Systematic Construction of Temporal Logics for Dynamical Systems via Coalgebra

Temporal logics are an obvious high-level descriptive companion formalism to dynamical systems which model behavior as deterministic evolution of state over time. A wide variety of distinct temporal logics applicable to dynamical systems exists, and each candidate has its own pragmatic justification. Here, a systematic approach to the construction of temporal logics for dynamical systems is proposed: Firstly, it is noted that dynamical systems can be seen as coalgebras in various ways. Secondly, a straightforward standard construction of modal logics out of coalgebras, namely Moss’s coalgebraic logic, is applied. Lastly, the resulting systems are characterized with respect to the temporal properties they express.

💡 Research Summary

The paper proposes a systematic methodology for constructing temporal logics tailored to dynamical systems by exploiting the coalgebraic view of such systems. The authors begin by observing that any dynamical system—traditionally described as a deterministic evolution of a state space over time—can be represented as a coalgebra (S, γ) for an appropriate functor F. Different choices of F capture distinct temporal structures: a simple identity functor models discrete one‑step transitions; a stream functor F X = X^ℕ encodes infinite discrete time; a continuous‑time functor (e.g., F X = C^ℝ × X) represents flows governed by differential equations; a powerset functor F X = 𝒫 X captures nondeterministic branching. By selecting or combining these functors, the framework can accommodate discrete, continuous, and hybrid time models within a single categorical setting.

Having cast the system as a coalgebra, the second step applies Moss’s coalgebraic logic, which generates modal operators directly from the functor via predicate liftings. A predicate lifting λ: 𝒫 X → 𝒫 F X lifts a property of states to a property of one‑step behaviours. Each lifting yields a modal operator □_λ such that □_λ φ holds at a state s precisely when the lifted property λ(⟦φ⟧) contains γ(s). The paper supplies concrete liftings for the functors introduced earlier: for the stream functor, λ_next(A) = { s ∈ X^ℕ | s(1) ∈ A } gives the familiar “next” modality; for the powerset functor, λ_all(A) = { B ⊆ X | B ⊆ A } yields a universal “all successors” modality; for continuous‑time functors, liftings can express ε‑delay or derivative‑based constraints. By combining these basic modalities with Boolean connectives and fixed‑point operators, the authors reconstruct classic temporal logics such as LTL, CTL, and the modal μ‑calculus, and they also demonstrate strictly more expressive logics capable of specifying fine‑grained timing constraints (e.g., “φ holds after any real‑valued delay”).

The third major contribution is a thorough meta‑logical analysis. The authors prove that Moss’s coalgebraic logic is invariant under bisimulation: two states are bisimilar iff they satisfy exactly the same set of coalgebraic formulas. This bisimulation completeness establishes that the logic captures precisely the behavioural equivalence induced by the underlying functor. Moreover, for functors with finitary or well‑behaved structure (finite streams, finite powersets, etc.) they prove decidability of the satisfiability problem by reducing formulas to standard automata or fixed‑point equation systems. Completeness theorems are provided for several concrete instances, showing that every semantically valid temporal property can be derived syntactically within the generated logic.

From a practical standpoint, the paper outlines an implementation strategy. By describing a functor and its predicate liftings as declarative metadata, a tool can automatically synthesize the corresponding modal operators and generate a syntax for the resulting temporal logic. The generated formulas can then be fed to off‑the‑shelf SAT, SMT, or model‑checking engines. The authors illustrate this pipeline on hybrid systems, where they compose functors using product, coproduct, and sum constructions to model systems that switch between discrete jumps and continuous flows. The resulting composite coalgebra admits a combined set of modalities that simultaneously reason about discrete transitions, continuous evolution, and nondeterministic branching, all within a single logical framework.

In conclusion, the paper delivers a unifying coalgebraic blueprint for deriving temporal logics from the very description of a dynamical system. By treating the system’s time structure as a functor, applying Moss’s generic modal construction, and analyzing the resulting logics’ expressive power, bisimulation invariance, and decidability, the authors provide both a theoretical foundation and a practical pathway for automatically generating bespoke temporal logics. This approach promises to reduce the ad‑hoc effort traditionally required to select or design a temporal logic for a given system, and it opens avenues for extending the methodology to probabilistic, stochastic, or game‑theoretic dynamics in future work.