---
title: Generalised Streett Supermartingales
url: https://www.emergentmind.com/topics/generalised-streett-supermartingales-gssms
type: topic
---

# Generalised Streett Supermartingales

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 [2512.00270][2405.17304].

## 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 [2512.00270][2405.17304].

## 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 [2512.00270].

## 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) [2405.17304].

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 [2512.00270].

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 [2512.00270].
- **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 [2512.00270][2405.17304].

Source: https://www.emergentmind.com/topics/generalised-streett-supermartingales-gssms