Strong Completeness for Iteration-Free Coalgebraic Dynamic Logics

Abstract

Abstract. We present a (co)algebraic treatment of iteration-free dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL), both without star. The main observation is that the program/game constructs of PDL/GL arise from monad structure, and the axioms of these logics correspond to certain compatibilty requirements between the modalities and this monad structure. Our main contribution is a general soundness and strong completeness result for PDL-like logics for T -coalgebras where T is a monad and the "program" constructs are given by sequential composition, test, and pointwise extensions of operations of T .

Weak Completeness of Coalgebraic Dynamic Logics

“…One also easily verifies that λ is diamond-like iff its Boolean dual is box-like. It is easy to see that if λ is diamond-like then it is also diamond-like according to our "old" definition in [7], similarly for box-like. However, it is no longer the case that every predicate lifting is either box-like or diamond-like, e.g., for T = P, λ X (U )…”

“…In earlier work [7], we introduced the notion of a coalgebraic dynamic logic for programs built from Kleisli composition, pointwise operations and tests. Here we extend this notion to also include iteration (Kleene star).…”

“…Kleisli-composition distributes over this τ-induced join since µ X and T f preserve it, for all functions f : X → Y , due to naturality of τ, and these maps being T -algebra morphisms. QED Note that any natural transformation τ : P ⇒ T yields a natural transformation 1 ⇒ P ⇒ T , where 1 ⇒ P picks out the empty set, such that T is pointed as defined in [7].…”

“…If σ : T n ⇒ T and χ : N n ⇒ N are such that λ • σ = χ • λ n , then we call σ positive if χ is positive. The axioms for positive pointwise operations of the form χ =δ ∧ρ are obtained by extending Definition 14 from [7] with a case for conjunction: . We define Λ = { α | α ∈ A} and let Ax A = α∈A Ax α where Ax α is the set of rank-1 axioms over the labelled modal language F (P 0 , A 0 , Σ) obtained by substituting α for ✸ in all the axioms in Ax.…”

“…Our main contribution is a completeness proof for coalgebraic dynamic logics with iteration. We introduced coalgebraic dynamic logics in our previous work [7] as a natural generalisation of PDL and GL with the aim to study various dynamic logics within a uniform framework that is parametric in the type of models under consideration, or -categorically speaking -parametric in a given monad. In [7] we presented an initial soundness and strong completeness result for such logics.…”