State complexity of operations on two-way deterministic finite automata over a unary alphabet

KUNC, Michal and Alexander OKHOTIN. State complexity of operations on two-way deterministic finite automata over a unary alphabet. In Markus Holzer, Martin Kutrib, Giovanni Pighizzini. Descriptional Complexity of Formal Systems: 13th International Workshop, DCFS 2011, Gießen/Limburg, Germany, July 25-27, 2011. Proceedings. Berlin: Springer, 2011, p. 222-234. ISBN 978-3-642-22599-4. Available from: https://dx.doi.org/10.1007/978-3-642-22600-7_18.

Basic information

• Original name: State complexity of operations on two-way deterministic finite automata over a unary alphabet
• Name in Czech: Stavová složitost operací na dvoucestných deterministických konečných automatech nad unární abecedou
• Authors: KUNC, Michal (203 Czech Republic, guarantor, belonging to the institution) and Alexander OKHOTIN (643 Russian Federation).
• Edition: Berlin, Descriptional Complexity of Formal Systems: 13th International Workshop, DCFS 2011, Gießen/Limburg, Germany, July 25-27, 2011. Proceedings, p. 222-234, 13 pp. 2011.
• Publisher: Springer

Other information

• Original language: English
• Type of outcome: Proceedings paper
• Field of Study: 10101 Pure mathematics
• Country of publisher: Germany
• Confidentiality degree: is not subject to a state or trade secret
• Impact factor: Impact factor: 0.402 in 2005
• RIV identification code: RIV/00216224:14310/11:00050222
• Organization unit: Faculty of Science
• ISBN: 978-3-642-22599-4
• ISSN: 0302-9743
• Doi: http://dx.doi.org/10.1007/978-3-642-22600-7_18
• Keywords (in Czech): konečné automaty; dvoucestné automaty; stavová složitost
• Keywords in English: finite automata; two-way automata; state complexity
• Tags: AKR, rivok
• Tags: International impact, Reviewed

Abstract

The paper determines the number of states in a two-way deterministic finite automaton (2DFA) over a one-letter alphabet sufficient and in the worst case necessary to represent the results of the following operations: (i) intersection of an m-state 2DFA and an n-state 2DFA requires between m + n and m + n + 1 states; (ii) union of an m-state 2DFA and an n-state 2DFA, between m + n and 2m + n + 4 states; (iii) Kleene star of an n-state 2DFA, (g(n) + O(n))^2 states, where g(n) = e^(sqrt(n.ln(n))(1 + o(1))) is the maximum value of lcm(p1,...,pk) for p1 +...+ pk <= n, known as Landau’s function; (iv) k-th power of an n-state 2DFA, between (k - 1)g(n) - k and k.(g(n) + n) states; (v) concatenation of an m-state and an n-state 2DFAs, e^((1 + o(1)).sqrt((m + n).ln(m + n))) states.

Abstract (in Czech)

V článku je určen počet stavů v dvoucestném deterministickém konečném automatu (2DFA) nad jednopísmennou abecedou dostatečný a v nejhorším případě nutný k reprezentování výsledků následujících operací: (i) průnik 2DFA o m stavech a 2DFA o n stavech vyžaduje mezi m + n a m + n + 1 stavy; (ii) sjednocení 2DFA o m stavech a 2DFA o n stavech mezi m + n a 2m + n + 4 stavy; (iii) Kleeneho hvězdička 2DFA o n stavech (g(n) + O(n))^2 stavů, kde g(n) = e^(sqrt(n.ln(n))(1 + o(1))) je největší hodnota lcm(p1,...,pk) pro p1 +...+ pk <= n, známá jako Landauova funkce; (iv) k-tá mocnina 2DFA o n stavech mezi (k - 1)g(n) - k a k.(g(n) + n) stavy; (v) zřetězení 2DFA o m stavech a o n stavech e^((1 + o(1)).sqrt((m + n).ln(m + n))) stavů.

Links

• GA201/09/1313, research and development project: Name: Algebraické metody v teorii automatů a formálních jazyků II

Investor: Czech Science Foundation, Algebraic Methods in Automata and Formal Language Theory II

• MSM0021622409, plan (intention): Name: Matematické struktury a jejich fyzikální aplikace

Investor: Ministry of Education, Youth and Sports of the CR, Mathematical structures and their physical applications