Progress-Measure Supermartingales in Verification

• Progress-measure supermartingales (PMSMs) are supermartingale-based certificates that verify almost-sure ω-regular properties in probabilistic systems using truncated lexicographic drift conditions.
• They extend deterministic progress measures from parity and Streett games to the probabilistic setting by enforcing expected, rather than pathwise, decrease in rank.
• Constraint-based synthesis of PMSMs has been validated on complex infinite-state benchmarks, demonstrating effective verification of advanced Markov chain properties.

Progress-measure supermartingales (PMSMs) are a class of supermartingale-based certificates for verifying the almost-sure satisfaction of $\omega$-regular properties in stochastic models, particularly Markov chains and probabilistic programs. PMSMs generalize deterministic progress measures of parity and Streett games to the probabilistic setting by enforcing truncated lexicographic decrease in expectation rather than pathwise. Their lexicographic extensions (LexPMSMs) provide increased verification power, matching a strict hierarchy established for $\omega$-regular verification. PMSMs enable mechanical synthesis via constraint solving and have been validated on challenging infinite-state benchmarks that are otherwise intractable by prior supermartingale-based methods (Kura et al., 29 Nov 2025, Abate et al., 2024).

1. Formal Definition and Theoretical Framework

Let $S$ be a measurable state space and $F: S \to G S$ a (possibly infinite-state) Markov chain, with $$X: \operatorname{Meas}(S,[0, \infty]) \to \operatorname{Meas}(S,[0, \infty])$$ as the next-time operator. The chain induces a probability measure $\mathbb{P}_{x_0}$ on infinite traces $S^{\omega}$, with the canonical filtration $\{\mathcal{F}_k\}_{k \ge 0}$.

A parity condition is a function $p: S \to \{1, \ldots, d\}$ assigning a priority to each state. A $d$-dimensional, nonnegative measurable ranking map $r: S \to [0, \infty)^d$ is associated to $p$.

Truncated lexicographic orderings on $\mathbb{R}^d$ are:

• $u \succeq_i u'$ iff the first $i$ entries satisfy $u_1 \ldots u_i \geq_{lex} u'_1 \ldots u'_i$
• $u \succ_i u'$ iff $u_1 \ldots u_i >_{lex} u'_1 \ldots u'_i$

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

• if $p(x)$ is even: $r(x) \succeq_{p(x)} Xr(x)$
• if $p(x)$ is odd: $r(x) \succ_{p(x)} Xr(x)$

Here $Xr(x) = \int r(y) \; dF(x)(y)$ is the expected next-state valuation under $F$.

This framework extends to deterministic $\omega$-regular objectives in Streett form by letting $k$-component progress measures correspond to the $k$ acceptance pairs $(G_j, R_j)$, requiring drift conditions per component.

2. Probabilistic Generalization of Deterministic Progress Measures

Deterministic parity progress measures (Jurdziński '00) assign natural-number vectors so that along every edge $(v, w)$ in a parity graph:

• if $p(v)$ is even: $r(v) \succeq_{p(v)} r(w)$
• if $p(v)$ is odd: $r(v) \succ_{p(v)} r(w)$

The PMSM generalization replaces the next-state rank $r(w)$ by its expectation $Xr(v)$:

• when $p(v)$ is even: $r(v) \succeq_{p(v)} Xr(v)$
• when $p(v)$ is odd: $r(v) \succ_{p(v)} Xr(v)$

The drift conditions are enforced in expectation, fundamentally adapting the strictly combinatorial progress measure to the stochastic setting. This enables handling probabilistic systems where the exact successor is not determined, only its distribution.

3. Soundness for Almost-Sure $\omega$-Regular Satisfaction

Progress-Measure Soundness Theorem:

Let $F$ be a Markov chain on $S$ with parity condition $p: S \to \{1, \ldots, d\}$. If there exists a PMSM $r : S \to [0, \infty)^d$, then for any initial state $x_0$:

$$\mathbb{P}_{x_0}[\mathrm{trace} \in \mathrm{Parity}(p)] = 1$$

A canonical stopping-time argument shows that any $\omega$-trace violating the parity condition would have to induce an infinite strictly descending sequence in the lexicographic value of $r$, contradicting its nonnegativity. Thus, PMSMs certify almost-sure satisfaction of the $\omega$-regular property associated with $p$.

For Streett acceptance, the key is that componentwise drift inequalities exactly encode the criteria of a nonnegative almost-supermartingale, allowing direct application of the Robbins–Siegmund convergence theorem to deduce almost-sure satisfaction (Abate et al., 2024).

