Generalised Streett Supermartingales

• GSSMs are defined as measurable functions satisfying drift conditions via least fixed-point characterization, ensuring finite expected A-visits before reaching B.
• They generalize classical Streett supermartingales by omitting constraints on set B, thereby certifying a broader range of ω-regular properties.
• They support efficient template-based synthesis approaches using LP, SDP, and SMT solvers for verifying almost-sure satisfaction in stochastic systems.

Generalised Streett Supermartingales (GSSMs) provide an order-theoretic and algorithmically tractable class of certificates for the verification and synthesis of almost sure satisfaction of $\omega$-regular properties, especially Streett objectives, for general discrete-time stochastic systems and Markov chains. GSSMs characterize positive recurrence with respect to Streett conditions via least fixed-point constructions, yielding strict generalizations of prior supermartingale-based certificates and enabling the verification of a broader class of $\omega$-regular properties, including those not certifiable by standard Streett supermartingales. Their main operational significance lies in offering sound and complete certificates for positive recurrence, exhibiting robustness under template-based synthesis, and admitting efficient semidefinite and SMT-based constraint solving methods for practical instances (Kura et al., 29 Nov 2025, Abate et al., 2024).

1. Formal Definition and Fixed-Point Characterisation

Let $(S, \Sigma)$ be a measurable state space, $F: S \rightarrow G S$ a Markov kernel (or a discrete-time stochastic process with transition operator $X$), and $(A, B)\subseteq S \times S$ a measurable Streett pair. The foundational operator is

$K_{\mathbb{E}} : \Meas(S, [0, \infty]) \to \Meas(S, [0, \infty])$

$$(K_{\mathbb{E}}\eta)(x) = \begin{cases} (X\eta)(x) + 1, & x \in A \setminus B \ 0, & x \in B \ (X\eta)(x), & x \notin A \cup B \end{cases}$$

The least fixed point $$\mu K_{\mathbb{E}} = \sup_{n\in\mathbb{N}} K_{\mathbb{E}}^n(0)$$ coincides with $x \mapsto \mathbb{E}_x[\#\{\text{$A$-visits before hitting$B$}\}]$.

Definition (GSSM): A measurable function $r : S \rightarrow [0, \infty)$ is a Generalised Streett Supermartingale for $(F, A, B)$ if

$$r \ge K_{\mathbb{E}} r \quad\text{and}\quad r(x)<\infty\ \forall x,$$

that is,

$\forall x\in A\setminus B : (Xr)(x) \le r(x) - 1,$

$\forall x \notin A \cup B : (Xr)(x) \le r(x).$

No condition is imposed for $x \in B$.

Fixed-Point Characterisation: The least fixed point of $K_{\mathbb{E}}$ characterizes the expected number of $A$-visits before reaching $B$: $$\mu K_{\mathbb{E}}(x) = \mathbb{E}_x[\mathrm{steps}^{(A,B)}].$$ Thus, a GSSM exists if and only if this expectation is finite, and any GSSM $r$ satisfies $r \ge \mu K_{\mathbb{E}}$.

2. $\omega$-Regular Properties and Streett Objectives

$\omega$-regular Streett conditions can be expressed as a finite set of pairs $(G_i, R_i)$. A trajectory satisfies $(G_i, R_i)$ if

$$\sum_{t=0}^{\infty} {\bf 1}_{G_i}(\tau_t) < \infty \quad \lor \quad \sum_{t=0}^{\infty} {\bf 1}_{R_i}(\tau_t) = \infty,$$

and the full Streett condition holds if this is met for every $i$.

GSSMs generalize all classical supermartingale-based principles:

• Safety: Setting $(G, R) = (B, \emptyset)$, GSSM drift conditions force only finitely many visits to $B$.
• Reachability: For $(S\setminus G, G)$, negative drift outside $G$ ensures eventual reachability.
• Persistence and Recurrence: Appropriate choices $(S, P)$ and $(\emptyset, R)$ yield certificates for almost sure persistence and recurrence.

Any trajectory in the product of a Markov chain and a deterministic Streett automaton satisfies the almost-sure acceptance condition $\mathbf{Streett}(A,B)$ precisely when the underlying chain is null-recurrent for $(A,B)$:

$$\forall x : \Pr_x[\mathrm{steps}^{(A,B)}<\infty]=1.$$

The existence of a GSSM implies almost-sure satisfaction (Kura et al., 29 Nov 2025, Abate et al., 2024).

