Reach-Avoid Probability Certificates (RAPCs)

• RAPCs are certificates that provide quantitative guarantees for systems to reach target sets while avoiding unsafe sets with a certified probability bound.
• They encompass formulations such as supermartingale, Bellman fixed-point, and Farkas certificates, ensuring soundness through rigorous proof techniques and duality principles.
• RAPCs enable both verification and controller synthesis across various model classes while balancing conservatism with practical certification needs.

Searching arXiv for relevant RAPC-related papers and terminology to ground the article in current literature. Reach-Avoid Probability Certificates (RAPCs) are certificate objects that establish quantitative guarantees for reach-avoid specifications: a system reaches a target set while avoiding an unsafe set, typically with a certified probability lower bound or upper bound. Across the literature, RAPCs appear in multiple mathematical forms rather than under a single canonical name. In stochastic nonlinear control, a prominent instance is the reach-avoid supermartingale (RASM), introduced as a formal infinite-horizon certificate for discrete-time stochastic systems (Žikelić et al., 2022). In finite Markov decision processes (MDPs), fixed-point and Farkas-style certificates play the same role after the standard reduction from reach-avoid to reachability (Chatterjee et al., 20 Jan 2025, Funke et al., 2019). More recent work broadens the landscape to robust dynamic-programming value certificates on abstractions (Schmid et al., 22 Jan 2026), observation-aware runtime certificates (Feng et al., 12 Nov 2025), and explicit time-varying and time-invariant reach-avoid certificates for continuous-space stochastic systems (Mazouz et al., 26 Mar 2026). Taken together, these works define RAPCs as a family of sound, generally conservative, proof artifacts for certifying probabilistic target-reaching under safety constraints.

1. Conceptual scope and formal semantics

A reach-avoid specification distinguishes an initial set, a target set, an unsafe set, and, in probabilistic formulations, a threshold $p$. In the discrete-time stochastic-control setting studied in "Learning Control Policies for Stochastic Systems with Reach-avoid Guarantees" (Žikelić et al., 2022), the objective is to ensure that for every initial condition $x_0\in X_0$,

$$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$

where success means that there exists a time at which the state enters the target set while all preceding states remain outside the unsafe set. This is an infinite-horizon property (Žikelić et al., 2022).

In finite MDPs, reach-avoid is usually reduced to ordinary reachability by making avoid states absorbing non-target states. That reduction is explicit in "Fixed Point Certificates for Reachability and Expected Rewards in MDPs" (Chatterjee et al., 20 Jan 2025), where the Bellman operator for reach-avoid becomes

$$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$

Under this reduction, RAPCs certify either maximal or minimal reach-avoid probabilities, and can provide lower bounds, upper bounds, or exact values when matching certificates are available (Chatterjee et al., 20 Jan 2025).

The same semantics recur in other model classes. For continuous-time linear stochastic systems with hierarchical control architectures, the certified quantity is a lower bound on the finite-horizon probability of reaching $\mathcal T$ before leaving $\mathcal S$, while also satisfying hard constraints $\mathcal G$ (Schmid et al., 22 Jan 2026). For runtime-observed stochastic systems, the certified quantity becomes conditional: $$\mathbb P_{x_0}(\mathrm{RA}(U,T)\mid X_{t_i}\in O_i,\forall i\le k)\ge 1-\frac pq,$$ with $p$ and $q$ certified separately by paired barrier functions (Feng et al., 12 Nov 2025).

This suggests that RAPC is best understood as a semantic category rather than a single formalism. A plausible implication is that the defining feature is not the algebraic form of the certificate but its certified statement: a sound bound on the probability of satisfying a reach-before-avoid property.

2. Principal certificate forms

The literature supports several distinct RAPC realizations.

