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"... With this work we aim to place dynamic modal logics such as Propositional Dynamic Logic (PDL) [1] and Game Logic (GL) [4] in a uniform coalgebraic framework. In our view, a dynamic system S consists of the following ingredients: 1. A set S which represents the global states of S. 2. An algebra L of ..."

Abstract
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With this work we aim to place dynamic modal logics such as Propositional Dynamic Logic (PDL) [1] and Game Logic (GL) [4] in a uniform coalgebraic framework. In our view, a dynamic system S consists of the following ingredients: 1. A set S which represents the global states of S. 2. An algebra L of labels (denoting actions, programs, games,...). 3. An interpretation of labels as G-coalgebras on the state space S. 4. A collection of labelled modalities [α], for α ∈ L, where intuitively [α]ϕ reads: “after α, ϕ holds”. Formally, the interpretation of labels is a map σ: L → (GS) S which describes how actions change the global system state. The algebraic structure on L describes how one can compose actions into more complex ones. The same type of algebraic structure should be carried by (GS) S, and we say that σ is standard, if σ is an algebra homomorphism, which means that the semantics of actions is compositional. By considering the exponential adjoint ̂σ: S → (GS) L we obtain a behavioural description of the system in the form of a G L-coalgebra. These two (equivalent) views of a dynamic system form the basis of our modelling. In short, σ describes structure and dynamics, and ̂σ describes behaviour and induces modalities. σ: L → (GS) S (algebraic view: structure, dynamics)