Abstract
Valuations — morphisms from (Σ *·λ) to ((0, ∞),·,1) —are a simple generalization of Bernoulli morphisms (distributions, measures) as introduced in [12, 20, 6, 4, 5, 21]. This paper shows that valuations are not only useful within the theory of codes, but also when dealing with ambiguity, especially in regular expressions and contextfree grammars, or for defining outer measures on the space of ω-words which are of some importance for the theory of fractals. These connections yield new formulae to determine the Hausdorff dimension of fractal sets (especially in Euclidean spaces) defined via regular expressions and contextfree grammars.
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L. M. Andersson. Recursive Construction of Fractals. PhD thesis, Helsinki: Suomalainen Tiedeakatemia, Aug. 1992. Annales Academiae Scientiarum Fennicae Series A, I. Mathematica, Dissertationes, 86.
M. F. Barnsley. Fractals Everywhere, Boston: Academic Press, 1988.
M. F. Barnsley, J. H. Elton, and D. P. Hardin. Recurrent iterated function systems. Constructive Approximation, 5:3–31, 1989.
J. Berstel and D. Perrin. Theory of Codes. Pure and Applied Mathematics. Orlando: Academic Press, 1985.
J. Berstel and C. Reutenauer. Rational Series and Their Languages, volume 12 of EATCS Monographs on Theoretical Computer Science. Berlin: Springer, 1988.
F. Blanchard and G. Hansel. Languages and subshifts. In M. Nivat and D. Perrin, editors, Automata on infinite words, volume 192 of LNCS, pages 138–146. Berlin: Springer, May 1984.
A. Brüggemann-Klein. Regular expressions into finite automata. In LATIN'92, volume 483 of LNCS, pages 87–98, 1992.
K. Čulik, II and S. Dube. Methods for generating deterministic fractals and image processing. In LNCS: 464; IMYCS, pages 2–28. Springer, 1990.
F. M. Dekking. Recurrent sets: A fractal formalism. Technical Report 82-32, Technische Hogeschool, Delft (NL), 1982.
S. Dube. Undecidable problems in fractal geometry. Manuscript, May 1993.
G. A. Edgar. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. New York: Springer, 1990.
S. Eilenberg. Automata, Languages, and Machines, Volume A. Pure and Applied Mathematics. New York: Academic Press, 1974.
K. J. Falconer. Fractal Geometry. Chichester: John Wiley & Sons, 1990.
H. Fernau. MRFS-Fraktale aus dem Blickwinkel der regulären ω-Sprachen. In W. Thomas, editor, 2. Theorietag ‘Automaten und Formale Sprachen', volume 9220 of Technische Berichte, pages 46–50. Universität Kiel, 1992.
H. Fernau. IIFS and codes (extended version). Technical Report 23/93, Universität Karlsruhe, Fakultät für Informatik, 1993.
H. Fernau. Infinite iterated function systems. Accepted for publication in Mathematische Nachrichten, 1994.
H. Fernau. Valuations of languages, with applications to fractal geometry. Accepted for publication in TCS, 1994.
H. Fernau. Valuations, regular expressions, and fractal geometry. Submitted for publication, Dec. 1993.
H. Fernau. Varianten Iterierter Funktionensysteme und Methoden der Formalen Sprachen. PhD thesis, Universität Karlsruhe (TH) (Germany), 1993.
G. Hansel and D. Perrin. Codes and Bernoulli partitions. Mathematical Systems Theory, 16:133–157, 1983.
G. Hansel and D. Perrin. Rational probability measures. Theoretical Computer Science, 65:171–188, 1989.
J. Hutchinson. Fractals and self-similarity. Indiana University Mathematics Journal, 30:713–747, 1981.
W. Kuich and A. Salomaa. Semirings, Automata, Languages, volume 5 of EATCS Monographs on Theoretical Computer Science. Berlin: Springer, 1986.
B. Mandelbrot. The Fractal Geometry of Nature. New York: Freeman, 1977.
R. D. Mauldin and S. C. Williams. Hausdorff dimension in graph directed constructions. Transactions of the American Mathematical Society, 309(2):811–829, Oct. 1988.
M. Nolle. Comparison of different methods for generating fractals. To appear in the Proceedings of the IMYCS'92, 1992.
H.-O. Peitgen, H. Jürgens, and D. Saupe. Fractals for the Classroom. Part One. Introduction to Fractals and Chaos. New York: Springer, 1992.
C. A. Rogers. Hausdorff Measures. Cambridge at the University Press, 1970.
A. K. Salomaa and M. Soittola. Automata-Theoretic Aspects of Formal Power Series. New York: Springer, 1978.
L. Staiger. On infinitary finite length codes. RAIRO Informatique théorique et Applications/Theoretical Informatics and Applications, 20(4):483–494, 1986.
L. Staiger. Quadtrees and the Hausdorff dimension of pictures. In A. Hübler et al., editors, Geobild'89, volume 51 of Mathematical Research, pages 173–178, Georgenthal, 1989.
L. Staiger. Kolmogorov complexity and Hausdorff dimension. Information and Computation (formerly Information and Control), 103:159–194, 1993.
K. Wicks. Fractals and Hyperspaces, volume 1492 of LNM. Berlin: Springer-Verlag, 1991.
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Fernau, H., Staiger, L. (1994). Valuations and unambiguity of languages, with applications to fractal geometry. In: Abiteboul, S., Shamir, E. (eds) Automata, Languages and Programming. ICALP 1994. Lecture Notes in Computer Science, vol 820. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58201-0_54
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DOI: https://doi.org/10.1007/3-540-58201-0_54
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