3. Soundness, Completeness, and Expressiveness

The soundness and completeness theorem for GSSMs asserts:

• There exists a GSSM $r$ if and only if the Markov chain $F$ is positively $(A,B)$-recurrent, i.e., $\mathbb{E}_x[\mathrm{steps}^{(A,B)}]<\infty$ for all $x$.
• If such $r$ exists, $\Pr_x[\mathbf{Streett}(A,B)] = 1$ for all $x$.
• If the positive recurrence criterion fails (i.e., for some $x$ the expectation diverges), then no GSSM exists.

GSSMs strictly generalize standard Streett supermartingales (SSMs). The essential distinctions are:

• SSMs enforce a global bounded increase in $r$ on $B$ with a uniform constant $M$, whereas GSSMs impose no constraint on $B$.
• GSSMs can yield certificates even in instances where SSM constraints are infeasible due to unbounded behavior at $B$.

A canonical counter-example demonstrates $r(x) = x$ is a GSSM for a Markov chain on $S = \mathbb{N}$ with $(A, B) = (\mathbb{N}\setminus\{0\}, \{0\})$ and $F$ enabling arbitrary large jumps out of $0$; no SSM exists for this system (Kura et al., 29 Nov 2025).

4. Synthesis and Algorithmic Aspects

Template-based synthesis is the primary algorithmic paradigm:

• Choose a functional template $r(x) = \sum_j c_j \phi_j(x)$ (linear or polynomial basis).
• Substitute the template into the GSSM drift inequalities:

$$\int r(y) F(x,dy) \le r(x) - 1, \quad x\in A\setminus B$$

$\int r(y) F(x,dy) \le r(x), \quad x \notin A\cup B$

• Reduce the problem to (depending on template choice):
  • Linear programming (LP) for linear templates,
  • Sum-of-squares or Positivstellensatz-based semidefinite programming (SDP) for polynomial templates.

If available, inductive invariants ("shields") can simplify constraints: with linear pieces, Farkas’ Lemma reduces universally quantified conditions to existential arithmetic constraints, typically suitable for SMT solvers (e.g., Z3) (Abate et al., 2024).

A fundamental completeness property is template-relative: if a GSSM of the chosen template exists, the synthesis algorithm is guaranteed to find it.

5. Experimental and Practical Considerations

Prototype tools implementing the above synthesis approach (focusing on the more expressive Lexicographic Progress-Measure Supermartingales, which subsume GSSMs) can efficiently certify positivity recurrence for a range of benchmark models:

• All benchmarks admitting a GSSM are handled straightforwardly by the tool using linear-template constraint solving.
• Simple stochastic recurrence examples, including models not certifiable via SSMs, can be decided in under one second.
• No known standard SSM-based tool succeeds on these instances (Kura et al., 29 Nov 2025).

A key modeling requirement for effective synthesis is the existence of a symbolic closed-form for the post-expectation operator $\Post f$—preferably polynomial or piecewise-linear—so that the drift constraint is expressible in first-order logic or transferable to LP/SDP solvers. For systems where the disturbance distribution does not permit a closed-form expectation, conservative approximations may be used.

6. Generalisations, Limitations, and Future Directions

GSSMs are a unifying abstraction for qualitative (almost-sure) verification of a spectrum of $\omega$-regular properties, encompassing classical rules for reachability, safety, persistence, recurrence, and their combinations.

Notable generalizations and future avenues include:

• Quantitative Analysis: Strengthening the martingale convergence arguments underlying GSSMs via concentration inequalities for probability bounds beyond almost-sure satisfaction.
• Relaxation Techniques: Integration with Positivstellensatz-based sum-of-squares relaxations for template synthesis, enabling SDP approaches.
• Learning-Augmented Synthesis: Use of neural-network-based templates for $f_i$ with subsequent verification via SMT or SDP solvers.
• Omega-regular Hierarchy: Within the supermartingale hierarchy, Distribution-Valued Streett Supermartingales (DVSSMs) provide completeness for null recurrence and handle the full spectrum of almost-sure $\omega$-regular objectives (Kura et al., 29 Nov 2025).
• Expressiveness Boundaries: The main expressiveness limitation is the requirement for positive $(A,B)$-recurrence; null recurrence necessitates more powerful certificates (e.g., DVSSMs).

GSSMs provide a formally validated and computationally practical method for certifying positive recurrence and verifying almost-sure $\omega$-regular properties in discrete-time stochastic models, with ongoing research exploring their theoretical boundaries and integration with scalable synthesis paradigms (Kura et al., 29 Nov 2025, Abate et al., 2024).