The supermartingale-style formulation is represented by the RASM of (Žikelić et al., 2022). A continuous function $x_0\in X_0$0 is a reach-avoid supermartingale if it satisfies nonnegativity, an initial bound, an unsafe-set threshold, and a strict expected decrease condition outside the target: $x_0\in X_0$1 and for states outside the target with $x_0\in X_0$2,

$x_0\in X_0$3

This certifies

$x_0\in X_0$4

for all $x_0\in X_0$5 (Žikelić et al., 2022).

The Bellman/fixed-point formulation appears in finite MDPs (Chatterjee et al., 20 Jan 2025). There, RAPCs are vectors $x_0\in X_0$6, sometimes accompanied by ranking functions $x_0\in X_0$7, satisfying Bellman inequalities and side conditions that eliminate spurious greatest fixed points. Upper bounds arise from inductive vectors $x_0\in X_0$8; lower bounds arise from coinductive vectors $x_0\in X_0$9 plus ranking witnesses (Chatterjee et al., 20 Jan 2025).

A dual linear-algebraic variant is given by Farkas certificates (Funke et al., 2019). For finite-state MDPs, lower and upper bounds on minimal and maximal reachability probabilities are certified by vectors $$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$0 over states or $$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$1 over state-action pairs. After the standard reach-avoid reduction, these become reach-avoid certificates. For example, a lower bound on maximal reach-avoid probability corresponds to a $$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$2-certificate satisfying

$$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$3

while a lower bound on minimal reach-avoid probability corresponds to a $$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$4-certificate with

$$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$5

(Funke et al., 2019).

A value-certificate formulation over abstractions is developed in "Maximizing Reach-Avoid Probabilities for Linear Stochastic Systems via Control Architectures" (Schmid et al., 22 Jan 2026). There the main certificate object is a robust abstract value function $$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$6, especially $$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$7, obtained by robust dynamic programming on a gridded abstraction of the closed-loop plant-plus-MPC system. Its guarantee is

$$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$8

so the abstract value is a certified lower bound on the actual achieved reach-avoid probability under the synthesized hierarchical controller (Schmid et al., 22 Jan 2026).

A runtime-conditioned formulation is given by observation-aware barrier functions (Feng et al., 12 Nov 2025). The observation-aware reach-avoid barrier function (ORBF) upper-bounds the joint probability of the bad event $$\mathbb{P}_{x_0}\big[\mathrm{ReachAvoid}(X_t,X_u)\big]\ge p,$$9 and the observation sequence, while the observation-aware barrier function (OBF) lower-bounds the observation likelihood. Their composition yields a conditional RAPC (Feng et al., 12 Nov 2025).

Finally, "Time-Varying Reach-Avoid Control Certificates for Stochastic Systems" (Mazouz et al., 26 Mar 2026) introduces explicit time-varying and time-invariant reach-avoid certificates for discrete-time continuous-space stochastic systems. These are value-function-like lower-bounding certificates with Bellman-relaxed inequalities, producing lower bounds for both finite- and infinite-horizon reach-avoid probabilities (Mazouz et al., 26 Mar 2026).

3. Soundness mechanisms and proof patterns

Despite their formal diversity, RAPCs share recurring proof structures.

In supermartingale-based certificates, soundness is obtained by converting the certificate into a nonnegative supermartingale or stopped supermartingale and then applying martingale inequalities. In the RASM framework, a stopped process is constructed from $$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$0, shown to be a nonnegative supermartingale, and combined with a maximal inequality

$$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$1

to derive the lower bound on reach-avoid success probability (Žikelić et al., 2022).

In Bellman/fixed-point certificates, soundness follows from monotonicity and least-fixed-point theory. "Fixed Point Certificates for Reachability and Expected Rewards in MDPs" (Chatterjee et al., 20 Jan 2025) bases its method on monotone Bellman operators over $$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$2, using Knaster–Tarski arguments for upper bounds and ranking-function side conditions for lower bounds. A central issue is that $$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$3 only yields a greatest-fixed-point lower bound unless additional qualitative information excludes spurious fixed points (Chatterjee et al., 20 Jan 2025).

