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Weighted Picture Automata and Weighted Logics

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Abstract

We investigate formal power series on pictures. These are functions that map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We establish a notion of a weighted MSO logics over pictures. The semantics of a weighted formula will be a picture series. We introduce weighted 2-dimensional on-line tessellation automata (W2OTA) and prove that for commutative semirings, the class of picture series defined by sentences of the weighted logics coincides with the family of picture series that are computable by W2OTA. Moreover, we show that the family of behaviors of W2OTA coincide precisely with the class of picture series characterized by weighted (quadrapolic) picture automata and consequently, the notion of weighted recognizability presented here is robust. However, the weighted structures can not be used to get better decidability properties than in the language case. For every commutative semiring, it is undecidable whether a given MSO formula has restricted structure or whether the semantics of a formula has empty support.

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References

  1. Anselmo, M., Giammarresi, D., Madonia, M., Restivo, A.: Unambiguous recognizable two-dimensional languages. R.A.I.R.O.—Inf. Théor. Appl. 40, 277–293 (2006)

    MATH  MathSciNet  Google Scholar 

  2. Berstel, J., Reutenauer, C.: Rational Series and Their Languages. EATCS Monographs on Theoretical Computer Science, vol. 12. Springer, Berlin (1988)

    MATH  Google Scholar 

  3. Blum, M., Hewitt, C.: Automata on a 2-dimensional tape. In: IEEE Symposium on Switching and Automata Theory, pp. 155–160 (1967)

  4. Bozapalidis, S., Grammatikopoulou, A.: Recognizable picture series. In: Droste M., Vogler H. (eds.) Special Issue on Weighted Automata, Presented at WATA 2004, Dresden, Journal of Automata, Languages and Combinatorics, vol. 10, pp. 159–183 (2005)

  5. Choffrut, C., Durak, B.: Collage of two-dimensional words. Theor. Comput. Sci. 340(1), 364–380 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Crespi-Reghizzi, S., Pradella, M.: Tile rewriting grammars and picture languages. Theor. Comput. Sci. 340(1), 257–272 (2005)

    Article  MathSciNet  Google Scholar 

  7. de Prophetis, L., Varricchio, S.: Recognizability of rectangular pictures by Wang systems. J. Autom. Lang. Combin. 2(4), 269–288 (1997)

    MATH  Google Scholar 

  8. Droste, M., Gastin, P.: Weighted automata and logics. In: 32nd ICALP. Lecture Notes in Computer Science, vol. 3580, pp. 513–525. Springer, Berlin (2005)

    Google Scholar 

  9. Droste, M., Gastin, P.: Weighted automata and weighted logics. Theor. Comput. Sci. 380, 69–86 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Droste, M., Rahonis, G.: Weighted automata and weighted logics on infinite words. In: DLT. Lecture Notes in Computer Science, vol. 4036, pp. 49–58. Springer, Berlin (2006)

    Google Scholar 

  11. Droste, M., Vogler, H.: Weighted tree automata and weighted logics. Theor. Comput. Sci. 366, 228–247 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dubois, D., Hüllermeier, E., Prade, H.: A systematic approach to the assessment of fuzzy association rules. Technical report, Philipps-Universität Marburg (2004)

  13. Eilenberg, S.: Automata, Languages, and Machines. vol. A. Academic Press, San Diego (1974)

    MATH  Google Scholar 

  14. Fichtner, I.: Characterizations of recognizable picture series. Dissertation, Universität Leipzig (2007)

  15. Fu, K.S.: Syntactic Methods in Pattern Recognition. Academic Press, New York (1974)

    MATH  Google Scholar 

  16. Giammarresi, D., Restivo, A.: Recognizable picture languages. In: Nivat, M., Saoudi, A., Wangs, P.S.P. (eds.) Special Issue on Parallel Image Processing, International Journal Pattern Recognition and Artificial Intelligence, vol. 6, pp. 241–256 (1992)

  17. Giammarresi, D., Restivo, A.: Two-dimensional finite state recognizability. Fundam. Inf. 25(3), 399–422 (1996)

    MATH  MathSciNet  Google Scholar 

  18. Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 215–267. Springer, Berlin (1997)

    Google Scholar 

  19. Giammarresi, D., Restivo, A., Seibert, S., Thomas, W.: Monadic second-order logic over rectangular pictures and recognizability by tiling systems. Inf. Comput. 125, 32–45 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Inoue, K., Nakamura, A.: Some properties of two-dimensional on-line tessellation acceptors. Inf. Sci. 13, 95–121 (1977)

