Soundness-Aware Level (SAL) Overview

Updated 24 October 2025

Soundness-Aware Level (SAL) is a framework that quantifies behavioral and semantic correctness in computational and logical systems.
It employs methods such as anti-pattern diagnostics, coverability analysis, and divergence metrics to measure soundness.
SAL drives practical applications in quantum complexity, workflow verification, static analysis, and neural model selection through rigorous stratification.

Soundness-Aware Level (SAL) formalizes and quantifies the extent to which a computational, logical, or process-based system can distinguish, guarantee, or leverage behavioral and semantic correctness at granular levels, enabling more efficient verification, analysis, or selection decisions. Across fields such as quantum complexity theory, workflow modeling, static analysis, assurance, modal logic, and neural architecture, the concept captures the spectrum from purely structural or logical soundness to sophisticated, nuanced measures of soundness discrimination. The following sections explore SAL by surveying technical definitions, core methodologies, computational results, representative models, and broad implications for rigorous system design and analysis.

1. Formal Definitions and Technical Scope

SAL encompasses multiple technical formulations, depending on the domain:

Workflow and Negotiation Models: In Petri nets and negotiations, the Soundness-Aware Level is demarcated by the ability to guarantee process completion or termination from any reachable state, often extended to include different concurrency levels (e.g., 1-soundness, k-soundness, generalised soundness). Notably, the presence or absence of anti-patterns (for negotiations) or the reachability and coverability conditions (for workflow nets) directly calibrate the SAL (Esparza et al., 2017, Blondin et al., 2022, Blondin et al., 6 Mar 2025).
Quantum Interactive Proofs: For quantum Merlin-Arthur (QMA) protocols with multiple provers, SAL refers to the system’s capability to detect unsatisfiability with well-quantified “soundness gaps.” These are proven to scale as Ω(1/N²) for the Blier-Tapp protocol and Ω(κ²/N) for the Chen-Drucker protocol, where κ is the number of provers (Chiesa et al., 2011).
Static Program Analysis: In code analysis for platforms such as Android, SAL is formalized as the empirical ratio or divergence between statically approximated and dynamically observed behaviors, quantifying the degree of unsoundness introduced by omissions in method coverage (Samhi et al., 10 Jul 2024).
Neural Reasoning Systems: In pre-trained LLMs, SAL quantifies the microscopic separation between internal logic rule distributions corresponding to different soundness levels (strict, plausible, noisy). This is accomplished by calculating the Jensen-Shannon Divergence (JSD) between transition probability distributions assigned to semantically-labeled rules (Wu et al., 17 Oct 2025).
Assurance and Confidence Arguments: In safety-critical systems, SAL refers to the vertical integration of logical, probabilistic, and dialectical rigor within assurance cases, such that doubts and defects are systematically identified, refuted, or managed as residual risks (Bloomfield et al., 16 Sep 2024).
Modal and Logical Frameworks: SAL designates the hierarchical stratification of ontological admissibility regimes within modal logic, with modalities indexed by levels of stability, thereby distinguishing local actualization from global possible world semantics (Nepvou, 12 Jun 2025).

2. Methodological Foundations and Analytical Techniques

SAL is supported by a variety of analytical and algorithmic frameworks:

Anti-Pattern Diagnostics: Deterministic negotiations employ anti-pattern-based structural checks (types B, F, C) for soundness (Esparza et al., 2017).
Coverability and Cyclicity Analysis: Soundness of workflow nets is determined via reachability and coverability queries, augmented by cyclicity checks, and boundedness conditions. Complexity ranges from NL-complete for deterministic cases to EXPSPACE- and PSPACE-complete for classical and generalized cases (Blondin et al., 2022).
Quantum Property Testing: Protocols utilize swap tests, consistency checks, and uniformity tests to probabilistically detect unsound proof states, with Fourier analysis and second-moment (Chebyshev-type) bounds providing quantitative rejection probabilities (Chiesa et al., 2011).
Constraint-Based Symbolic Methods: Data-aware processes employ transformation to colored Petri nets and constraint graphs (with representative abstraction for data domains) to enable tractable reachability analysis and soundness verification using SMT solvers (Leoni et al., 2018, Felli et al., 2022).
Empirical Metric Derivation: The soundness-aware level in static analysis is computed as the proportion of dynamically executed methods captured in static models, with diverse call graph construction algorithms contributing to differential coverage (Samhi et al., 10 Jul 2024).
Distributional Divergence Metrics: Neural model SAL is quantified via the JSD over rule distribution histograms across semantic soundness categories, leveraging cross-layer sparse autoencoders to extract internal feature logic (Wu et al., 17 Oct 2025).
Canonical Model Theory: In modal logic, completeness and soundness relative to stratified actualization regimes are established using maximal consistent sets and indexed accessibility relations (Nepvou, 12 Jun 2025).

3. Complexity Landscape and Decidability

SAL stratifies systems not only by their soundness guarantees but also by the computational tractability of verification:

Domain	Soundness Property	Complexity Status
Petri nets/workflow	Classical, structural	EXPSPACE-complete
	Generalised	PSPACE-complete
Reset workflow nets	1-soundness, generalized	Undecidable
	1-in-between (℘₁)	Decidable (intermediate)
Negotiations	Deterministic	NL-complete
	Weakly non-deterministic	Polynomial time

Notably, for reset workflow nets, a decidable intermediate property ("1-in-between soundness") is situated between two undecidable properties, revealing non-monotonic complexity (Blondin et al., 6 Mar 2025).