In Farkas-style certificates, soundness is a consequence of LP duality and an MDP-specific use of Farkas’ Lemma (Funke et al., 2019). The infeasibility of the negated threshold condition is converted into a short dual witness. The resulting certificate polytopes also have structural meaning: sparse certificates correspond to witnessing subsystems (Funke et al., 2019).

In abstraction-based value certificates, the proof principle is simulation-style coupling. The robust abstract value function $$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$4 of (Schmid et al., 22 Jan 2026) lower-bounds the concrete closed-loop value because center trajectories and concrete cell trajectories remain within a radius-$$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$5 neighborhood under common inputs and disturbances, while robust Bellman backups take worst-case continuation values over that neighborhood (Schmid et al., 22 Jan 2026).

In observation-aware certificates, soundness is obtained by constructing a submartingale for the denominator and a supermartingale up to a stopping time for the numerator, then applying Bayes’ rule. This yields the bound $$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$6 on conditional safety or conditional reach-avoid probability (Feng et al., 12 Nov 2025).

A plausible implication is that RAPCs can be classified by their proof principle: supermartingale, Bellman fixed point, dual linear witness, robust abstraction coupling, or conditional barrier decomposition.

4. Synthesis and verification workflows

RAPCs are used both for verification of fixed policies and for controller synthesis.

In neural supermartingale synthesis for stochastic nonlinear systems, the method of (Žikelić et al., 2022) jointly trains a policy network $$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$7 and a certificate network $$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$8. The learner-verifier loop alternates between optimizing loss terms corresponding to initial, unsafe, and decrease conditions, and formally checking strengthened conditions over the full state space using discretization and interval analysis. If verification fails, counterexamples are added and training repeats (Žikelić et al., 2022). The same framework supports three uses: learn a policy from scratch, verify a fixed policy, or fine-tune a pre-trained policy (Žikelić et al., 2022).

In time-varying and time-invariant reach-avoid certificates (Mazouz et al., 26 Mar 2026), verification of a fixed controller reduces to SOS optimization over a polynomial certificate $$(B^{opt}_{T,A}(x))(s)= \begin{cases} 1 & s\in T,\ 0 & s\in A,\ \underset{a\in Act(s)}{opt}\sum_{s'}P(s,a,s')x(s') & s\in S\setminus(T\cup A). \end{cases}$$9. For a time-varying bounded-horizon certificate, the certified lower bound is

$\mathcal T$0

subject to Bellman-relaxed inequalities on target, unsafe, safe, and initial sets (Mazouz et al., 26 Mar 2026). Joint synthesis optimizes both $\mathcal T$1 and the controller to maximize the certificate-derived lower bound (Mazouz et al., 26 Mar 2026).

In robust abstraction-based planning (Schmid et al., 22 Jan 2026), the workflow is hierarchical. Offline, one chooses discretizations, constructs tightened sets and a robust terminal set, partitions the planning state space, estimates abstract transition probabilities, and runs robust value iteration. Online, the current cell is located, the abstract policy chooses a command, and an inner robust MPC tracks the corresponding reference (Schmid et al., 22 Jan 2026).

In finite MDP model checking, certificate generation can be layered atop standard exact or approximate algorithms. Exact policy iteration, rational search, linear programming, and interval iteration can be augmented to emit fixed-point certificates with modest overhead (Chatterjee et al., 20 Jan 2025). Storm has been extended to produce such certificates, and an Isabelle/HOL-verified Standard ML checker validates them using exact rational arithmetic (Chatterjee et al., 20 Jan 2025).

Observation-aware runtime certificates (Feng et al., 12 Nov 2025) decompose synthesis into offline and online phases. A time-invariant tail barrier $\mathcal T$2 is synthesized offline by SOS/SDP, while OBF and ORBF or OSBF are updated online through backward iteration as new observations arrive (Feng et al., 12 Nov 2025). This gives a runtime-refinable certificate architecture rather than a static one.

5. Relation to adjacent certificate families

RAPCs overlap with but are not identical to several established certificate notions.

