniz Zentrum für Informatik |doi=10.4230/LIPIcs.ICALP.2018.138|doi-access=free }}</ref>); and at most <math>\sqrt{n+1} \cdot 2^{0.5n}</math> states, see Indzhev and Kiefer.<ref>{{cite journal |last1=Indzhev |first1=Emil |last2=Kiefer |first2=Stefan |title=On complementing unambiguous automata and graphs with many cliques and cocliques |journal=Information Processing Letters |date=1 August 2022 |volume=177 |page=106270 |doi=10.1016/j.ipl.2022.106270 |s2cid=234741832 |url=https://wwwora.sciencedirectox.comac.uk/scienceobjects/articleuuid:a36b96e8-fa8e-4ef9-b45b-2a625366cf54/piifiles/S0020019022000278rrj4305198 |access-date=29 May 2022 |ref=IndzhevKiefer22 |language=en |issn=0020-0190|doi-access=free |arxiv=2105.07470 }}</ref>

* SVFA: <math>n</math> states, by exchanging accepting and rejecting states.

* 2DFA: at least <math>n</math> and at most <math>4n</math> states, see Geffert, Mereghetti and Pighizzini.<ref name="GeffertMereghetti2007">{{cite journal|last1=Geffert|first1=Viliam|last2=Mereghetti|first2=Carlo|last3=Pighizzini|first3=Giovanni|title=Complementing two-way finite automata|journal=Information and Computation|volume=205|issue=8|year=2007|pages=1173–1187|issn=0890-5401|doi=10.1016/j.ic.2007.01.008|doi-access=free}}</ref>

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* DFA: <math>n</math> states.

* NFA: <math>g(n)+O(n^2)</math> states, see Holzer and Kutrib.<ref name="HolzerKutrib2003" />

* UFA: at least <math>n^{(\log \log \log n)^{\Theta(1)}}</math> states, see Raskin,<ref name="Raskin2018">{{cite conference |last1=Raskin|first1=Michael|title=A superpolynomial lower bound for the size of non-deterministic complement of an unambiguous automaton|pages=138:1–138:11|book-title=Proc. ICALP 2018|year=2018|doi=10.4230/LIPIcs.ICALP.2018.138|doi-access=free }}</ref> and at most <math>e^{\Theta(\sqrt[3]{n (\ln n)^2})}</math> states, see Okhotin.<ref name="Okhotin2012" />

* 2DFA: at least <math>n</math> and at most <math>2n+3</math> states, see Kunc and Okhotin.<ref name="KuncOkhotin2012" />

* 2NFA: at least <math>n</math> and at most <math>O(n^8)</math> states. The upper bound is by implementing the method of the [[Immerman–Szelepcsényi theorem]], see Geffert, Mereghetti and Pighizzini.<ref name="GeffertMereghetti2007"/>

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{{reflist}}

[[Category:Finite-state automatamachines]]