Completeness Evaluation Module

• Completeness Evaluation Module is a component that quantifies the sufficiency and thoroughness of a system or dataset against defined semantic standards.
• It integrates symbolic, statistical, and algorithmic approaches to assess coverage, accuracy, and diagnostic outputs across diverse domains.
• Applications span data cleaning, knowledge base maintenance, safety assessments, and formal verification in both hardware and software testing.

A Completeness Evaluation Module (CEM) is a rigorously specified component or subsystem—algorithmic, statistical, symbolic, or hybrid—designed to quantify, certify, or argue for the sufficiency or thoroughness of a system, dataset, procedure, or artifact with respect to an intended semantic or operational standard. Completeness evaluation manifests across data cleaning, knowledge base maintenance, interpretable machine learning, scenario-based safety assessment, V&V for reactive systems, formal logics, and hardware testbench construction. This article surveys foundational principles, formalizations, methodologies, and empirical results drawn from the literature, illustrating the breadth and the precise technical content of state-of-the-art completeness evaluation modules.

1. Formal Definitions and Conceptual Distinctions

Completeness evaluation requires context- and domain-specific definitions. Several paradigms illustrate the diversity:

• Vision-Language Data (HMGIE): Completeness (𝓗₍cₒₘₚ₎) measures the semantic coverage or richness of an image caption, contrasting with accuracy (𝓗ₐcc), which measures correctness of information present. Coverage is operationalized as the proportion of structured semantic nodes (e.g., objects, attributes, relations) examined during hierarchical QA (Zhu et al., 7 Dec 2024).
• Evolving Knowledge Bases: For a class C and property p in an RDF KB, completeness is defined via a longitudinal comparison of normalized frequencies:

$$\text{Completeness}_t(p,C) = \begin{cases} 1 & \text{if }NF_t(p,C)\ge NF_{t-1}(p,C) \ 0 & \text{otherwise} \end{cases}$$

and averaged over all properties to obtain a class-level score (Rashid et al., 2018).

• Reasoning in LLMs (RACE): Explanation completeness quantifies the overlap between an LLM-generated rationale and the interpretable, high-importance lexical features as ranked by a logistic regression baseline, with coverage measured at different lexical granularities and partitioned by “supporting” vs. “contradicting” roles (Patil, 23 Oct 2025).
• Scenario Completeness (Automotive Domains): Given scenario class catalog $C \subseteq S$ (universal scenario space), completeness means $$\forall s \in S,\ \exists c \in C\ \text{such that}\ s$$ matches $c$, a strictly logical requirement. In contrast, coverage quantifies the empirical fraction of real or simulated samples that fall into some class in $C$ (Glasmacher et al., 2 Apr 2024).
• Logic Programming: Completeness of a program $P$ w.r.t. a specification $S$ means the least Herbrand model $\mathcal{M}_P \supseteq S$, i.e., all specified answers are semantically entailed by $P$ (Drabent, 2014).
• Test Suites for Finite-State Systems: $T$ is $m$-complete for a specification FSM $M$ if every non-equivalent implementation of up to $m$ states will be detected (i.e., fail a test from $T$). Under blocking tests, perfectness generalizes this to require detection of both behavioral and domain mismatches (Bonifacio et al., 2015).

2. Mathematical Foundations and Scoring Functions

Completeness metrics are typically quantitative (scoring) or Boolean (predicate-style):

• Multi-level Weighted Scoring (HMGIE): Given L levels, $N_l$ slots per level, and weights $\alpha_l$,

$$\mathcal{H}_{(\mathrm{comp})} = \sum_{l=1}^{L} \alpha_l \frac{n_l}{N_l}$$

where $n_l$ is the number of distinct semantic items addressed at level $l$ (Zhu et al., 7 Dec 2024).

• Evolving KBs: The per-property indicator is piecewise, but the class-level average is scalar in $[0,1]$.
• LLM Reasoning (RACE): For matcher $\tau$ (token, exact, edit),

$$\mathrm{support\_cov}_{i,\tau} = \frac{1}{|\mathcal{S}_i|} \sum_{f \in \mathcal{S}_i} m_\tau(f, r_i)$$

and correspondingly for contradicting features, aggregated according to correctness (Patil, 23 Oct 2025).

