6 research outputs found
Moss' logic for ordered coalgebras
We present a finitary coalgebraic logic for -coalgebras, where is a
locally monotone endofunctor of the category of posets and monotone maps that
preserves exact squares and finite intersections. The logic uses a single cover
modality whose arity is given by the dual of the coalgebra functor , and the
semantics of the modality is given by relation lifting. For the finitary
setting to work, we need to develop a notion of a base for subobjects of .
This in particular allows us to talk about a finite poset of subformulas for a
given formula, and of a finite poset of successors for a given state in a
coalgebra. The notion of a base is introduced generally for a category equipped
with a suitable factorisation system.
We prove that the resulting logic has the Hennessy-Milner property for the
notion of similarity based on the notion of relation lifting. We define a
sequent proof system for the logic and prove its completeness
Moss' logic for ordered coalgebras
We present a finitary version of Moss' coalgebraic logic for -coalgebras,
where is a locally monotone endofunctor of the category of posets and
monotone maps. The logic uses a single cover modality whose arity is given by
the least finitary subfunctor of the dual of the coalgebra functor
, and the semantics of the modality is given by relation
lifting. For the semantics to work, is required to preserve exact squares.
For the finitary setting to work, is required to preserve
finite intersections. We develop a notion of a base for subobjects of . This in particular allows us to talk about the finite poset of subformulas
for a given formula. The notion of a base is introduced generally for a
category equipped with a suitable factorisation system.
We prove that the resulting logic has the Hennessy-Milner property for the
notion of similarity based on the notion of relation lifting. We define a
sequent proof system for the logic, and prove its completeness
Proof systems for Moss’ coalgebraic logic
We study Gentzen-style proof theory of the finitary version of the coalgebraic logic introduced by L. Moss. The logic captures the behaviour of coalgebras for a large class of set functors. The syntax of the logic, defined uniformly with respect to a finitary coalgebraic type functor T , uses a single modal operator ∇T∇T of arity given by the functor T itself, and its semantics is defined in terms of a relation lifting functor View the MathML sourceT¯. An axiomatization of the logic, consisting of modal distributive laws, has been given together with an algebraic completeness proof in work of C. Kupke, A. Kurz and Y. Venema. In this paper, following our previous work on structural proof theory of the logic in the special case of the finitary powerset functor, we present cut-free, one- and two-sided sequent calculi for the finitary version of Moss' coalgebraic logic for a general finitary functor T in a uniform way. For the two-sided calculi to be cut-free we use a language extended with the boolean dual of the nabla modality
Proof systems for Moss’ coalgebraic logic
We study Gentzen-style proof theory of the finitary version of the coalgebraic logic introduced by L. Moss. The logic captures the behaviour of coalgebras for a large class of set functors. The syntax of the logic, defined uniformly with respect to a finitary coalgebraic type functor T , uses a single modal operator ∇T∇T of arity given by the functor T itself, and its semantics is defined in terms of a relation lifting functor View the MathML sourceT¯. An axiomatization of the logic, consisting of modal distributive laws, has been given together with an algebraic completeness proof in work of C. Kupke, A. Kurz and Y. Venema. In this paper, following our previous work on structural proof theory of the logic in the special case of the finitary powerset functor, we present cut-free, one- and two-sided sequent calculi for the finitary version of Moss' coalgebraic logic for a general finitary functor T in a uniform way. For the two-sided calculi to be cut-free we use a language extended with the boolean dual of the nabla modality