Abstract
A category of one-step semantics is introduced to unify different approaches to coalgebraic logic parametric in a contravariant functor that assigns to the state space its collection of predicates with propositional connectives. Modular constructions of coalgebraic logic are identified as colimits, limits, and tensor products, extending known results for predicate liftings. Generalised predicate liftings as modalities are introduced. Under common assumptions, the logic of all predicate liftings together with a complete axiomatisation exists for any type of coalgebras, and it is one-step expressive for finitary functors. Colimits and compositions of one-step expressive coalgebraic logics are shown to remain one-step expressive.
Highlights
Two syntax-oriented approaches to coalgebraic modal logic — Moss’ cover modality [23] and Pattinson’s predicate liftings [24, 25, 26] — are successful in producing a wide range of modal logics parametric in a Set functor
Markov processes are coalgebras of the Giry monad, and propositional connectives for measurable spaces can be specified by the contravariant functor S : Meas → MSL mapping a space to its σ-algebra, considered as a meet semilattice
We would like to suggest that our work provides the right level of abstraction for understanding coalgebraic modal logic
Summary
Two syntax-oriented approaches to coalgebraic modal logic — Moss’ cover modality [23] and Pattinson’s predicate liftings [24, 25, 26] — are successful in producing a wide range of modal logics parametric in a Set functor. Multi-modal logic is expressive for image-finite A-labelled Kripke frames, but Klin’s condition does not cover this case, since its corresponding type functor, the A-fold product PA, is not necessarily finitary. Another important line of research investigated modularity of predicate liftings [7, 27]: expressiveness and completeness are stable under certain constructions. The categorical viewpoint allows us to formulate general preservation principles for coalgebraic expressiveness that apply to all logics in CoLog. The framework is parametric in a contravariant functor mapping state spaces to “algebras” for the base logic about which very little needs to be assumed to cover a large variety of examples. This paper summaries the first author’s PhD thesis [6], to which we point for most of the proofs
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