Search

In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically, this problem is in general undecidable. More precisely, we show that there is no algorithm that classifies the logic of a finite consistent Hilbert calculus in the Leibniz and in the Frege hierarchies. [ABSTRACT FROM AUTHOR]

Abstract: We generalize stochastic Kripke models and Markov transition systems to stochastic right coalgebras. These are coalgebras for a functor with as an endofunctor on the category of analytic spaces, and is the subprobability functor. The modal operators are generalized through predicate liftings which are set-valued natural transformations involving the functor. Two states are equivalent iff they cannot be separated by a formula. This equivalence relation is used to construct a cospan for logical equivalent coalgebras under a separation condition for the set of predicate liftings. Consequently, behavioral and logical equivalence are really the same. From the cospan we construct a span. The central argument is a selection argument giving us the dynamics of a mediating coalgebra from the domains of the cospan. This construction is used to establish that behavioral equivalent coalgebras are bisimilar, yielding the equivalence of all three characterizations of a coalgebra’s behavior as in the case of Kripke models or Markov transition systems. [Copyright &y& Elsevier]

Relation algebras were invented by Tarski and his collaborators in the middle of the 20th century. The concept of integrality arose naturally early in the history of the subject, as did various constructions of finite integral relation algebras. Later the concept of finite-dimensionality was introduced for classifying nonrepresentable relation algebras. This concept is closely connected to the number of variables used in proofs in first-order logic. Some results on these topics are presented in chronological order. [Copyright &y& Elsevier]