Clock systems for stochastic and non-deterministic categorical systems theories

Abstract: One of the characteristic features of categorical systems theory is that the behavior of systems can be characterized by certain morphisms into them. In other words, behaviors form a representable covariant functor to Set. And more generally, in the compositional setting, behaviors form a representable double functor to Span. Clock systems are convenient because behavior functors represented by clock systems are automatically well-behaved. It was previously not known whether stochastic and non-deterministic systems theories have clock systems. In this paper, we show that indeed they do have clock systems. Moreover, the clock systems for non-deterministic systems point to generalized notions of behavior for non-linear time.

• The paper presents a novel categorical framework extending deterministic systems to stochastic and nondeterministic contexts via explicit clock systems.
• It develops clock constructions for stochastic Moore machines using probabilistic filtrations and for nondeterministic machines via powerset monads to yield representable behavior functors.
• The framework leverages double category structures and sheaf theory to ensure modular compositionality and opens pathways for hybrid and imprecise probability systems.

Clock Systems for Stochastic and Nondeterministic Categorical Systems Theories

Introduction and Motivation

The paper "Clock systems for stochastic and non-deterministic categorical systems theories" (2603.29573) extends the categorical framework for systems theory by constructing clock systems that enable representable behavior functors in stochastic and nondeterministic contexts. These results generalize the well-developed deterministic case, where behaviors of systems (e.g., trajectories of automata) are classically captured by representable functors indexed by a universal “clock.” The principal innovations lie in defining explicit clock systems for stochastic and nondeterministic Moore machines and identifying the precise connection between categorical compositionality and established notions from probability theory and nondeterministic computation.

Theoretical Framework

The study is grounded in the rich interface between category theory, coalgebra, and systems theory. Systems are traditionally described via functors $\mathsf{Sys} \to \mathsf{Set}$, mapping each system to its space of behaviors, often representable by a fixed “clock” object. For deterministic Moore machines, the clock is essentially an incrementing natural number object, capturing all possible system trajectories.

The authors extend this premise to generalized Moore machines parameterized by a strong monad $M$, encompassing determinism ($M = \text{id}$), nondeterminism ($M = P$ powerset), and stochasticity ($M = D$ probability distributions in $\mathsf{Set}$, or the Giry monad $G$ in $\mathsf{Meas}$). The structure is embedded in a double categorical framework, encoding both composition patterns (horizontal morphisms) and interface morphisms (vertical morphisms), capturing the behavior of open and interconnected systems.

Clock Systems for Stochastic Moore Machines

The construction of clock systems in the stochastic context resolves a previously open question: how to model behaviors (trajectories) as morphisms from a universal clock, capturing the correct filtration and conditional independence structure central to stochastic processes.

• Stochastic Clock Construction: The authors define, for a filtered probability space $(\Omega, \mathcal{F}_\Omega, P, \{\mathcal{F}_i\})$, a clock system whose state space is $$\tilde{\Omega} = \sum_{i \in \mathbb{N}} (\Omega, \mathcal{F}_i)$$, with transitions defined via conditional probabilities and filtration increments. This clock generates trajectories corresponding exactly to $M$0-adapted Markov processes, with the representable behavior functor aligning with classical law-based definitions in probability.
• Theoretical Consequence: The main technical theorem establishes a canonical bijection between system morphisms out of the stochastic clock and behaviors defined via adapted processes and the Kolmogorov consistency conditions for Markov processes, subject to almost-everywhere equivalence.
• Presheaf/Sheaf Structure: The construction is shown to extend to a functor $M$1, where the site consists of filtered probability spaces and immersions of filtrations (i.e., morphisms preserving martingale structure and conditionals). Behaviors thus form a probability sheaf, clarifying the gluing conditions and almost-sure identifications as colimits in the appropriate topos.

Clock Systems for Nondeterministic Moore Machines

For nondeterministic systems, the powerset monad replaces the Giry monad, and the clock system encodes all possible nondeterministic evolutions.

• Nondeterministic Linear and Nonlinear Clocks: The authors introduce the concept of nondeterministic clocks whose state space is $M$2 for a set $M$3, where transitions append any possible element in $M$4 at each step. This construction extends to nonlinear clocks parameterized by graphs, encoding decorated paths, thus allowing the treatment of behaviors over more general time-indexing structures.
• Functorial Representation of Behaviors: As in the stochastic case, behaviors are representably functorial: maps from the clock system to a given system corepresent the set of possible behaviors, encapsulating the branching structure of nondeterministic evolution. Unlike the stochastic setting, there is not a classical measure-theoretic notion of behavior; the authors’ approach thus provides a definition with strong categorical justification.

Compositionality and Double Categories

A salient advantage of representable behaviors via clock systems is the inheritance of compositionality properties. The behavior functors respect the double categorical structure, enabling modular reasoning about interconnected open systems. Specifically:

• The behavior functor lifts from $M$5 to a morphism between systems theories, preserving both the action on interfaces and the interaction with composition patterns.
• This machinery ensures that decomposing a system and composing the behaviors of its parts agrees with the behavior of the composite, aligning with formal compositionality theorems established in prior work.

Sheaf-Theoretic and Topos-Theoretic Insights

The identification of behaviors as sheaves on the site of filtered probability spaces (or their nondeterministic analogs) connects categorical system semantics with the logic of probability and stochastic processes as developed in the topos-theoretic literature (notably [simpson-2017-probability]). This perspective allows for:

• Systematic treatment of behaviors up to almost-sure equivalence.
• Canonical gluing and localization, underpinning variable (context-dependent) probability spaces and filtrations.
• An abstract (but concrete in this framework) way to phrase and prove results about lifting behaviors and their compatibility under morphisms.

Implications and Future Directions

The categorical establishment of clock systems for stochastic and nondeterministic systems has several implications:

• Modular Reasoning in Open Probabilistic/Nondeterministic Systems: The explicit construction of clock objects enables the extension of compositional verification, control, and synthesis techniques from deterministic to stochastic and nondeterministic models.
• Integration with Imprecise Probability and Hybrid Systems: The paper outlines future avenues including the adaptation of these results to infradistribution monads (imprecise probabilities) and to continuous-time (stochastic differential equation) settings, where the clock would be parameterized by spaces of continuous paths and the update operation informed by stochastic integration (e.g., Brownian motion increments).
• Categorical Approach to Markov and Martingale Theory: The framework allows recasting central concepts from the theory of stochastic processes — such as Markov properties, filtrations, and conditional independence — as categorical structures and universal properties, potentially impacting the way compositional stochastic analysis is formalized.
• Potential for Higher-Order and Nonlinear Clocks: The generality of the clock construction, especially its parametrization by graphs for nonlinear time, suggests applicability in settings ranging from concurrency theory to dynamic epistemic logic.

Conclusion

The paper establishes that behaviors for stochastic and nondeterministic Moore machines admit a rigorous categorical semantics via representable functors defined by clock systems, naturally extending deterministic cases. By explicitly constructing stochastic and nondeterministic clocks, the work connects categorical system theory, probability theory, and logic of computation. It provides both practical tools for reasoning about modular stochastic/nondeterministic systems and foundational advances in the semantics of complex behavioral theories, laying groundwork for further extensions to hybrid, continuous-time, and imprecise probability settings (2603.29573).