4. Representative Models and Formulas

SAL is captured with precise mathematical formalism across domains:

Workflow Nets:

$∀ m, { i : k } →^* m \implies m →^* { f : k }$

with generalised soundness requiring the property for all $k \in \mathbb{N}_+$ .

LLM Reasoning (Horn Clause Extraction, Transition Probabilities, SAL Metric):

\alpha_{c_1} \land \ldots \land \alpha_{c_M} \rightarrow \alpha_{c_q}
\
\hat{p}(Q|P) = \frac{\text{count}(P, Q) + \beta}{\text{count}(P) + 2\beta}
\
SAL := JSD(\{\rho_y\}_{y \in \mathcal{Y}}) = \frac{1}{|\mathcal{Y}|} \sum_{y \in \mathcal{Y}} KL(\rho_y \| m)

Stratified Modal Logic Models:
- Underlying frame: $F = (W, \leq, \{R_\alpha\}_{\alpha \in I})$
- Modal axiom: $□_\alpha \phi \rightarrow □_\beta \phi$ for $\alpha \leq \beta$
Negotiation Anti-patterns: Diagnostic checks for forbidden topological configurations; algorithms operate in NL (Esparza et al., 2017).

5. Impact and Applications

SAL plays a significant role in both theoretical computer science and practical system design:

Quantum Complexity: Enables QMA protocols with multiple unentangled quantum proofs to attain sublinear total proof length and robust soundness gaps, facilitating efficient verification of NP-complete problems amenable to quantum proof systems (Chiesa et al., 2011).
Process Mining and BPM: Supports verification of decision-aware and data-aware workflow models. Scalable techniques (e.g., SMT-based, representative abstraction) make practicality feasible for high-complexity processes (Leoni et al., 2018, Felli et al., 2022).
Object-Centric Process Models: SAL ensures correctness at the artifact level, important for systems supporting concurrent, interacting cases (Lomazova et al., 2021).
Static and Dynamic Analysis: Quantitative SAL metrics enable informed selection or trust calibration in code analysis tools, driving improvements in security-critical application domains (Samhi et al., 10 Jul 2024).
Assurance Systems: Modular, stratified soundness levels align with rigorous, indefeasible safety and security case design, blending logical deduction, probabilistic metrics, and dialectical risk management (Bloomfield et al., 16 Sep 2024).
Modal Logic Foundations: SAL formalizes ontological stratification for modalities, yielding new insights in temporal logic, quantum branching, and metaphysics (Nepvou, 12 Jun 2025).
Neural Model Selection and Engineering: The SAL metric enables prediction and selection of pre-trained models with high potential for robust reasoning, informing RLVR interventions and architecture/dataset design (Wu et al., 17 Oct 2025).

6. Methodological Innovations and Future Directions

Several methodological advances coalesce in SAL:

Soundness “Stratification”: The development of soundness-aware levels provides a framework for hierarchical or graded verification, contributing rigorous stratification for both logical and probabilistic assurance.
Empirical Law for Neural Reasoning: SAL scores exhibit a strong predictive relationship with post-training reasoning error rates:

$\varepsilon = \exp(-\alpha \cdot s^\beta)$

with observed $R^2 = 0.87$ to $0.985$, directly connecting microscopic distributional structure to macroscopic performance (Wu et al., 17 Oct 2025).

Decidability by Surrogate Properties: The discovery and algorithmic implementation of intermediate decidable soundness proxies (e.g., 1-in-between soundness) enable practical soundness verification where direct analysis is provably infeasible (Blondin et al., 6 Mar 2025).
Cross-Domain Generalization: SAL’s underlying concept appears in diverse domains, suggesting a generalizable framework for measuring, propagating, and leveraging soundness in computational and logical systems.

7. Challenges, Limitations, and Open Questions

SAL addresses tractability and granularity but is subject to domain-specific limitations:

Tractability: Complexity bounds (EXPSPACE, PSPACE, undecidability) often constrain direct verification, requiring surrogate metrics or approximation schemes (e.g., ℘₁ properties, anti-pattern checks).
Modeling Omissions: Empirical SAL in static analysis shows high levels of unsoundness in practical tools, especially for event-driven and framework-dependent systems (Samhi et al., 10 Jul 2024).
Divergence between Theory and Implementation: Quantitative soundness metrics are sensitive to modeling choices (e.g., call graph construction algorithms), and in neural systems, labeling soundness categories depends on external LLM judgment.
Logical-Probabilistic Integration: Assurance frameworks highlight persistent challenges in blending deductive and probabilistic confidence, with risk of implausible results in naive combinations (Bloomfield et al., 16 Sep 2024).

A plausible implication is continued need for research into more expressive, scalable soundness-aware frameworks, including abstraction techniques and automated metrics that retain provable guarantees across complex, dynamic, or data-rich systems.

SAL thus provides a rigorous conceptual and quantitative apparatus for classifying, verifying, and optimizing system soundness at multiple levels of granularity and abstraction, grounding both theoretical advances and practical solutions in diverse areas of computer science, logic, and artificial intelligence.