Stochastic Modality Selection in Hybrid Systems

• Stochastic modality selection is an adaptive approach that integrates stochastic and deterministic behaviors for efficient decision-making in complex hybrid systems.
• It employs decomposed (race-based) and composed (two-kernel) scheduling to model event timings and transition probabilities in formal system frameworks.
• Recent developments like the MARS policy leverage learned gating mechanisms to balance expressivity and computational efficiency in sequential control tasks.

Stochastic modality selection refers to the adaptive, context-dependent invocation of stochasticity within a decision or generative process, such that multimodal behaviors are produced only when necessary, while deterministic behavior is retained when a task is essentially single-modal or unambiguous. This paradigm arises in both formal system modeling (e.g., stochastic hybrid automata) and modern machine learning for control and imitation learning. Recent advances, such as the Modality-Adaptive Robot Sampling (MARS) policy, explicitly operationalize stochastic modality selection via learned gating mechanisms, balancing expressivity and computational efficiency in sequential prediction and control (Jia et al., 28 May 2026). Historical approaches in the modeling of timed and hybrid systems also reveal two archetypes: decomposed (“race-based”) scheduling and composed (“two-kernel”) scheduling, each implementing stochastic event selection via distinct architectural principles (Willemsen et al., 2023).

1. Formalization in Hybrid Systems

Two principal frameworks have been established for introducing stochasticity into hybrid automata, each corresponding to different modalities of stochastic choice:

• Decomposed (Race-Based) Scheduling: Every possible discrete event transition is equipped with an associated, independently sampled random variable (clock). The time to the next event is the minimum realization among these clocks—known as the “race” paradigm. The event with the smallest clock fires, and its clock is then resampled, with remaining clocks decremented by the elapsed interval. The density for the $j$-th event firing at time $t$ is $f_j(t) \prod_{i\neq j}(1-F_i(t))$, where $F_i$ is the CDF of the $i$-th clock (Willemsen et al., 2023).
• Composed (Two-Kernel) Scheduling: The time delay until the next event is sampled globally from a location-indexed distribution. At this sampled time, the transition to be executed is chosen stochastically from a kernel over enabled events. This model can generate uniform or arbitrarily structured distributions over event timings and choices, with the probability of a transition $e$ at time $t$ given by $\text{Dist}(\sigma, dt) \cdot \Pi((\ell, x+t), e)$ (Willemsen et al., 2023).

This distinction is foundational: the decomposed paradigm maps onto classical models such as Singular Automata with Random Clocks or Piecewise Deterministic Markov Processes with competing exponentials, while the composed paradigm underpins frameworks employing stochastic kernels, including Continuous-Time Stochastic Hybrid Automata (CTSHA) and probabilistic timed automata extensions.

2. Expressivity, Equivalence, and Limitations

The formal relationship between decomposed and composed stochastic scheduling regimes is characterized by a strict expressivity hierarchy:

• Expressivity Result: Every decomposed (race-based) hybrid automaton can be encoded as a composed (two-kernel) automaton, i.e., for every decomposed HA $D$ there exists a composed HA $C$ that is Pr-equivalent (matching all finite traces and conditional time-to-event laws).
• Incomparability: There exist composed HAs for which no Pr-equivalent decomposed HA exists. Specifically, arbitrary randomization over next-jump time and event choice (such as uniform event timing with fixed split probabilities) cannot be realized by any race of independent