Mathematical Safety Supervisor

• Mathematical safety supervisors are formally specified mechanisms that guarantee adherence to safety constraints through rigorous mathematical certification.
• They integrate control theory, optimization, formal verification, and statistical inference to maintain system invariance under uncertainty and adversarial conditions.
• Key frameworks like barrier certificates, chance constraints, and automata-based synthesis enable real-time decision-making and risk quantification.

A mathematical safety supervisor is a formally specified supervisory mechanism, applicable to both continuous and discrete systems, that guarantees adherence to safety constraints over operational trajectories by bounding (exactly or probabilistically) the risk of violating specified unsafe sets. It achieves this through rigorous mathematical analysis—often combining control theory, optimization, formal verification, and statistical inference—enabling real-time decision-making or certification in settings exhibiting uncertainty, partial observability, or adversarial inputs.

1. Formal Definition and Scope

A mathematical safety supervisor is defined as a supervisory control or monitoring architecture that provides certified guarantees, typically in the form of invariance or probabilistic bounds, that the system state avoids a prescribed unsafe set for all time or with quantifiable risk. This construct spans diverse system types: stochastic differential equations (SDEs) (Feng et al., 2020), control-affine deterministic or stochastic systems (Zhao et al., 2022, Teuwen et al., 3 Dec 2025), nonlinear systems with adversarial disturbances (Lin et al., 2023), hybrid/discrete-event systems (finite/infinite-state DES) (Cui et al., 6 Oct 2025, Zhang et al., 2017), learning-enabled and software-controlled systems (Huang et al., 2022, Hegna, 2015), and systems with partial observability or non-determinism (POMDPs) (Zhang et al., 2017).

Mathematical safety supervision can appear in the form of:

• State-feedback or output-feedback control policies designed via barrier certificates, reachability analysis, or invariance arguments.
• Online safety filters that project potentially unsafe decisions into the safety-enforcing admissible set.
• Supervisory controllers represented as (possibly minimal) automata encoding permissible decision sequences.
• Embedded risk monitors for software, quantifying empirical or modeled risk using operational test data.

2. Key Methodological Frameworks

Several families of supervisory methods exist, characterized as follows:

a. Barrier Certificate and Invariance-based Supervision

For continuous or stochastic dynamics, safety can be enforced by constructing a barrier or safety index function (polynomial or otherwise) whose sublevel sets are forward-invariant under admissible control. In the stochastic setting, exponential barrier certificates are used to bound tail probabilities of unsafe set violations, with supermartingale properties enabling exponential decay bounds in time (Feng et al., 2020). For deterministic control-affine systems, the existence of a barrier function satisfying a Lie-derivative condition guarantees the safety set’s invariance (Zhao et al., 2022).

Barrier conditions can be enforced up to the limits of actuator constraints by encoding local manifold or pointwise conditions, typically using the Positivstellensatz and sum-of-squares (SOS) programming, resulting in tractable (convex or semidefinite) optimization for polynomial systems (Zhao et al., 2022, Huang et al., 2022, Lin et al., 2023).

b. Chance-Constrained Safety and Probabilistic Filters

In stochastic or uncertain systems, safety supervisors may impose one-step or finite-horizon chance constraints, often using control barrier functions generalized to the probabilistic setting (PCBFs). The supervisor uses quantile statistics, possibly extracted from scenario-based or data-driven order statistics, to ensure that the violation probability remains below a prescribed level, even under arbitrary disturbance distributions (Teuwen et al., 3 Dec 2025). This framework captures both distribution-free and Gaussian/moment-based invariance techniques.

c. Supervisory Synthesis and Automata-Theoretic Techniques

For discrete-event systems and partially observable MDPs, safety supervision is realized as a language-theoretic or automata-based synthesis problem (Cui et al., 6 Oct 2025, Zhang et al., 2017, Zhang et al., 2017). Safety supervisors are typically constructed as DFA, za-DFA, or maximally permissive automata, whose behavior is synthesized to ensure satisfaction of properties specified in logics such as PCTL.

Modern approaches employ machine learning (e.g., $L^*$ algorithm for regular-language learning via counterexample-guided refinement) to synthesize minimal supervisors guaranteeing PCTL-bounded risk (Zhang et al., 2017). The safety game construction in prediction-based DES extends such synthesis to guarantees over sets of predicted possible future behaviors (Cui et al., 6 Oct 2025).

d. Safety Monitoring and Statistical Risk Certification

For software and embedded applications, safety supervisors can operate as runtime monitors that gather operational traces, execute random-cone-based demonstration tests, and compute upper statistical risk bounds (e.g., via compound Poisson process models) with formal confidence levels (Hegna, 2015). Supervisory logic is implemented atop real-time automaton execution, integrating operational profile monitoring and test-based indemnification to support both product and process assurance.

e. Supervision in Learning-Enabled and Hybrid Systems

Contemporary learning-based controllers require post-hoc wrapping or filtering to ensure safety due to the lack of formal guarantees in the learning process. Control barrier functions and SOS-program-based filters can be applied to enforce invariance, sometimes realizing controller compensation through optimization or policy adjustment (Huang et al., 2022). Hamilton-Jacobi reachability theory, when combined with conformal prediction and ensemble learning, can yield trajectory-level, statistically calibrated safety supervisors, restoring formal guarantees to RL-driven systems (Tabbara et al., 11 Nov 2025).

