An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal” sets of formulas that have natural syntactic closure properties.
The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if the deducibility relation is generated by countably many inference rules.
A counter-example to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable. 相似文献
We define a modal logic whose models are coalgebras of a polynomial functor. Bisimilarity turns out to be the same as logical equivalence. Ideas and concepts of modal logic are directly applied to the theory of coalgebras: we give an axiomatization and define canonical coalgebras. That leads to a completeness result. Each canonical coalgebra proves to be terminal in a certain class of coalgebras. The approach also yields a functional characterization of the terminal coalgebra of all coalgebras with respect to a given polynomial functor. 相似文献
In this paper we show a categorical treatment of general time systems using the categorization method presented in our previous paper. Various concepts about general time systems are categorized in the unified framework. Some category theoretical tools for the investigation of such time systems are presented. Using those tools some basic properties of time systems are explored in our framework. In particular, a conceptual equivalence between the causality and the state space representability is proved in the categorical terms. These results show that our method can be a universal tool for a categorization and a categorical treatment of mathematically defined general systems. 相似文献
In a class of categories, including E. Manes's assertional ones, the control structures if-then-else and repeat-until are modeled as natural transformations of suitable functors. This context show how the three basic pieces of any structured programming language (concatenation, conditional and recursion) share naturality. 相似文献