Two Modal Logics for Coalgebras

Nowadays, information has become one of the most valuable sources for hu- mankind. The development of computers has seriously increased the amount of information that we can handle. Other developments, like internet, have made the interexchange of information much easier. Even more, transferring large quantities of information is now a daily activity. However, there are some tech- nical issues. For example: (1) not all computers use the same operative systems, and (2) not all users use the same methods (programs) to handle information. These issues leads to possible failures in the transmission of data with infor- mation. Developing new programs and trying to impose them to all users is just a temporal solution. Those programs will eventually become obsolete and information will have to be moved (transmitted) again. Furthermore, such kind of solution will have a high cost each time that data are moved. Another possible solution would be to store data in such a way that it will be compatible with all possible forms of visualization, i.e. to store information independently of its representations. In order to do this, a new kind kind of programing language should be developed, a generic language. Generic languages should have the property that the meaning does not depend on the representation and all representations of a concept have the same meaning. Such kind of languages have been already developed for data concerning black box systems", i.e. systems where information is modified by an user with some external interaction (buttons), but the user does not know how the systems works. For example, the management of a bank account using an ATM. Those systems are called coalgebraic systems. Languages for those systems are called (generic) coalgebraic languages. In this paper we aim to present two kinds of generic coalgebraic languages. Those two languages are, so far, the main languages developed to describe coal- gebras: they can describe the evolution of a state in a coalgebraic system, and most other coalgebraic languages are modications or extensions of those. The structure of the talk will be as follows: We will first introduce the basic theory of coalgebras and define coalgebraic languages. Using these, we will illustrate how coalgebraic languages describe the behavior of a coalgebraic system. Formally we will prove that existence of an expressive coalgebraic language is equivalent to the existence of a final coalgebra. We then proceed to present the first class of languages; we call those Moss' languages. After we present the second class of languages; we call those Languages of predicate liftings. We finish this section showing how Moss' languages and languages of predicate liftings describe the behavior of a system.  

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