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Semi-linear Lattices and Right One-Way Jumping Finite Automata (Extended Abstract)

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Implementation and Application of Automata (CIAA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11601))

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Abstract

Right one-way jumping automata (ROWJFAs) are an automaton model that was recently introduced for processing the input in a discontinuous way. In [S. Beier, M. Holzer: Properties of right one-way jumping finite automata. In Proc. 20th DCFS, number 10952 in LNCS, 2018] it was shown that the permutation closed languages accepted by ROWJFAs are exactly those with a finite number of positive Myhill-Nerode classes. Here a Myhill-Nerode equivalence class \([w]_L\) of a language L is said to be positive if w belongs to L. Obviously, this notion of positive Myhill-Nerode classes generalizes to sets of vectors of natural numbers. We give a characterization of the linear sets of vectors with a finite number of positive Myhill-Nerode classes, which uses rational cones. Furthermore, we investigate when a set of vectors can be decomposed as a finite union of sets of vectors with a finite number of positive Myhill-Nerode classes. A crucial role is played by lattices, which are special semi-linear sets that are defined as a natural way to extend “the pattern” of a linear set to the whole set of vectors of natural numbers in a given dimension. We show connections of lattices to the Myhill-Nerode relation and to rational cones. Some of these results will be used to give characterization results about ROWJFAs with multiple initial states. For binary alphabets we show connections of these and related automata to counter automata.

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References

  1. Beier, S., Holzer, M.: Decidability of right one-way jumping finite automata. In: Hoshi, M., Seki, S. (eds.) DLT 2018. LNCS, vol. 11088, pp. 109–120. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98654-8_9

    Chapter  Google Scholar 

  2. Beier, S., Holzer, M.: Properties of right one-way jumping finite automata. In: Konstantinidis, S., Pighizzini, G. (eds.) DCFS 2018. LNCS, vol. 10952, pp. 11–23. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94631-3_2

    Chapter  Google Scholar 

  3. Beier, S., Holzer, M., Kutrib, M.: On the descriptional complexity of operations on semilinear sets. In: Csuhaj-Varjú, E., Dömösi, P., Vaszil, G. (eds.) AFL 2017. EPTCS, vol. 252, pp. 41–55. Debrecen, Hungary (2017). https://doi.org/10.4204/EPTCS.252.8

    Article  MathSciNet  Google Scholar 

  4. Chigahara, H., Fazekas, S., Yamamura, A.: One-way jumping finite automata. Int. J. Found. Comput. Sci. 27(3), 391–405 (2016). https://doi.org/10.1142/S0129054116400165

    Article  MathSciNet  MATH  Google Scholar 

  5. Chistikov, D., Haase, C.: The taming of the semi-linear set. In: Chatzigiannakis, I., Mitzenmacher, M., Rabani, Y., Sangiorgi, D. (eds.) ICALP 2016. LIPIcs, vol. 55, pp. 128:1–128:13 (2016). https://doi.org/10.4230/LIPIcs.ICALP.2016.128

  6. Conway, J., Horton, S., Neil, J.A.: Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290, 3rd edn. Springer, New York (1999). https://doi.org/10.1007/978-1-4757-6568-7

    Book  Google Scholar 

  7. Gill, A., Kou, L.T.: Multiple-entry finite automata. J. Comput. System Sci. 9(1), 1–19 (1974). https://doi.org/10.1016/S0022-0000(74)80034-6

    Article  MathSciNet  MATH  Google Scholar 

  8. Ginsburg, S., Spanier, E.H.: Bounded ALGOL-like languages. Trans. Am. Math. Soc. 113, 333–368 (1964). https://doi.org/10.2307/1994067

    Article  MathSciNet  MATH  Google Scholar 

  9. Ginsburg, S., Spanier, E.H.: Bounded regular sets. Proc. Am. Math. Soc. 17(5), 1043–1049 (1966). https://doi.org/10.1090/S0002-9939-1966-0201310-3

    Article  MathSciNet  MATH  Google Scholar 

  10. Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas, and languages. Pac. J. Math. 16(2), 285–296 (1966). https://doi.org/10.2140/pjm.1966.16.285

    Article  MathSciNet  MATH  Google Scholar 

  11. Gubeladze, J., Michalek, M.: The poset of rational cones. Pac. J. Math. 292(1), 103–115 (2018). https://doi.org/10.2140/pjm.2018.292.103

    Article  MathSciNet  MATH  Google Scholar 

  12. Haase, C., Hofman, P.: Tightening the complexity of equivalence problems for commutative grammars. In: Ollinger, N., Vollmer, H. (eds.) STACS 2016. LIPIcs, vol. 47, pp. 41:1–41:14 (2016). https://doi.org/10.4230/LIPIcs.STACS.2016.41

  13. Meduna, A., Zemek, P.: Jumping finite automata. Int. J. Found. Comput. Sci. 23(7), 1555–1578 (2012). https://doi.org/10.1142/S0129054112500244

    Article  MathSciNet  MATH  Google Scholar 

  14. Parikh, R.J.: On context-free languages. J. ACM 13(4), 570–581 (1966). https://doi.org/10.1145/321356.321364

    Article  MATH  Google Scholar 

  15. Studený, M.: Convex cones in finite-dimensional real vector spaces. Kybernetika 29(2), 180–200 (1993). http://www.kybernetika.cz/content/1993/2/180

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Authors and Affiliations

  1. Institut für Informatik, Universität Giessen, Arndtstr. 2, 35392, Giessen, Germany

    Simon Beier & Markus Holzer

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  1. Simon Beier
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  2. Markus Holzer
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Correspondence to Markus Holzer .

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Editors and Affiliations

  1. Slovak Academy of Sciences, Košice, Slovakia

    Michal Hospodár

  2. Slovak Academy of Sciences, Košice, Slovakia

    Galina Jirásková

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Beier, S., Holzer, M. (2019). Semi-linear Lattices and Right One-Way Jumping Finite Automata (Extended Abstract). In: Hospodár, M., Jirásková, G. (eds) Implementation and Application of Automata. CIAA 2019. Lecture Notes in Computer Science(), vol 11601. Springer, Cham. https://doi.org/10.1007/978-3-030-23679-3_6

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  • DOI: https://doi.org/10.1007/978-3-030-23679-3_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-23678-6

  • Online ISBN: 978-3-030-23679-3

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