The World of Unary Languages. * A Quick Tour Carlo Mereghetti Dip. di Informatica, Sist. e Com. Univ. degli Studi di Milano – Bicocca via Bicocca degli Arcimboldi 8 20126 Milano, Italy mereghetti@disco.unimib.it Giovanni Pighizzini Dip. di Scienze dell’Informazione Univ. degli Studi di Milano via Comelico 39 20135 Milano, Italy pighizzi@dsi.unimi.it Abstract We give two flashes from the world of unary languages related to the study of tight computational lower bounds for nonregular language acceptance. They show both an interesting dissymmetry with the general case, and the flavor and some typical number theoretic tools of unary computations. 1 Introduction In mathematics, “simplifications” often lead to meaningful and interest- ing results. This is exactly the case even for theoretical computer science and, particularly, for language and complexity theory. In such realms, one of the most investigated simplification is that provided by unary lan- guages. A unary language is simply a language built over a single–letter input alphabet. Several results in the literature witness the relevance of dealing with unary inputs, often emphasizing sharp dissymmetries with the general case of languages on alphabets with two or more symbols. However, it is worth remarking that simplifications do not necessarily lead to easier problems to cope with, and this is the case even when considering unary languages. The total absence of structure in input strings forces our computation techniques to be much more sophisticated and sometimes tricky. In particular, as one may expect, it is interesting to record the constant and nice use of Number Theory which provides a valuable source of nontrivial tools and properties to operate with. In this work we are going to briefly survey some results showing the importance and the typical flavor of computing with unary languages. Specifically, we focus on the world of Turing machines that work within * Partially supported by MURST, under the project “Modelli di calcolo innovativi: metodi sintattici e combinatori”. 1