This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic p...

Iterative theories introduced by Calvin Elgot formalize potentially infinite computations as solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with “iterative algebras”, i. e., algeb...

Modal logic is extended by partially ordering the modalities. The modalities are normal, i.e., commute with either conjunctions or disjunctions and preserve either Truth or Falsity (respectively). The partial order does not conflict with type of modality (K, S4, etc.) although this paper will concentrate on S4 since partially ordered S4 systems appear to be numerous. The partially-ordered norma...

This paper introduces coalgebraic monads as a unified model of term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary coalgebraic monads which allows us to generalise the notion of rational term and term graph to categories other than Set. As an application we sketch part of the c...

We introduce an axiomatization for the coalgebraic fixed point logic which was introduced by Venema as a generalization, based on Moss’ coalgebraic modality, of the well-known modal mucalculus. Our axiomatization can be seen as a generalization of Kozen’s proof system for the modal mu-calculus to the coalgebraic level of generality. It consists of a complete axiomatization for Moss’ modality, e...

A partial component is a process which fails or dies at some stage, thus exhibiting a finite, more ephemeral behaviour than expected (e.g. operating system crash). Partiality — which is the rule rather than exception in formal modelling — can be treated mathematically via totalization techniques. In the case of partial functions, totalization involves error values and exceptions. In the context...

Coalgebraic modal logic serves as a unifying framework to study a wide range of modal logics beyond the relational realm, including probabilistic and graded logics as well as conditional logics and logics based on neighbourhoods and games. Coalgebraic predicate logic (CPL), a generalization of a neighbourhoodbased first-order logic introduced by Chang, has been identified as a natural first-ord...

CoLoSS, the Coalgebraic Logic Satisfiability Solver, decides satisfiability of modal formulas in a generic and compositional way. It implements a uniform polynomial space algorithm to decide satisfiability for modal logics that are amenable to coalgebraic semantics. This includes e.g. the logics K, KD, Pauly’s coalition logic, graded modal logic, and probabilistic modal logic. Logics are easily...