On the membership problem for finite automata over symmetric groups
-
Arthur A. Khashaev
Abstract
We consider automata in which transitions are labelled with arbitrary permutations. The language of such an automaton consists of compositions of permutations for all possible admissible computation paths. The membership problem for finite automata over symmetric groups is the following decision problem: does a given permutation belong to the language of a given automaton? We show that this problem is NP-complete. We also propose an efficient algorithm for the case of strongly connected automata.
Originally published in Diskretnaya Matematika (2021) 33,№1, 82–90 (in Russian).
Acknowledgment
The author thanks V. A. Zakharov for assistance in preparing this paper.
References
[1] Benois M., “Parties rationnelles du groupe libre”, C. R. Acad. Sci. Paris, Ser. A, 269 (1969), 1188-1190.Search in Google Scholar
[2] Davey B. A, Priestley H. A., Introduction to Lattices and Order, Cambridge Univ. Press, 2002.10.1017/CBO9780511809088Search in Google Scholar
[3] Eder E., “Properties of substitutions and unifications”, J. Symb. Comput., 1:1 (1985), 31-46.10.1007/978-3-642-69391-5_18Search in Google Scholar
[4] Eilenberg S., Automata, Languages, and Machines. Volume A, Acad. Press, 1974,451 pp.Search in Google Scholar
[5] Furst M., Hopcroft J., Luks E., “Polynomial-time algorithms for permutation groups”, 21st Annu. Symp. Found. Comput. Sci., IEEE Computer Soc., 1980, 36-41.10.1109/SFCS.1980.34Search in Google Scholar
[6] Grunschlag Z., Algorithms in Geometric Group Theory, PhD thesis, Univ. California, Berkeley, 1999.Search in Google Scholar
[7] Jerrum M., “A compact representation for permutation groups”, J. Algorithms, 7:1 (1986), 60-78.10.1109/SFCS.1982.52Search in Google Scholar
[8] Kambites M., Silva P. V., Steinberg B., “On the rational subset problem for groups”, J. Algebra, 309:2 (2007), 622-639.10.1016/j.jalgebra.2006.05.020Search in Google Scholar
[9] Lohrey M., “The rational subset membership problem for groups: a survey”, Groups St Andrews 2013, London Math. Soc. Lect. Note Ser., Cambridge Univ. Press, 2015,368-389.10.1017/CBO9781316227343.024Search in Google Scholar
[10] Sakarovitch J., Elements of Automata Theory, Cambridge Univ. Press, 2009.10.1017/CBO9781139195218Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Some cardinality estimates for the set of correlation-immune Boolean functions
- Invertible matrices over some quotient rings: identification, generation, and analysis
- Synthesis of reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs
- Contents
- On a class of irregular languages
- On the membership problem for finite automata over symmetric groups
- On the asymptotic normality conditions for the number of repetitions in a stationary random sequence
- On implementation of Boolean functions by contact circuits of minimal uniform width
- Retraction
- Some cardinality estimates for the set of correlation-immune Boolean functions
- Synthesis of reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs
- Invertible matrices over some quotient rings: identification, generation, and analysis
Articles in the same Issue
- Some cardinality estimates for the set of correlation-immune Boolean functions
- Invertible matrices over some quotient rings: identification, generation, and analysis
- Synthesis of reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs
- Contents
- On a class of irregular languages
- On the membership problem for finite automata over symmetric groups
- On the asymptotic normality conditions for the number of repetitions in a stationary random sequence
- On implementation of Boolean functions by contact circuits of minimal uniform width
- Retraction
- Some cardinality estimates for the set of correlation-immune Boolean functions
- Synthesis of reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs
- Invertible matrices over some quotient rings: identification, generation, and analysis
