Distribution-Valued Streett Supermartingales (DVSSMs)

• DVSSMs are distribution-valued extensions of Streett supermartingales that provide complete certificates for verifying all ω-regular properties.
• They generalize deterministic progress measures by substituting next-state values with expectations and enforcing truncated lexicographic decrease.
• Synthesis techniques for LexPMSMs utilize polynomial templates and quantifier elimination, demonstrating practical performance on benchmark probabilistic models.

A progress-measure supermartingale (PMSM) is a vector-valued function on the state space of a Markov process that certifies almost sure satisfaction of $\omega$-regular properties—specifically, parity and Streett acceptance conditions—by generalizing classical deterministic progress measures and enforcing appropriate decrease conditions in expectation. PMSMs and their lexicographic extension (LexPMSM) serve as sound and (relatively) complete certificates for verifying $\omega$-regular properties in probabilistic (possibly infinite-state) discrete-time models. In the hierarchy of supermartingale-based certificates, LexPMSMs strictly subsume existing Streett supermartingales and match the power of lexicographic generalized Streett supermartingales (LexGSSMs) (Kura et al., 29 Nov 2025, Abate et al., 2024).

1. Formal Definition of Progress-Measure Supermartingales

Let $(S, \Sigma)$ be a measurable state space, $F:S \to G S$ the Markov transition kernel, and $$X : \operatorname{Meas}(S, [0, \infty)) \to \operatorname{Meas}(S, [0, \infty))$$ the next-time (post-expectation) operator, defined as

$X r(x) = \int r(y)\,\mathrm{d}F(x)(y)$

for measurable $r : S \to [0, \infty)$.

Consider a parity condition encoded as $p : S \to \{1, \ldots, d\}$, assigning to each state a priority. A $d$-dimensional, nonnegative, vector-valued measurable map $r : S \to [0, \infty)^d$ serves as the ranking function.

For $u, u' \in [0, \infty)^d$ and $1 \leq i \leq d$, define lexicographic relations: $$u \succeq_i u' \iff (u_1, \ldots, u_i) \geq_{\text{lex}} (u'_1, \ldots, u'_i), \qquad u \succ_i u' \iff (u_1, \ldots, u_i) >_{\text{lex}} (u'_1, \ldots, u'_i)$$ where $\geq_{\text{lex}}$ and $>_{\text{lex}}$ denote the standard non-strict and strict lexicographic orders.

A measurable map $r : S \to [0, \infty)^d$ is a progress-measure supermartingale (PMSM) for $(F, p)$ if, for every $x \in S$,

$$\begin{cases} p(x)\text{ even} &\implies r(x) \succeq_{p(x)} Xr(x)\ p(x)\text{ odd} &\implies r(x) \succ_{p(x)} Xr(x) \end{cases}$$

A lexicographic PMSM (LexPMSM) is the lexicographic extension using vectors-of-vectors in $$[0,\infty)^{d_1} \times \cdots \times [0,\infty)^{d_k}$$ to achieve further expressiveness (Kura et al., 29 Nov 2025).

2. Probabilistic Extension of Deterministic Progress Measures

Deterministic parity progress measures (e.g., Jurdziński 2000) assign a natural-number vector to each vertex of a parity graph, enforcing

$$\begin{cases} p(v)\text{ even}: & r(v) \succeq_{p(v)} r(w)\ p(v)\text{ odd}: & r(v) \succ_{p(v)} r(w) \end{cases}$$

for each edge $(v, w)$. PMSMs extend this to Markov processes by replacing the next-state map $r(w)$ with its expectation $X r(v)$: $$\begin{cases} p(v)\text{ even}: & r(v) \succeq_{p(v)} Xr(v)\ p(v)\text{ odd}: & r(v) \succ_{p(v)} Xr(v) \end{cases}$$ This replacement establishes a rigorous bridge from deterministic progress measure theory to probabilistic verification, where ranking decreases are enforced in expectation and stratified by truncated lexicographic orderings.

