Towards a Coalgebraic Interpretation of Propositional Dynamic Logic

The interpretation of propositional dynamic logic (PDL) through Kripke models requires the relations constituting the interpreting Kripke model to closely observe the syntax of the modal operators. This poses a significant challenge for an interpretation of PDL through stochastic Kripke models, because the programs’ operations do not always have a natural counterpart in the set of stochastic relations. We use rewrite rules for building up an interpretation of PDL. It is shown that each program corresponds to an essentially unique irreducible tree, which in turn is assigned a predicate lifting, serving as the program’s interpretation. The paper establishes and studies this interpretation. It discusses the expressivity of probabilistic models for PDL and relates properties like logical and behavioral equivalence or bisimilarity to the corresponding properties of a Kripke model for a closely related non-dynamic logic of the Hennessy-Milner type.

💡 Research Summary

The paper tackles the long‑standing problem of giving a semantics to Propositional Dynamic Logic (PDL) that works with stochastic Kripke models rather than the classic deterministic Kripke frames. In ordinary Kripke semantics each program operator—choice (∪), sequential composition (;), and iteration ()—must be interpreted by a binary relation that mirrors the syntactic structure of the program. This requirement becomes problematic when the underlying transition relation is probabilistic, because there is no natural way to define operations such as Kπ∪Kπ′ or Kπ on probability kernels.

To overcome this obstacle the authors introduce a term‑rewriting system for programs. Programs are generated from a set U of primitive actions (including the empty program ε) by the grammar
π ::= ε | π₁∪π₂ | π₁;π₂ | π*.
The iteration operator is treated as a primitive infinite‑arity operator W, and a collection of rewrite rules (dₗ, dᵣ, dε, d*) together with a set of equations (idₗ, idᵣ, associativity, commutativity, idempotence, infinite‑choice law (dis∞), and a transposition law (transp)) is used to normalize any program.

A weight function w(π) is defined recursively; if w(π) is finite, repeated application of the rewrite rules always terminates, yielding an irreducible term (or tree) β. The irreducible terms form a set I(U) that is closed under ∪ and the infinite‑arity operator W, and any sequential composition of two irreducible terms can be collapsed again to a single irreducible term. Consequently every program π is provably equivalent (modulo the congruence generated by the rewrite rules and equations) to a unique irreducible tree β. In this tree the only remaining operators are nondeterministic choice (∪) and the explicit form of iteration (W); primitive actions appear only as finite blocks a₁;…;aₖ.

The second major contribution is to map each irreducible tree to a predicate lifting, i.e. a natural transformation that gives a coalgebraic semantics. The state space X is equipped with a Borel σ‑algebra 𝔅(X). A stochastic transition is a measurable map K : X → 𝔅(X) (a probability kernel). The authors interpret ∪ as the σ‑countable sum of kernels and interpret the infinite‑arity operator W as a fixed‑point construction on the Borel functor. This requires the underlying measurable space to be closed under the Souslin operation; the paper notes that this closure holds automatically for finite (Polish) spaces but must be assumed or enforced for general spaces. By constructing appropriate natural transformations for ∪ and W, the authors obtain a well‑defined coalgebraic model for every irreducible program tree, and thus for every original PDL program.

Having established a semantics, the paper investigates expressivity and equivalence notions. It shows that stochastic models are at least as expressive as ordinary Kripke models; in fact, when primitive actions are interpreted via a monad (e.g., the Giry monad), the sequential composition coincides with Kleisli composition. The authors then compare three notions of equivalence: logical equivalence (same PDL formulas hold), behavioural equivalence (states cannot be distinguished by any observation), and bisimilarity (standard coalgebraic bisimulation). By introducing a non‑dynamic Hennessy‑Milner style logic that uses only primitive programs and atomic propositions, they prove that, provided the sets of primitive actions and atomic propositions are countable, the three notions coincide for the stochastic PDL models constructed.

Technical subtleties are highlighted: the definition of W relies on the Souslin‑closed property, which fails for many infinite state spaces; thus the results are most straightforward for finite or Polish spaces. Moreover, the rewrite system must handle infinite trees with potentially infinite fan‑out, but the authors show that the irreducible form always exists and is unique up to the congruence.

In summary, the paper presents a pipeline: (1) rewrite any PDL program into a canonical irreducible tree, (2) assign to each tree a predicate lifting using measurable‑theoretic constructions, (3) obtain a stochastic coalgebraic model that respects the original PDL syntax, and (4) analyze expressivity and equivalence, relating the stochastic setting back to classic Kripke semantics via a Hennessy‑Milner correspondence. This work bridges the gap between dynamic logics and probabilistic coalgebraic semantics, opening avenues for further research on infinite state spaces, alternative probability monads, and concrete applications in probabilistic verification.