Valuations and unambiguity of languages, with applications to fractal geometry

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.