Abstract
We give a survey of simple undecidability results and open problems concerning matrices of low order with integer entries. Connections to the theory of finite automata (with multiplicities) are also provided.
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References
Albert, M. H., Lawrence, J.: A proof of Ehrenfeucht’s Conjecture. Theoret. Comput. Sci. 41 (1985) 121–123.
Berstel, J., Mignotte, M.: Deux propriétés décidables des suites récurrentes linéaires. Bull. Soc. Math. France 104 (1976) 175–184.
Berstel, J., Reutenauer, C.: “Rational Series and Their Languages”, Springer-Verlag, 1988.
Blondel, V. D., Tsitsiklis, J. N.: When is a pair of matrices mortal. Inf. Proc. Letter 63 (1997) 283–286.
Cassaigne, J., Karhumäki, J.: Examples of undecidable problems for 2-generator matrix semigroups. Theoret. Comput. Sci. 204 (1998) 29–34.
Cassaigne, J., Harju, T., Karhumäki, J.: On the undecidability of freeness of matrix semigroups. Int. J. Algebra Comput. 9 (1999) 295–305.
Claus, V.: Some remarks on PCP(k) and related problems. Bull. EATCS 12 (1980) 54–61.
Claus, V.: The (n,k)-bounded emptiness-problem for probabilistic acceptors and related problems. Acta Inform. 16 (1981) 139–160.
Eilenberg, S.: “Automata, Languages, and Machines”, Vol A, Academic Press, New York, 1974.
Guba, V. S.: The equivalence of infinite systems of equations in free groups and semigroups with finite subsystems. Mat. Zametki 40 (1986) 321–324 (in Russian).
Halava, V.: “Decidable and Undecidable Problems in Matrix Theory”, Master’s Thesis, University of Turku, 1997.
Halava, V., Harju, T.: Mortality in matrix semigroups. Amer. Math. Monthly 108 (2001) 649–653.
Hansel, G.: Une démonstration simple du théorème de Skolem-Mahler-Lech. Theoret. Comput. Sci. 43 (1986) 1–10.
Harju, T., Karhumäki, J.: The equivalence problem of multitape automata. Theoret. Comput. Sci. 78 (1991) 347–355.
Harju, T., Karhumäki, J.: Morphisms. In Handbook of Formal Languages,Vol. 1, (A. Salomaa and G. Rozenberg, eds), Springer-Verlag, 1997, pp. 439–510.
Harju, T., Karhumäki, J., Plandowski, W.: Independent system of equations, in M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, to appear.
Jacob, G.: La finitude des reprèsentations linéaires de semi-groupes est décidable. J. Algebra 52 (1978) 437–459.
Klarner, D. A., Birget, J.-C., Satterfield, W.: On the undecidability of the freeness of integer matrix semigroups. Int. J. Algebra Comp. 1 (1991) 223–226.
Krom, M.:An unsolvable problem with products of matrices. Math. System. Theory 14 (1981) 335–337.
Mandel, A., Simon, I.: On finite semigroups of matrices. Theoret. Comput. Sci. 5 (1977) 101–111.
Manna, Z.: “Mathematical Theory of Computations”, McGraw-Hill, 1974.
Markov, A. A.: On certain insoluble problems concerning matrices. Doklady Akad. Nauk SSSR (N.S.) 57 (1947) 539–542 (in Russian).
Markov, A. A.: On the problem of representability of matrices. Z. Math. Logik Grundlagen Math. 4 (1958) 157–168 (in Russian).
Matiyasevich, Y., Sénizergues, G.: Decision problems for semi-Thue systems with a fewrules. In Proceedings of the 11th IEEE Symposium on Logic in Computer Science, pages 523–531, 1996.
Miller III, C. F.: “On Group Theoretic Decision Problems and Their Classification”, Annals of Math. Study 68, Princeton Univ. Press 1971.
Miller III, C. F.: Decision problems for groups-Survey and reflections, in “Algorithms and Classification in Combinatorial Group Theory” (G. Baumslag and C. F. Miller III, eds.), Springer-Verlag, 1992, 1–59.
Nielsen, J.: Die Gruppe der dreidimensionalen Gittertransformationen. Danske Vid. Selsk. Math.-Fys. Medd. V 12 (1924) 1–29.
Paterson, M. S.: Unsolvability in 3×3-matrices. Studies in Appl. Math. 49 (1970) 105–107.
Post, E.:A variant of a recursively unsolvable problem. Bulletin of Amer. Math. Soc. 52 (1946) 264–268.
Schultz, P.: Mortality of 2 × 2-matrices. Amer. Math. Monthly 84 (1977) 463–464.
Schupp, P. E.: Embeddings into simple groups. J. London Math. Soc. 13 (1976) 90–94.
Turakainen, P.: On multistochastic automata. Inform. Control 23 (1973) 183–203.
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Harju, T. (2002). Decision Questions on Integer Matrices. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds) Developments in Language Theory. DLT 2001. Lecture Notes in Computer Science, vol 2295. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46011-X_5
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DOI: https://doi.org/10.1007/3-540-46011-X_5
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