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Improved matrix pair undecidability results

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Abstract

We improve the undecidability bounds for problems involving two integer matrices by showing that Scalar Reachability, Zero in the Right Upper Corner, Vector Reachability, and Zero in the Left Upper Corner are undecidable for dimensions of 9, 10, 11, and 13, respectively. Problems Scalar Reachability, Zero in the Right Upper Corner, and Vector Reachability were previously known undecidable for dimensions 18, 18, and 16, respectively.

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References

  1. Bell, P., Potapov, I.: Lowering undecidability bounds for decision questions in matrices. In: Lecture Notes in Computer Science 4036 (Proceedings of DLT’06), pp. 375–385 (2006)

  2. Blondel V.D. and Tsitsiklis J.N. (1997). When is a pair of matrices mortal?. Inf. Process. Lett. 63(5): 283–286

    Article  MathSciNet  Google Scholar 

  3. Cassaigne J., Harju T. and Karhumäki J. (1999). On the undecidability of freeness of matrix semigroups. Int. J. Algebra Comput. 9: 295–305

    Article  MATH  Google Scholar 

  4. Cassaigne J. and Karhumäki J. (1998). Examples of undecidable problems for 2-generator matrix semigroups. Theoret. Comput. Sci. 204: 29–34

    Article  MATH  MathSciNet  Google Scholar 

  5. Claus V. (1980). Some remarks on PCP(k) and related problems. Bull. EATCS 12: 54–61

    Google Scholar 

  6. Eilenberg, S.: Automata, Languages, and Machines, vol. A. Academic, New York

  7. Gaubert S. and Katz R.D. (2006). Reachability problems for products of matrices in semirings. Int. J. Algebra Comput. 16(3): 603–627

    Article  MATH  MathSciNet  Google Scholar 

  8. Halava, V., Harju, T., Hirvensalo, M.: Undecidability bounds for integer matrices using claus instances. TUCS Technical Report 766 (2006)

  9. Harju, T., Karhumäki, J.: Morphisms. In: Rozenberg, G., Salomaa, A. (eds) Handbook of Formal Languages. Springer, Heidelberg (1997)

  10. Matiyasevich Y. and Sénizergues G. (2005). Decision problems for semi-Thue systems with a few rules. Theoret. Comput. Sci. 330(1): 145–169

    Article  MATH  MathSciNet  Google Scholar 

  11. Paterson M.S. (1970). Unsolvability in 3 ×  3 matrices. Stud. Appl. Math. 49: 105–107

    MathSciNet  MATH  Google Scholar 

  12. Post E.L. (1946). A variant of a recursively unsolvable problem. Bull. Am. Math. Soc. 52: 264–268

    Article  MATH  MathSciNet  Google Scholar 

  13. Turing A. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. Ser. 2(42): 230–265

    Google Scholar 

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Authors and Affiliations

  1. TUCS-Turku Centre for Computer Science and Department of Mathematics, University of Turku, 20014, Turku, Finland

    Vesa Halava & Mika Hirvensalo

Authors
  1. Vesa Halava
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  2. Mika Hirvensalo
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Correspondence to Mika Hirvensalo.

Additional information

The authors are supported by the Academy of Finland under grants 208414 and 208797, respectively.

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Halava, V., Hirvensalo, M. Improved matrix pair undecidability results. Acta Informatica 44, 191–205 (2007). https://doi.org/10.1007/s00236-007-0047-y

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  • DOI: https://doi.org/10.1007/s00236-007-0047-y

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