4. Hierarchy and Relationship to Other Supermartingale Certificates

A strict hierarchy of supermartingale-based certificates for $\omega$-regular verification emerges:

Certificate Type	Inclusion Relationship	Principle Features
Streett-SM (SSM)	⊂ GSSM	Handles Streett pairs, requires expectation bound on "good" set
Generalized Streett SM	⊂ LexGSSM and LexPMSM	Drops bound on "good" set, matches positive recurrence
LexGSSM	= LexPMSM	Lexicographic vector of GSSMs, extends to null recurrence
PMSM	= LexGSSM via extension	Parity-analogue of GSSM; lexicographic extension matches LexGSSM
Distribution-valued SM	⊇ LexPMSM	Distribution-valued, complete for all $\omega$-regulars

GSSMs and LexGSSMs (generalized Streett supermartingales and their lexicographic vector-valued extension) handle broader classes of verification problems than SSMs. LexPMSMs, the lexicographic variant of PMSMs, match the power of LexGSSMs. Distribution-valued Streett supermartingales (DVSSMs) are strictly more powerful in theory but do not currently admit practical synthesis algorithms (Kura et al., 29 Nov 2025, Abate et al., 2024).

5. Algorithmic Synthesis of Lexicographic PMSMs

The synthesis of LexPMSMs is accomplished via a constraint-based approach:

• Input: Parametric Control Flow Graph (pCFG) with location set $L$, real-valued variables $x \in \mathbb{R}^n$, priority partition $P_{l, i}$ for each $i \in \{1, \ldots, d\}$.
• Output: LexPMSM map consisting of dimensions $m_1, \ldots, m_{\lceil d/2 \rceil}$, level-assignment function $\operatorname{lev}$, and polynomial template components $r_{(l, i), j, k}(x) \ge 0$.

At a high level, the algorithm:

• Iteratively solves (for increasing priority levels and template indices) hard constraints enforcing nonnegative drift and soft constraints enforcing strict decrease, via Quantified Polynomial-Entailment (PQE) solvers.
• Associates each $(l, i)$ with the first (j, k) tuple that removes it by satisfying all constraints, or with a failure marker if none found.
• Terminates after at most $\lceil d/2 \rceil$ outer and $|L| \cdot d$ inner iterations.

For polynomial templates, complexity is polynomial in template size and the cost of each PQE solver invocation (usually handled by Positivstellensatz or Satisfiability Modulo Theories, SMT, relaxations).

Soundness is guaranteed by construction; any returned map satisfies all LexPMSM inequalities. Relative completeness holds with respect to the template space: if a polynomial-template LexPMSM exists, a complete solver will find it (Kura et al., 29 Nov 2025).

6. Experimental Validation and Benchmark Performance

A prototype implementation, using linear polynomial templates and the PolyQEnt solver (Chatterjee et al., ATVA 2026), synthesizes LexPMSMs by encoding constraints as universally quantified polynomial inequalities and solving them via SMT.

Benchmarks include:

• Examples from Abate CAV ’24: ex_3_8, ex_3_9, ex_4_11
• Separating examples: EvenOrNegative, PersistRW, RecurRW
• Extended case studies: GuaranteeRW, Temperature2

Key findings:

• All benchmarks, including examples where Streett-SMs fail (e.g., RecurRW, a one-dimensional symmetric random walk, and the two-loop counterexample), are successfully certified.
• Synthesis times are between 0.8 and 12 seconds for all instances.

This demonstrates the practical effectiveness of LexPMSMs for certifying almost-sure $\omega$-regular properties in probabilistic infinite-state systems, confirming theoretical expectations from the established hierarchy (Kura et al., 29 Nov 2025).

7. Context and Significance in $\omega$-Regular Verification

PMSMs and LexPMSMs extend the scope of supermartingale-based verification previously limited to reachability, safety, and basic recurrence persistence properties, now encompassing the full generality of $\omega$-regular and temporal logic objectives. By bridging deterministic small progress measure techniques and stochastic verification (via drift-in-expectation and lexicographic orderings), PMSMs provide rigorous certificates that are both sound and, via template synthesis, relatively complete for a broad class of probabilistic models.

The verified hierarchy, computational effectiveness, and capacity to handle null recurrence and positive recurrence in Markov chains position PMSMs as a central tool for advanced stochastic $\omega$-regular verification (Kura et al., 29 Nov 2025, Abate et al., 2024).