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Geometrical regular languages and linear Diophantine equations. (English) Zbl 1341.68083

Holzer, Markus (ed.) et al., Descriptional complexity of formal systems. 13th international workshop, DCFS 2011, Gießen/Limburg, Germany, July 25–27, 2011. Proceedings. Berlin: Springer (ISBN 978-3-642-22599-4/pbk). Lecture Notes in Computer Science 6808, 107-120 (2011).
Summary: We present a new method for checking whether a regular language over an arbitrarily large alphabet is semi-geometrical or whether it is geometrical. This method makes use first of the partitioning of the state diagram of the minimal automaton of the language into strongly connected components and secondly of the enumeration of the simple cycles in each component. It is based on the construction of systems of linear Diophantine equations the coefficients of which are deduced from the the set of simple cycles. This paper addresses the case of a strongly connected graph.
For the entire collection see [Zbl 1218.68017].

MSC:

68Q45 Formal languages and automata
05C40 Connectivity
05C90 Applications of graph theory
11D04 Linear Diophantine equations
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[1] Blanpain, B., Champarnaud, J.-M., Dubernard, J.P.: Geometrical languages. In: Martin-Vide, C. (ed.) International Conference on Language Theory and Automata (LATA 2007). GRLMC Universitat Rovira I Virgili, vol. 35(07), pp. 127–138 (2007)
[2] Brauer, A.: On a problem of partitions. Amer. J. Math. 64, 299–312 (1942) · Zbl 0061.06801 · doi:10.2307/2371684
[3] Carpentier, F.: Systèmes d’équations diophantiennes et test de géométricité sur un langage rationnel. Rapport de master, Université de Rouen, France (2008)
[4] Champarnaud, J.-M., Dubernard, J.P., Jeanne, H.: Geometricity of binary regular languages. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 178–189. Springer, Heidelberg (2010) · Zbl 1284.68342 · doi:10.1007/978-3-642-13089-2_15
[5] Champarnaud, J.-M., Dubernard, J.P., Jeanne, H.: Regular geometrical languages and tiling the plane. In: Domaratzki, M., Salomaa, K. (eds.) CIAA 2010. LNCS, vol. 6482, pp. 69–78. Springer, Heidelberg (2011) · Zbl 1297.68114 · doi:10.1007/978-3-642-18098-9_8
[6] Champarnaud, J.-M., Hansel, G.: Puissances des matrices booléennes. Unsubmitted manuscript, Université de Rouen, France (2005)
[7] Chrobak, M.: Finite automata and unary languages. Theor. Comput. Sci. 47(3), 149–158 (1986) · Zbl 0638.68096 · doi:10.1016/0304-3975(86)90142-8
[8] d’Alessandro, F., Intrigila, B., Varricchio, S.: On some counting problems for semi-linear sets. CoRR, abs/0907.3005 (2009)
[9] Eilenberg, S.: Automata, languages and machines, vol. B. Academic Press, New York (1976) · Zbl 0359.94067
[10] Geldenhuys, J., van der Merwe, B., van Zijl, L.: Reducing nondeterministic finite automata with SAT solvers. In: Yli-Jyrä, A., Kornai, A., Sakarovitch, J., Watson, B. (eds.) FSMNLP 2009. LNCS, vol. 6062, pp. 81–92. Springer, Heidelberg (2010) · Zbl 05771734 · doi:10.1007/978-3-642-14684-8_9
[11] Geniet, D., Largeteau, G.: WCET free time analysis of hard real-time systems on multiprocessors: A regular language-based model. Theor. Comput. Sci. 388(1-3), 26–52 (2007) · Zbl 1143.68034 · doi:10.1016/j.tcs.2007.03.054
[12] Golomb, S.W.: Polyominoes: Puzzles, patterns, problems, and packings. Princeton Academic Press, London (1996) · Zbl 0831.05020
[13] LinBox Group. Linbox project: Exact computational linear algebra (2002), http://www.linalg.org
[14] Holladay, J.C., Varga, R.S.: On powers of non negative matrices. Proc. Amer. Math. Soc. 9(4), 631–634 (1958) · Zbl 0096.00805 · doi:10.1090/S0002-9939-1958-0097416-8
[15] Kleene, S.: Representation of events in nerve nets and finite automata. Automata Studies 34, 3–41 (1956); Ann. Math. Studies
[16] Mordell, L.: Diophantine equations. Academic Press, London (1969) · Zbl 0188.34503
[17] Mulders, T., Storjohann, A.: Certified dense linear system solving. J. Symb. Comput. 37(4), 485–510 (2004) · Zbl 1137.11361 · doi:10.1016/j.jsc.2003.07.004
[18] Myhill, J.: Finite automata and the representation of events. WADD TR-57-624, 112–137 (1957)
[19] Nerode, A.: Linear automata transformation. Proceedings of AMS 9, 541–544 (1958) · Zbl 0089.33403 · doi:10.1090/S0002-9939-1958-0135681-9
[20] Paranthoën, T.: Génération aléatoire et structure des automates à états finis. PhD, Université de Rouen, France (2004)
[21] Parikh, R.: On context-free languages. J. ACM 13(4), 570–581 (1966) · Zbl 0154.25801 · doi:10.1145/321356.321364
[22] Robinson, R.W.: Counting strongly connected finite automata. In: Alavi, Y., et al. (eds.) Graph Theory with Applications to Algorithms and Computer Science, pp. 671–685. Wiley, New York (1985)
[23] Schrijver, A.: Theory of Linear and Integer Programming. John Wiley and Sons, New York (1986) · Zbl 0665.90063
[24] Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972) · Zbl 0251.05107 · doi:10.1137/0201010
[25] Tarjan, R.E.: Enumeration of the elementary circuits of a directed graph. SIAM J. Comput. 2, 211–216 (1973) · Zbl 0274.05106 · doi:10.1137/0202017
[26] The Polylib Team. Polylib User’s Manual. IRISA, France (2002), www.irisa.fr/polylib/doc/
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