    Article  MathSciNet  Google Scholar 

  21. Inoue, K., Takanami, I.: A survey of two-dimensional automata theory. Inf. Sci. 55, 99–121 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kari, J., Moore, C.: New results on alternating and non-deterministic two-dimensional finite-state automata. In: 18th STACS. Lecture Notes in Computer Science, vol. 2010, pp. 396–406. Springer, Berlin (2001)

    Google Scholar 

  23. Kuich, W., Salomaa, A.: In: Semirings, Automata, Languages. EATCS Monographs on Theoretical Computer Science, vol. 5. Springer, Berlin (1986)

    Google Scholar 

  24. Latteux, M., Simplot, D.: Recognizable picture languages and domino tiling. Theor. Comput. Sci. 178, 275–283 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lindgren, K., Moore, C., Nordahl, M.: Complexity of two-dimensional patterns. J. Stat. Phys. 91(5–6), 909–951 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mathissen, Ch.: Definable transductions and weighted logics for texts. In: Developments in Language Theory (DLT 2007). Lect. Notes in Comput. Sci., vol. 4588, pp. 324–336. Springer, Berlin (2007)

    Chapter  Google Scholar 

  27. Mathissen, Ch.: Weighted logics for nested words and algebraic formal power series. In: 35th ICALP. Lecture Notes in Computer Science, vol. 5126, pp. 221–232. Springer, Berlin (2008)

    Google Scholar 

  28. Matz, O.: Regular expressions and context-free grammars for picture languages. In: 14th STACS. Lecture Notes in Computer Science, vol. 1200, pp. 283–294. Springer, Berlin (1997)

    Chapter  Google Scholar 

  29. Matz, O.: On piecewise testable, starfree, and recognizable picture languages. In: van Emde Boas, P. (eds.) FoSSaCS. Lecture Notes in Computer Science, vol. 1378, pp. 203–210. Springer, Berlin (1998)

    Google Scholar 

  30. Mäurer, I.: Recognizable and rational picture series. In: Conference on Algebraic Informatics, pp. 141–155. Aristotle University of Thessaloniki Press (2005)

  31. Mäurer, I.: Weighted picture automata and weighted logics. In: Durand, B., Thomas, W. (eds.) STACS 2006. Lecture Notes in Computer Science, vol. 3884, pp. 313–324. Springer, Berlin (2006)

    Chapter  Google Scholar 

  32. Mäurer, I.: Characterizations of recognizable picture series. Theor. Comput. Sci. 374, 214–228 (2007)

    Article  MATH  Google Scholar 

  33. Meinecke, I.: Weighted logics for traces. In: Computer Science—Theory and Applications (Proceedings of CSR 2006). Lect. Notes in Comput. Sci., vol. 3976, pp. 235–246. Springer, Berlin (2006)

    Google Scholar 

  34. Minski, M., Papert, S.: Perceptron. MIT Press, Cambridge (1969)

    Google Scholar 

  35. Post, E.L.: A variant of a recursively unsolvable problem. Bull. Am. Math. Soc. 52 (1946)

  36. Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science. Springer, Berlin (1978)

    MATH  Google Scholar 

  37. Simplot, D.: A characterization of recognizable picture languages by tilings by finite sets. Theor. Comput. Sci. 218(2), 297–323 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  38. Smith, R.A.: Two-dimensional formal languages and pattern recognition by cellular automata. In: 12th IEEE FOCS Conference Record, pp. 144–152 (1971)

  39. Wilke, T.: Star-free picture expressions are strictly weaker than first-order logic. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP’97 Proceedings. Lecture Notes in Computer Science, vol. 1256, pp. 347–357. Springer, Berlin (1997)

    Google Scholar 

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  1. Institut für Informatik, Universität Leipzig, Johannisgasse 26, 04103, Leipzig, Germany

    Ina Fichtner

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  1. Ina Fichtner
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Correspondence to Ina Fichtner.

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Supported by the Ph.D. program 446/3 “Wissensrepräsentation” of the German Research Foundation.

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Fichtner, I. Weighted Picture Automata and Weighted Logics. Theory Comput Syst 48, 48–78 (2011). https://doi.org/10.1007/s00224-009-9225-3

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