3. Mathematical Mechanisms and Certificate Construction

Mathematical safety supervisors are typically constructed by searching for certificate functions and control/filter policies subject to tractable relaxations of invariance or risk constraints. Key mathematical mechanisms include:

• Exponential or time-dependent barrier certificates, ensuring supermartingale conditions for SDEs and yielding explicit failure probability bounds (Feng et al., 2020).
• Sum-of-squares programming using the S-procedure and the Positivstellensatz to search for polynomial certificates and associated polynomial controllers enforcing forward invariance or risk conditions (Zhao et al., 2022, Lin et al., 2023).
• Quantile-based enforcement for one-step and multi-step probabilistic CBFs, translating data-driven or analytical quantiles of disturbance-induced barrier violations into sufficient safety conditions enforced via MIQP or tractable QP heuristics (Teuwen et al., 3 Dec 2025).
• Safety games over information-state spaces synthesized using greatest fixed-point iterations or expansion/pruning algorithms, as in prediction-based DES supervision or POMDP regular-language synthesis (Cui et al., 6 Oct 2025, Zhang et al., 2017).
• Statistical upper bounding of failure or risk rates using Clopper-Pearson intervals, operational profile estimation, and runtime test logging in embedded software (Hegna, 2015).

4. Realization in Practical Supervisory Architectures

The formal supervisory logic is embedded into runtime architectures as either online safety filters, certificate-verifying modules, backup control supervisors, or self-contained automata. Representative implementations include:

• MPC-based safety supervisors that check the feasibility of proposed inputs over a finite or infinite time horizon, substituting backup controls in the event of impending infeasibility, with robustness to model mismatch achieved via constraint tightening (Nezami et al., 2022, Nezami et al., 2022).
• Online barrier- or QP-based safety filters that minimally modify nominal or RL-generated actions to maintain state invariance, using controller blending or compensation strategies (Huang et al., 2022).
• Automata supervisors synthesized as DFA or za-DFA, applied to discrete/hybrid systems, with the supervisor logic enforcing only admissible actions compatible with the safety certificate (Zhang et al., 2017, Zhang et al., 2017).
• Statistical runtime supervision in software, integrating operational monitoring, periodic random testing, and upper risk bound computation tied directly to shutdown or intervention logic (Hegna, 2015).

The following table organizes major mathematical safety supervisor frameworks:

System Class	Certification Technique	Computational Tooling
SDEs	Exponential/time-dependent barrier cert.	SOS/SDP, Doob's inequality
Deterministic affine	Barrier index, Lie-derivative test	SOS, Positivstellensatz, QP
Stochastic/discrete	Chance constraint, quantile bounds	MIQP/QP, scenario sampling
DES/POMDP	Supervisor automata, safety games	$L^*$, DFA construction, fixed-point
Software/embedded	Statistical demonstration, CPP risk	Random-cone test, Clopper-Pearson

5. Guarantee Types and Theoretical Properties

Supervisors are constructed to achieve one or more of the following guarantees:

• Invariance: System state is kept in a safe set for all time, robust to control and disturbance constraints.
• Probabilistic risk bounds: The probability of unsafe set violation is bounded explicitly, possibly with exponential tail decay (Feng et al., 2020, Teuwen et al., 3 Dec 2025).
• Finite-time or time-to-failure extension: If infinite-horizon safety is infeasible, the supervisor maximizes the time-to-violation, producing soft-landing or graceful degradation (Li et al., 2023).
• Maximal permissiveness: For discrete-event and automata-theoretic supervision, the synthesized automaton admits all and only the behaviors that are compatible with the safety specification, i.e., is maximally permissive (Cui et al., 6 Oct 2025, Zhang et al., 2017).
• Statistical confidence: Supervisors can return risk (rate or cumulative loss) bounds at user-specified statistical confidence, updating these metrics in real time (Hegna, 2015).

6. Empirical and Illustrative Case Studies

Numerous empirical studies validate the practical effectiveness of mathematical safety supervisors:

• SDE population risk bounds: Explicit probability upper bounds for unsafe excursions, computed via SDP/SOS in real time (Feng et al., 2020).
• RL agent safety: Maintaining invariance in inverted pendulum experiments through barrier-based online compensation (Huang et al., 2022).
• Robotic manipulation and vehicle navigation: SOS-synthesized safety indices and controllers deployed on high-DOF manipulators and vehicular systems, ensuring non-collision across thousands of random initializations (Zhao et al., 2022, Teuwen et al., 3 Dec 2025).
• Highway/road traffic management: MPC- and tube-MPC-supervised maneuvering, robust to disturbance and road geometry uncertainty, with empirical success rates approaching 100% (Nezami et al., 2022, Nezami et al., 2022).
• Software risk: Embedded risk monitors in safety-critical code providing on-the-fly statistical upper bounds for failure rates and acting on risk threshold exceedances (Hegna, 2015).

7. Advanced Topics and Future Directions

Recent research explores integrating formal safety supervision with learning-based controllers using conformal prediction and ensembles for sharply calibrated guarantees (Tabbara et al., 11 Nov 2025), synthesizing supervisors in partially observable, adversarial, or information-constrained settings (Cui et al., 6 Oct 2025, Zhang et al., 2017), and unifying protection and extension into a single, convex optimization-based supervisor (Li et al., 2023).

There is increasing emphasis on scaling computational approaches (e.g., exploiting sparsity in SOS programs, heuristic MIQP decompositions for high-dimensional risk filters) and on developing interpretable and adaptive supervisor structures to handle model uncertainty, unknown disturbances, and human-in-the-loop decision-making.

Mathematical safety supervision constitutes a foundational paradigm for trustworthy autonomous, embedded, and cyber-physical systems, marrying formal guarantees with tractable and practical certification architectures (Feng et al., 2020, Teuwen et al., 3 Dec 2025, Zhao et al., 2022, Cui et al., 6 Oct 2025, Hegna, 2015).