Relative to Lyapunov functions, RASMs replace deterministic decrease by expected decrease. The sublevel sets of a Lyapunov function are deterministic invariants; the level-set semantics of a RASM are probabilistic and derive from supermartingale tail bounds (Žikelić et al., 2022).

Relative to barrier certificates, RAPCs add target reachability to avoidance. A safety barrier may certify non-entry into an unsafe set, whereas a reach-avoid certificate must also encode progress toward a target. This is explicit in both the RASM formulation (Žikelić et al., 2022) and the time-varying/time-invariant reach-avoid certificates of (Mazouz et al., 26 Mar 2026).

Relative to dynamic-programming value functions, some RAPCs are exact or approximate value functions rather than auxiliary barriers. The robust abstract value function $\mathcal T$3 in (Schmid et al., 22 Jan 2026) and the exact Bellman value functions $\mathcal T$4 for distributionally robust reach-avoid in (Chen et al., 6 Jan 2025) fall in this category. By contrast, RASMs and ORBFs are auxiliary certificate functions whose soundness is indirect (Žikelić et al., 2022, Feng et al., 12 Nov 2025).

Relative to $\mathcal T$5-regular verification, reach-avoid can be embedded into richer automata-based certificate frameworks. "Supermartingale Certificates for Quantitative Omega-regular Verification and Control" (Henzinger et al., 24 May 2025) introduces limit-deterministic Büchi supermartingales (LDBSMs), which explicitly generalize prior supermartingale certificates for quantitative reachability, safety, and reach-avoid. Reach-avoid formulas such as $\mathcal T$6 are handled as special $\mathcal T$7-regular cases via product constructions with a limit-deterministic Büchi automaton (Henzinger et al., 24 May 2025).

Relative to controlled reach-avoid sets in deterministic systems, some works provide structural analogues rather than probabilistic certificates. The controlled reach-avoid set computation of (Wu et al., 7 Jun 2025) uses a probabilistic perspective to compute deterministic controlled reach-avoid sets via 0-threshold reach-avoid sets, thereby yielding a qualitative RAPC-style positivity certificate. This suggests a continuum between existential deterministic reachability certificates and quantitative probabilistic RAPCs.

6. Model classes, guarantees, and limitations

The surveyed RAPC literature spans several model classes.

For discrete-time nonlinear stochastic control systems, (Žikelić et al., 2022) gives infinite-horizon lower-bound certificates via RASMs, with neural synthesis and interval-based verification. Its assumptions include compact state space, Lipschitz continuity of the dynamics, disjoint Borel target and unsafe sets, and distributional conditions for expectation verification. The method covers $\mathcal T$8, and $\mathcal T$9 only when the unsafe set is empty; it does not handle $\mathcal S$0 with nonempty unsafe set (Žikelić et al., 2022).

For finite MDPs, (Chatterjee et al., 20 Jan 2025) and (Funke et al., 2019) provide exact certificate frameworks after reach-avoid reduction. These are state-explicit and confined to finite models, but they admit formally verified checking and sparse witness extraction (Chatterjee et al., 20 Jan 2025, Funke et al., 2019).

For continuous-time linear stochastic systems with continuous state space, (Schmid et al., 22 Jan 2026) certifies lower bounds through robust abstraction and hierarchical control. The guarantee is finite-horizon, and the abstract-value lower bound depends on accurate transition-kernel estimation. The paper notes that the formal lower-bound theorem applies to the exact abstract kernel, not to a finite-sample estimate, so statistical uncertainty in transition estimation is not certified (Schmid et al., 22 Jan 2026).

For distributionally robust systems under Wasserstein ambiguity, (Chen et al., 6 Jan 2025) provides exact finite-horizon Bellman value functions for reach-avoid and explicit control barrier certificates only for safety, not reach-avoid. This is relevant because it highlights a limitation: explicit barrier-style distributionally robust RAPCs for reach-avoid remain open in that framework (Chen et al., 6 Jan 2025).

