Abstract
The rank of a word in a deterministic finite automaton is the size of the image of the whole state set under the mapping defined by this word. We study the length of shortest words of minimum rank in several classes of complete deterministic finite automata, namely, strongly connected and Eulerian automata. A conjecture bounding this length is known as the Rank Conjecture, a generalization of the well known Černý Conjecture. We prove upper bounds on the length of shortest words of minimum rank in automata from the mentioned classes, and provide several families of automata with long words of minimum rank. Some results in this direction are also obtained for automata with rank equal to period (the greatest common divisor of lengths of all cycles) and for circular automata.
Jarkko Kari is supported by the Academy of Finland grant 296018. Anton Varonka is supported by Poland’s National Science Centre (NCN) grant no. 2016/21/D/ST6/00491.
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References
Almeida, J., Steinberg, B.: Matrix Mortality and the Černý-Pin Conjecture. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 67–80. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02737-6_5
Ananichev, D., Gusev, V., Volkov, M.: Slowly synchronizing automata and digraphs. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 55–65. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15155-2_7
Béal, M., Perrin, D.: A quadratic algorithm for road coloring. Discrete Appl. Math. 169, 15–29 (2014). https://doi.org/10.1016/j.dam.2013.12.002
Berlinkov, M.V.: On two algorithmic problems about synchronizing automata. In: Shur, A.M., Volkov, M.V. (eds.) DLT 2014. LNCS, vol. 8633, pp. 61–67. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09698-8_6
Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1994)
Carpi, A., D’Alessandro, F.: Strongly transitive automata and the černý conjecture. Acta Informatica 46(8), 591–607 (2009)
Carpi, A., D’Alessandro, F.: On the hybrid Černý-Road coloring problem and Hamiltonian paths. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 124–135. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14455-4_13
Černý, J., Pirická, A., Rosenauerova, B.: On directable automata. Kybernetika 7(4), 289–298 (1971)
Černý, J.: Pozńamka k homoǵennym eksperimentom s konečńymi automatami, Matematicko-fyzikalny Casopis Slovensk. Akad. Vied 14(3), 208–216 (1964)
Cole, R., Ost, K., Schirra, S.: Edge-coloring bipartite multigraphs in \(O(E \log D)\) time. Combinatorica 21(1), 5–12 (2001)
Dubuc, L.: Sur les automates circulaires et la conjecture de černý. RAIRO - Theor. Inform. Appl. 32(1–3), 21–34 (1998)
Gusev, V.V.: Lower bounds for the length of reset words in eulerian automata. Int. J. Found. Comput. Sci. 24(2), 251–262 (2013). https://doi.org/10.1142/S0129054113400108
Heap, B.R., Lynn, M.S.: The structure of powers of nonnegative matrices: I. The index of convergence. SIAM J. Appl. Math. 14(3), 610–639 (1966)
Kari, J.: A counter example to a conjecture concerning synchronizing words in finite automata. Bull. EATCS 73, 146 (2001)
Kari, J.: Synchronizing finite automata on Eulerian digraphs. Theor. Comput. Sci. 295(1), 223–232 (2003)
Klyachko, A.A., Rystsov, I.K., Spivak, M.A.: In extremal combinatorial problem associated with the bound on the length of a synchronizing word in an automaton. Cybernetics 23(2), 165–171 (1987)
Pin, J.: On two combinatorial problems arising from automata theory. In: Berge, C., Bresson, D., Camion, P., Maurras, J., Sterboul, F. (eds.) Combinatorial Mathematics, North-Holland Mathematics Studies, vol. 75, pp. 535–548. North-Holland, Amsterdam (1983)
Pin, J.E.: Le problème de la synchronisation et la conjecture de Cerný. In: Luca, A.D. (ed.) Non-commutative structures in algebra and geometric combinatorics, vol. 109, pp. 37–48. Quaderni de la Ricerca Scientifica, CNR (Consiglio nazionale delle ricerche, Italy) (1981)
Szykuła, M., Vorel, V.: An extremal series of Eulerian synchronizing automata. In: Brlek, S., Reutenauer, C. (eds.) DLT 2016. LNCS, vol. 9840, pp. 380–392. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53132-7_31
Volkov, M.V.: Synchronizing automata and the Černý conjecture. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 11–27. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-88282-4_4
Vorel, V.: Subset synchronization and careful synchronization of binary finite automata. Int. J. Found. Comput. Sci. 27(5), 557–577 (2016). https://doi.org/10.1142/S0129054116500167
Wielandt, H.: Unzerlegbare, nicht negative matrizen. Mathematische Zeitschrift 52(1), 642–648 (1950)
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Kari, J., Ryzhikov, A., Varonka, A. (2019). Words of Minimum Rank in Deterministic Finite Automata. In: Hofman, P., Skrzypczak, M. (eds) Developments in Language Theory. DLT 2019. Lecture Notes in Computer Science(), vol 11647. Springer, Cham. https://doi.org/10.1007/978-3-030-24886-4_5
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