Entropies and Co–entropies for Incomplete Information Systems ⋆ Daniela Bianucci, Gianpiero Cattaneo, and Davide Ciucci Dipartimento di Informatica, Sistemistica e Comunicazione Universit`a di Milano – Bicocca Via Bicocca degli Arcimboldi 8, I–20126 Milano, Italia {bianucci,cattang,ciucci}@disco.unimib.it Abstract. A partitioning approach to the problem of dealing with the entropy of incomplete information systems is explored. The aim is to keep into account the incompleteness and at the same time to obtain a probabilistic partition of the information system. For the resulting probabilistic partition, measures of entropy and co–entropy are defined, similarly to the entropies and co–entropies defined for the complete case. Keywords: entropy, co–entropy, incomplete information system. 1 Introduction: Qualitative and Quantitative Valuations of Roughness for Complete Information Systems In this work, we discuss the entropy of incomplete information systems as an ex- tension of the approach based on partitions from complete information systems. In order to introduce an approach of probability partition from an incomplete information system, let us first recall how one gets a partition from a complete information system, and thus how one can apply a measure of rough entropy when dealing with an information system. Let us recall that the original Pawlak approach to rough sets is essentially based on an approximation space, i.e., a pair 〈X, π〉 where X is a (finite) set, called the universe of objects, and π = {A 1 ,A 2 ,...,A N } is a partition of X , in general induced by the indistinguishability equivalence relation from a complete information system [1]. The subsets A j are the elementary sets (or also events ), each of which can be interpreted as a granule of knowledge supported by the partition. We denote by gr π (x) the granule (equivalence class) from π which contains the point x ∈ X . In the rough set theory, once fixed a partition π of X , any of its subsets H can be approximated from the bottom and from the top by the two lower and upper approximations defined respectively as: l π (H ) := ∪{A i ∈ π : A i ⊆ H } and u π (H ) := ∪{A j ∈ π : H ∩ A j = ∅}, producing the rough approximation of H defined as the pair r π (h)=(l π (H ),u π (H )) (with trivially l π (H ) ⊆ H ⊆ u π (H )), see [2] for a complete discussion. We can also ⋆ The author’s work has been supported by MIUR\PRIN project ”Automata and Formal languages: mathematical and application driven studies.” J.T. Yao et al. (Eds.): RSKT 2007, LNAI 4481, pp. 84–92, 2007. c  Springer-Verlag Berlin Heidelberg 2007