Abstract
A two-dimensional code is defined as a set X ⊆ Σ** such that any picture over Σ is tilable in at most one way with pictures in X. The codicity problem is undecidable. The subclass of prefix codes is introduced and it is proved that it is decidable whether a finite set of pictures is a prefix code. Further a polynomial time decoding algorithm for finite prefix codes is given. Maximality and completeness of finite prefix codes are studied: differently from the one-dimensional case, they are not equivalent notions. Completeness of finite prefix codes is characterized.
Partially supported by MIUR Project “Aspetti matematici e applicazioni emergenti degli automi e dei linguaggi formali”, by 60 % Projects of University of Catania, Roma “Tor Vergata”, Salerno.
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References
Aigrain, P., Beauquier, D.: Polyomino tilings, cellular automata and codicity. Theoretical Computer Science 147, 165–180 (1995)
Anselmo, M., Giammarresi, D., Madonia, M.: Deterministic and unambiguous families within recognizable two-dimensional languages. Fund. Inform. 98(2-3), 143–166 (2010)
Anselmo, M., Giammarresi, D., Madonia, M., Restivo, A.: Unambiguous Recognizable Two-dimensional Languages. RAIRO: Theoretical Informatics and Applications 40(2), 227–294 (2006)
Anselmo, M., Jonoska, N., Madonia, M.: Framed Versus Unframed Two-dimensional Languages. In: Nielsen, M., Kučera, A., Miltersen, P.B., Palamidessi, C., Tůma, P., Valencia, F. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 79–92. Springer, Heidelberg (2009)
Anselmo, M., Madonia, M.: Deterministic and unambiguous two-dimensional languages over one-letter alphabet. Theoretical Computer Science 410-16, 1477–1485 (2009)
Beauquier, D., Nivat, M.: A codicity undecidable problem in the plane. Theoret. Comp. Sci. 303, 417–430 (2003)
Berstel, J., Perrin, D., Reutenauer, C.: Codes and Automata. Cambridge University Press (2009)
Blum, M., Hewitt, C.: Automata on a two-dimensional tape. In: IEEE Symposium on Switching and Automata Theory, pp. 155–160 (1967)
Book, R.V., Otto, F.: String-rewriting Systems. Springer (1993)
Bozapalidis, S., Grammatikopoulou, A.: Picture codes. ITA 40(4), 537–550 (2006)
Giammarresi, D., Restivo, A.: Recognizable picture languages. Int. Journal Pattern Recognition and Artificial Intelligence 6(2 & 3), 241–256 (1992)
Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., et al. (eds.) Handbook of Formal Languages, vol. III, pp. 215–268. Springer (1997)
Grammatikopoulou, A.: Prefix Picture Sets and Picture Codes. In: Procs. CAI 2005, pp. 255–268 (2005)
Kolarz, M., Moczurad, W.: Multiset, Set and Numerically Decipherable Codes over Directed Figures. In: Smyth, B. (ed.) IWOCA 2012. LNCS, vol. 7643, pp. 224–235. Springer, Heidelberg (2012)
Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)
Lindgren, K., Moore, C., Nordahl, M.: Complexity of two-dimensional patterns. Journal of Statistical Physics 91(5-6), 909–951 (1998)
Moczurad, M., Moczurad, W.: Some Open Problems in Decidability of Brick (Labelled Polyomino) Codes. In: Chwa, K.-Y., Munro, J.I. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 72–81. Springer, Heidelberg (2004)
Simplot, D.: A Characterization of Recognizable Picture Languages by Tilings by Finite Sets. Theoretical Computer Science 218-2, 297–323 (1991)
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Anselmo, M., Giammarresi, D., Madonia, M. (2013). Two Dimensional Prefix Codes of Pictures. In: Béal, MP., Carton, O. (eds) Developments in Language Theory. DLT 2013. Lecture Notes in Computer Science, vol 7907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38771-5_6
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