For runtime-observed stochastic systems, (Feng et al., 12 Nov 2025) gives conditional lower bounds on reach-avoid probability under a fixed policy and known model. The reach-avoid theorem additionally assumes almost-sure eventual entry into $\mathcal S$1 (Feng et al., 12 Nov 2025).

For polynomial stochastic differential equations, (Xue et al., 2022) provides open-horizon reach-avoid analysis by constructing a bounded value function $\mathcal S$2 whose strict $\mathcal S$3-superlevel set equals the $\mathcal S$4-reach-avoid set in the exact theory, and a polynomial under-approximation $\mathcal S$5 whose superlevel set yields a certified inner approximation (Xue et al., 2022). This is an infinite-horizon continuous-time RAPC in value-function form, but it depends on polynomial dynamics, semialgebraic sets, and SOS tractability (Xue et al., 2022).

Common limitations recur. RAPCs are typically sufficient rather than complete. Neural and SOS verifiers are conservative. Scalability deteriorates with dimension, polynomial degree, or network size. Many certificates are model-relative: for Bayesian neural-network dynamics, the certified probability is sound relative to the learned posterior model rather than the unknown environment (Wicker et al., 2023). A plausible implication is that “certificate soundness” and “model adequacy” should be distinguished sharply in any encyclopedic treatment of RAPCs.

7. Empirical patterns and current directions

The reported empirical record indicates several trends.

Neural supermartingale certificates substantially improve certified reach-avoid thresholds over naive reachability-first extensions. On three stochastic nonlinear tasks, the RASM method verified larger reach-avoid probabilities than a naive extension of ranking supermartingales, including 93.3% versus 83.4% on a 2D system and 92.1% versus 47.9% on a stochastic inverted pendulum (Žikelić et al., 2022).

For Bayesian neural-network dynamics, backward-recursive lower-bound certificates significantly expand the number of certifiable states and the average guaranteed reach-avoid probability relative to purely learned policies. On the Zigzag benchmark, synthesized policies achieved an average lower bound of 0.710 with coverage 0.910, versus 0.189 and 0.193 for learned policies (Wicker et al., 2023).

Time-varying reach-avoid certificates outperform time-invariant ones in several continuous-state stochastic benchmarks, often achieving strong lower bounds at much lower polynomial degree, though with higher runtime (Mazouz et al., 26 Mar 2026). This suggests that certificate expressivity in horizon index can be as important as function-class richness.

Observation-aware runtime certificates show that conditioning on observations can both increase and decrease certified reach-avoid probability, rather than monotonically improving it. For the $\mathcal S$6 reach-avoid benchmark, the lower bound decreased from 0.357 offline to 0.087 as more observations were incorporated, while in other systems observations increased the bound (Feng et al., 12 Nov 2025). This demonstrates that runtime RAPCs are genuinely informational rather than merely optimistic.

Robust abstraction-based value certificates for a 12D quadcopter model show strong conservatism in cluttered environments with narrow passages. In several scenarios, the certified lower bound $\mathcal S$7 was far below empirical Monte Carlo success, e.g. $\mathcal S$8 versus $\mathcal S$9 in Zigzag (Schmid et al., 22 Jan 2026). This supports the general observation that correctness-preserving set tightenings and worst-case neighborhood backups can be very conservative.

Current directions therefore include richer temporal objectives via automata products (Henzinger et al., 24 May 2025), runtime and conditional certification (Feng et al., 12 Nov 2025), distributional robustness (Chen et al., 6 Jan 2025), verified certificate checking for finite models (Chatterjee et al., 20 Jan 2025), and explicit continuous-state finite- and infinite-horizon reach-avoid certificates (Mazouz et al., 26 Mar 2026). This suggests that RAPCs are evolving from a narrow supermartingale or barrier notion into a broader family of certificate-based probabilistic proof systems spanning control synthesis, verification, abstraction, runtime monitoring, and formal trust architectures.