Abstract
A two-dimensional code is defined as a set \(X\subseteq \Sigma ^{**}\) such that any picture over \(\Sigma \) is tilable in at most one way with pictures in \(X\). It is in general undecidable whether a set \(X\) of pictures is a code also in the finite case. Very recently in [3] strong prefix picture codes were defined as a decidable subclass that generalizes prefix string codes. Here a characterization for strong prefix codes that results in an effective procedure to construct them is presented. As a consequence there are also proved interesting results on the measure of strong prefix codes and a connection with the family of string prefix codes.
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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.: Strong Prefix Codes of Pictures. In: Muntean, T., Poulakis, D., Rolland, R. (eds.) CAI 2013. LNCS, vol. 8080, pp. 47–59. Springer, Heidelberg (2013)
Anselmo, M., Giammarresi, D., Madonia, M.: Two Dimensional Prefix Codes of Pictures. In: Béal, M.-P., Carton, O. (eds.) DLT 2013. LNCS, vol. 7907, pp. 46–57. Springer, Heidelberg (2013)
Anselmo, M., Giammarresi, D., Madonia, M.: Prefix picture codes: a decidable class of two-dimensional codes. Int. Jou. of Foundations of Computer Science (to appear, 2015)
Anselmo, M., Giammarresi, D., Madonia, M., Restivo, A.: Unambiguous recognizable two-dimensional languages. RAIRO -ITA 40(2), 227–294 (2006)
Anselmo, M., Madonia, M.: Deterministic and unambiguous two-dimensional languages over one-letter alphabet. Theoretical Computer Science 410(16), 1477–1485 (2009)
Anselmo, M., Jonoska, N., Madonia, M.: Framed Versus Unframed Two-Dimensional Languages. In: Nielsen, M., Kučera, A., Miltersen, P.B., Palamidessi, C., T\r{u}ma, P., Valencia, F. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 79–92. Springer, Heidelberg (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 2-dimensional tape. In: SWAT (FOCS). pp. 155–160 (1967)
Bozapalidis, S., Grammatikopoulou, A.: Picture codes. RAIRO - ITA 40(4), 537–550 (2006)
Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., (ed.) Handbook of Formal Languages, Vol. III, pp. 215–268. Springer (1997)
Kari, J., Salo, V.: A Survey on Picture-Walking Automata. In: Kuich, W., Rahonis, G. (eds.) Algebraic Foundations in Computer Science. LNCS, vol. 7020, pp. 183–213. Springer, Heidelberg (2011)
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)
Mauceri, S., Restivo, A.: A family of codes commutatively equivalent to prefix codes. Inf. Process. Lett. 12(1), 1–4 (1981)
Otto, F., Mráz, F.: Extended Two-Way Ordered Restarting Automata for Picture Languages. In: Dediu, A.-H., Mart\’ın-Vide, C., Sierra-Rodr\’ıguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 541–552. Springer, Heidelberg (2014)
Pradella, M., Cherubini, A., Crespi-Reghizzi, S.: A unifying approach to picture grammars. Inf. Comput. 209(9), 1246–1267 (2011)
Pr\r{u}ša, D., Mráz, F., Otto, F.: New Results on Deterministic Sgraffito Automata. In: Béal, M.-P., Carton, O. (eds.) DLT 2013. LNCS, vol. 7907, pp. 409–419. Springer, Heidelberg (2013)
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. (2015). Structure and Measure of a Decidable Class of Two-dimensional Codes. In: Dediu, AH., Formenti, E., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2015. Lecture Notes in Computer Science(), vol 8977. Springer, Cham. https://doi.org/10.1007/978-3-319-15579-1_24
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