NOETHER: A Constructive Framework for Metamorphic Pattern Discovery from Operator Algebras
Authors: Meng Li, Xiaohua Yang, Jie Liu, Shiyu Yan
Organizations: School of Computing, University of South China, Hengyang, 421001, China · Hunan Engineering Research Center of Software Evaluation and Testing for Intellectual Equipment, Hengyang, 421001, China · CNNC Key Laboratory on High Trusted Computing, Hengyang, 421001, China · School of Computing, University of South China, China, Hunan Engineering Research Center of Software Evaluation and Testing for Intellectual Equipment, China, and CNNC Key Laboratory on High Trusted Computing, China · Hunan Engineering Research Center of Software Evaluation and Testing for Intellectual Equipment, China, and CNNC Key Laboratory on High Trusted Computing, China
Context. Metamorphic Testing is recognised in IEEE/ISO software-testing standards and increasingly recommended for AI systems, but its progress is bottlenecked by metamorphic relation (MR) identification: existing approaches (structured frameworks, mining and evolutionary pipelines, LLM-assisted methods, MetaPattern catalogues) share an inductive grounding that leaves three foundational questions open: origin, closure, and transferability. Objective. We propose a framework whose downstream step from program-induced operator algebra to MetaPattern set is mechanical and provable, while the upstream curation of the algebra is a stated empirical hypothesis with explicit scope precondition. Method. NOETHER is a two-layer framework. The upstream layer is an eight-block decomposition over recurrent mathematical structures (symmetry, order, self-adjoint, time-reversal, limit, qualitative-dynamics, method-comparison, relational equivalence). The downstream CONSTRUCT-MP algorithm produces a MetaPattern set with algebraic-closure (Theorem 1) and polynomial-time decidability (Theorem 2) guarantees. We test the framework on three operator-algebraic domains. Results. On Boltzmann reactor physics NOETHER systematises a prior inductive catalogue; on equivariant ML it derives executable MRs for rotation invariance, adjoint duality, and training-trajectory reversibility; on relational query optimisers it exercises the relational-equivalence block. The central falsifiable prediction (L*-blindness on homogeneity-preserving mutators) holds on the in-scope substrate. The absolute-completeness conjecture (Theorem 1') is falsified on PWR core diffusion via two pairwise-independent counterexamples that identify five Translate-extension dimensions. Conclusion. Induction is relocated from per-program MR sampling to a per-domain algebraic layer; the downstream step is deductive and mechanical.
Context. Metamorphic Testing addresses the test-oracle problem in scientific computing, but classical Mutation Score operates on syntactic AST mutations and misses domain semantics. Objective. We propose the Semantic Mutation Score (SMS), built on five domain-semantic operators (Conservation Erosion, Operator Substitution, Hyperparameter, Trajectory Flip, Structural Injection). SMS degenerates almost everywhere to MS in a characterised limit, so any SMS-based conclusion remains consistent with prior mutation-testing literature in the classical regime. Method. A 12-PUT x 5-MP design over four single-output float-to-float classes (numeric, probabilistic, surrogate, machine-learning) is paired with a three-layer attribution classifier separating true semantic faults from tolerance, OOD, statistical, and artefact categories. A same-source / cross-source ablation under an identical prompt isolates the LLM-source-diversity contribution. LLM-generated mutants are compared against a default-configuration cosmic-ray syntactic pool at the AST-normalised level. Results. The pre-registered large-effect threshold for Cliff's delta is not met under the point-estimate criterion; the observed effect lies in the medium-effect range. Cross-source pooling under an identical prompt does not appreciably shift delta, indicating that LLM identity is not the lever within this design. AST-level overlap between LLM-generated and default cosmic-ray syntactic mutants is small; the Hyperparameter, Structural Injection, and Trajectory Flip classes are unreachable under default first-order syntactic configurations. Conclusion. SMS is a backward-compatible adequacy metric for domain-semantic metamorphic-relation sets in scientific computing. The first-order unreachability evidence is independent of the effect-size question.
Scientific machine-learning (SciML) surrogates approximate expensive simulations, but exact expected outputs for arbitrary inputs are unavailable (the oracle problem). Metamorphic testing checks relations across executions, yet a candidate relation is not automatically valid: its preconditions, output mapping, and the numerical floor of the scoring operator determine whether a violation is meaningful. We study how candidate metamorphic relations (MRs) can be screened for domain validity and turned into executable, oracle-free test assets for SciML surrogates. We propose (i) a domain-validity rubric that admits a candidate only when its tolerance dominates the operator's numerical floor and its preconditions hold; (ii) an MR-card executable-asset format recording source cases, transformations, metrics, tolerances, and typed relation-level verdicts; and (iii) a case-study protocol on MeshGraphNets cylinder-flow surrogates, with a claim ledger binding every result to a tracked artifact. On a MeshGraphNets checkpoint, node permutation holds to machine precision, mirror-y is a bounded out-of-distribution stress finding rather than an exact symmetry, and absolute conservation stays deferred while a reference-relative guard passes. The same readings hold across held-out trajectories, a checkpoint roster, three further architectures, and PhysicsNeMo. On a second CFD task (compressible airfoil) the predicate instead rejects incompressible continuity on physical grounds, showing it reasons about domain validity rather than running a fixed checklist. On a second PDE family, FNO Burgers and heat surrogates run full admit/reject/execute verdicts. The evidence spans two CFD tasks and a second PDE family, supporting a validity-aware bridge from candidate MRs to auditable SciML test assets that separates model-level violations from out-of-domain applications.
As REST APIs become an increasingly significant part of software systems, their validation is becoming more critical. Hence, testing and uncovering underlying issues are of utmost importance for improving software quality. However, testing REST APIs is challenging mainly due to the difficulty of assessing whether the output of an API call is correct, i.e., the test oracle problem. Metamorphic testing is a specification-based testing approach for situations where correct outputs are unknown or not specified explicitly. To check the correctness of a system, relations between the different outputs are specified. We present ARMeta, a tool-supported approach that uses an LLM-based multi-agent workflow to support metamorphic testing of REST APIs documented with OpenAPI. The agentic workflow is used to identify metamorphic test scenarios and specify them in the Given-When-Then format. These scenarios are automatically implemented as executable tests and executed against the system under test. We evaluate ARMeta on two publicly available web applications that expose REST interfaces and compare its performance with a scenario-based testing baseline. The results show that ARMeta explores behaviors that serve as a complement to existing scenario-based testing approaches.
Shehroz Khan, Abdullah Mughees, Gaadha Sudheerbabu +2