The entropy of Łukasiewicz-languages

The paper presents an elementary approach for the calculation of the entropy of a class of languages. This approach is based on the consideration of roots of a real polynomial and is also suitable for calculating the Bernoulli measure. The class of languages we consider here is a generalisation of the Łukasiewicz language

Valuations Of Languages, With Applications To Fractal Geometry

Valuations -- morphisms from (\Sigma ; \Delta; ) to ((0; 1); \Delta; 1)--- are a simple generalization of Bernoulli morphisms (distributions, measures) as introduced in [28, 41, 11, 9, 10, 42]. This paper shows that valuations are useful not only within the theory of codes, but also when dealing with ambiguity, especially in contextfree grammars, or for defining outer measures on the space of !- words which are of some importance to the theory of fractals. These connections yield new formulae to determine the Hausdorff dimension of fractal sets (especially in Euclidean spaces) defined via formal languages. The class of fractals describable with contextfree languages strictly includes that of MRFS-fractals introduced in [58, 18]. Some of the results of this paper also appear as part of the author's PhD thesis [36] and in [30, 31]