• Bidirectional Attention Coverage ($C^3$): Visual $\bar{c}_{\text{region}}$ and semantic $\bar{c}_{\text{attr}}$ mean coverage scores are combined via harmonic mean:

$$S_\text{complete} = \frac{2\bar{c}_\text{region} \bar{c}_\text{attr}}{\bar{c}_\text{region}+\bar{c}_\text{attr}}$$

(Zhang et al., 9 Nov 2025).

• Statistical Completeness for Astronomical Catalogues:

$$T_c = \sum_{i=1}^N \frac{\zeta_i - 1/2}{\sqrt{\mathrm{Var}(\zeta_i)}}$$

where each $\zeta_i$ is a positionally-adaptive, signal-to-noise controlled quantile (Teodoro et al., 2010).

3. Module Architecture and Algorithmic Implementation

Comprehensive CEMs incorporate input normalization, automated scoring, and diagnostic capabilities:

• HMGIE (Vision-Language): Accepts image + caption pairs, builds a semantic graph, generates hierarchical QA nodes per level, computes coverage, and outputs a semantic completeness explanation. Hyperparameters (N, L, $\alpha$) tune the stringency and depth. The score is used for downstream filtering or feedback (Zhu et al., 7 Dec 2024).
• KB Evolution (RDF/Linked Data): Ingests consecutive KB snapshots, executes a small sequence of SPARQL or one-pass relational scans for class and property profiling, flags drops in normalized property frequencies, and outputs per-class statistics (optionally, SHACL/ML-based validation) (Rashid et al., 2018).
• RACE: Embeds LLM-generated rationales, aligns to baseline top-$k$ features with string normalization and hierarchical matching, aggregates coverage by correctness partition, and supports both real-time and batch evaluation via a dedicated metric engine (Patil, 23 Oct 2025).
• Scenario-Based Argumentation: Constructs a GSN decomposition (“goal-structured notation”) with top-level, layered, and per-scenario class goals; evidential assessment includes both knowledge-based expert reviews and data-driven scenario detection (Glasmacher et al., 2 Apr 2024).
• Polymorphic Gate Sets: The completeness module runs a phase-based construction algorithm, recursively synthesizing AND, OR, NOT “cells” for $m$ modes by closed-world enumeration of combinatorially generated sub-circuits (Li et al., 2017).
• Logic Programming: Checks coverage of the specification atoms by the program, applies program schemas (recurrent, acceptable), and validates under pruning/cut rules. Diagnostic messages are returned for uncovered atoms or incompatibility with splittings (Drabent, 2014).

4. Empirical Results and Diagnostic Output

Completeness evaluation is coupled to reporting and operational filtering:

• Vision-Language Data Cleansing: Filtering by $\mathcal{H}_{(\mathrm{comp})}$ thresholds (e.g., 0.5) identifies under-specified captions; explanations are synthesized that enumerate coverage by semantic level (Zhu et al., 7 Dec 2024).
• Knowledge Base Quality Control: Reported precision of flagged “incomplete” properties is 94–95% in studied DBpedia/3cixty cases. The approach scales to millions of triples; false positives may occur due to schema redesign or class population shocks (Rashid et al., 2018).
• RACE for LLMs: Empirical results confirm substantial gaps between correct and incorrect LLM predictions—correct examples cover more supporting features (e.g., $0.61$ vs. $0.34$ with edit matching in Wiki Ontology), confirming the metric's diagnostic value (Patil, 23 Oct 2025).
• Scenario Catalogs (inD Dataset): All event time-steps are exhaustively assigned (coverage = 1.0 at layer 4); scenario-type saturation and parameter coverage curves empirically plateau, supporting completeness claims within domain (Glasmacher et al., 2 Apr 2024).
• Milky Way Redshift Surveys: Adaptive S/N-controlled $T_c$, $T_v$ estimators identify the “true” faint flux cutoff via robust “roll-off” detection; improper (non-adaptive) estimators are shown to result in misleading completeness passes owing to shot noise (Teodoro et al., 2010).