3. Soundness: Almost-Sure Satisfaction of $\omega$-Regular Properties

Let $F$ be a Markov chain on $S$ with parity map $p$, and suppose there exists a PMSM $r : S \to [0, \infty)^d$. Then, for any initial state $x_0$,

$$\mathbb{P}_{x_0}[\text{the trace satisfies the parity condition associated with }p] = 1$$

This is established via a stopping-time argument: any infinite trace that violates the parity objective must induce an almost-sure lexicographic descent in $r$, which is impossible on the nonnegative orthant. Analogous reasoning applies to Streett supermartingales (SSMs), using the Robbins–Siegmund theorem: for each component, the drift inequality ensures the accepting condition is satisfied almost surely (Kura et al., 29 Nov 2025, Abate et al., 2024).

4. Hierarchy and Comparison with Other Supermartingale Certificates

The following strict hierarchy of certificate classes for almost-sure $\omega$-regular verification holds (Kura et al., 29 Nov 2025):

Certificate Type	Expressiveness	Notable Feature
Streett-SM (SSM)	Strict subset	Classical, cannot handle some cases
GSSM	Strictly larger	Captures positive recurrence
LexGSSM, LexPMSM	Equal/strictly larger	Handles null-recurring patterns
DVSSM	Most powerful	Complete for all $\omega$-regulars

Streett supermartingales handle Streett objectives with both decrease and bounded-expectation requirements, but cannot certify certain processes. GSSMs generalize SSMs by dropping aggressive bounds on "good" sets, and fully capture positive recurrence. LexGSSMs leverage a vector of GSSMs, overcoming limitations of GSSMs for some null-recurrent cases; LexPMSMs match their strength for parity conditions. DVSSMs, the distribution-valued class, can, in theory, capture all verifiable $\omega$-regular properties but currently lack practical synthesis techniques (Kura et al., 29 Nov 2025).

5. Synthesis Algorithms for LexPMSMs

The synthesis of LexPMSMs employs constraint-solving over polynomial templates. Given a probabilistic control-flow graph (pCFG) with locations $L$ and continuous variables, define priority regions $P_{l, i}$ for each $i=1, \ldots, d$. The synthesis algorithm constructs, for each relevant region and dimension, a template ranking map using polynomial expressions, enforcing hard constraints (nonnegative and non-increasing in expectation) and soft constraints (strict decrease when required by parity).

The process iterates over possible dimensions, refining templates, and removes regions where constraints are satisfied, while employing quantified polynomial-inequality solvers (PQE) to search template parameters. Complexity is polynomial in template size times the PQE solver's cost. Soundness holds by construction; relative completeness holds up to the expressiveness of the chosen templates and the power of the PQE solver (Kura et al., 29 Nov 2025).

6. Experimental Validation

A prototype implementation using linear (degree 1) polynomial templates and the PolyQEnt quantified polynomial-inequality solver demonstrated practical performance and expressiveness. Hard and soft constraints are encoded as universally quantified implications, reducible to PQE, SMT, QCP, or—in some cases—LP solvers.

Benchmarks include standard examples (from Abate CAV '24), new separating examples (EvenOrNegative, PersistRW, RecurRW), and extended case studies. On all tested benchmarks, LexPMSM synthesis succeeded within 0.8–12 seconds. Notably, the synthesis succeeded on instances (e.g., one-dimensional symmetric random walk, two-loop counterexample) where standard Streett-SMs fail, confirming the strict increase in certificate power. This suggests that PMSMs, and especially LexPMSMs, constitute an effectively implementable, strictly more expressive class of probabilistic $\omega$-regular certificates than previously available (Kura et al., 29 Nov 2025).

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PMSMs generalize deterministic progress measures to the probabilistic setting by substituting “next-state rank” with its expectation and imposing truncated lexicographic decrease. Their lexicographic extension (LexPMSM) matches the certificate power of LexGSSMs, thereby strictly subsuming earlier SSMs and enabling practical synthesis for robust $\omega$-regular verification in infinite-state probabilistic models (Kura et al., 29 Nov 2025, Abate et al., 2024).