5. Integration, Hyperparameters, and Practical Specifications

CEM deployment requires careful tuning and integration with existing pipelines:

• Weighting and Granularity: Tunable parameters ($\alpha_l$, $N_l$, depth $L$) in hierarchical schemes determine emphasis on coarse versus fine completeness. Geometric progressions for weight assignment are typical (Zhu et al., 7 Dec 2024).
• Profiling and Performance: Batch SPARQL queries or direct one-pass scans are favored for large KBs. Completeness evaluation is strictly comparative and linear in the number of class–property pairs; machine learning is used only for post-hoc validation of flagged items (Rashid et al., 2018).
• Coverage Thresholds: No hard thresholds are imposed internally in most modules. Instead, completeness scores are supplied for downstream filtering, alerting, or cost signals, e.g., in Markov Decision Process-based reasoning control (Zhang et al., 9 Nov 2025).
• Specification Engineering and Approximation: Where specifications are imprecise, approximate completeness pairs $(S_\mathrm{compl}, S_\mathrm{corr})$ are tracked; coverage checking up to grounding depth is used in logic programming to guide either diagnosis or certification (Drabent, 2014).
• Adversarial and Theoretical Limits: Structurally, all test suites have an inherent bound: for a maximal non-extensible test of length $\ell$ in an FSM with $|S|$ states, no suite is $n$-complete for $n > (\ell+1)|S|$ (Bonifacio et al., 2015).

6. Comparative Analysis, Limitations, and Domain Adaptation

CEMs must be critically evaluated with respect to recall, robustness, and context sensitivity:

• Recall and False Positives: Evolving-KB CEMs exhibit high precision but may miss incompleteness arising purely from schema extension or population discontinuity—completeness is measured strictly as stability of property frequencies (Rashid et al., 2018).
• Expressivity and Limitations of Baselines: In LLM reasoning, reliance on first-order lexical feature baselines means higher-order compositional or semantic evidence may be missed, and edit-distance or exact match-based scoring does not account for antonymy or idiomatic overlap (Patil, 23 Oct 2025).
• Empirical Scope: Scenario completeness modules validated on inD or related datasets demonstrate empirical sufficiency only within the ODD and at the levels of abstraction considered; broader or evolving domains require systematic scenario catalog expansion and re-validation (Glasmacher et al., 2 Apr 2024).
• Adaptivity to Instance Structure: Non-parametric estimators in astronomy (e.g., $T_c$, $T_v$) automatically adapt smoothing windows based on survey density—models that do not adapt to shot-noise are susceptible to over- or under-estimation (Teodoro et al., 2010).
• Generalization to New Domains: CEM methodologies are portable across RDF, relational, and even spatial data, given suitable translation of completeness statements to query satisfiability or containment problems, with tractability tied to the complexity of the corresponding fragment (e.g., $\Pi_2^P$-completeness for RDF critical queries (Darari et al., 2016)).

7. Synthesis and Outlook

Completeness Evaluation Modules are foundations for data quality assessment, formal system testing, automated scenario argumentation, and machine learning explanation verification. Their rigor stems from formal definitions, explicit scoring, coverage decomposition, and, where necessary, theoretical bounding arguments. State-of-the-art CEMs combine symbolic, statistical, logical, and hybrid techniques, always grounded in domain-specific semantics but unified by their computational and mathematical treatment of what it means to be "complete".

Further advances include compositional completeness checking (as in modular proofs for KAT/NetKAT (Pous et al., 2022)), semantically enriched or context-aware completeness metrics (e.g., embedding similarity for LLM rationales), and real-time adaptive CEMs for streaming and evolving data. Cross-domain standardization of completeness statements and APIs for integrating CEM logic into data pipelines, model evaluation, and safety cases remain important avenues for development.