ProbSPARQL: Querying Knowledge Graphs with Multi-dimensional, Uncertain Numeric Data

# ProbSPARQL: Querying Knowledge Graphs with Multi-dimensional, Uncertain Numeric Data
Source: [https://arxiv.org/html/2607.18262](https://arxiv.org/html/2607.18262)
11institutetext:Institute for Artificial Intelligence, University of Stuttgart, Germany
11email:\{jingcheng\.wu, ratan\.thapa, daniel\.hernandez, steffen\.staab\}@ki\.uni\-stuttgart\.de22institutetext:Bosch Corporate Research, Renningen, Germany
22email:hongkuan\.zhou@de\.bosch\.com33institutetext:Web and Internet Science Research Group, University of Southampton, United Kingdom###### Abstract

The SFB 1574 “Circular Factory” is building a shared knowledge graph infrastructure for integrating data about returned products\. A central challenge is that circular\-factory data include numeric measurements that \(i\) originate from sensors or are derived from sensor\-based measurements, \(ii\) are frequently multi\-dimensional, and \(iii\) are inherently uncertain, while downstream triage, validation, reliability\-modeling, and reassembly\-planning modules require queryable uncertainty representations\. Current RDF and SPARQL technologies lack native support for harmonized querying and analysis of such uncertain numeric measurement data\. To address this gap, we present ProbSPARQL, an upward\-compatible SPARQL extension developed as an early\-stage query\-layer pilot for this infrastructure\. ProbSPARQL models uncertain numeric values as random variables whose distributions are encoded by probabilistic RDF literal datatypes, and supports distribution\-aware expressions, probabilistic filters, and divergence\-based joins\. We implement ProbSPARQL on Apache Jena ARQ and expose it through a Fuseki\-compatible execution layer\. We assess real\-data applicability using project\-derived measurement fragments covering GMM\-encoded uncertainty and histogram\-based empirical roughness distributions, and evaluate scalability separately on controlled ontology\-conformant benchmarks with up to 5,000 angle\-grinder instances and 1\.5M triples\. The results show feasible in\-engine execution, filter\-pushdown speedups over application\-layer post\-processing, and latency–accuracy trade\-offs among divergence\-join decision strategies\.

## 1Introduction

Knowledge graphs\[[17](https://arxiv.org/html/2607.18262#bib.bib7)\], represented in RDF\[[44](https://arxiv.org/html/2607.18262#bib.bib27)\]and queried via SPARQL\[[43](https://arxiv.org/html/2607.18262#bib.bib26)\], have been employed for data interoperability in manufacturing\[[38](https://arxiv.org/html/2607.18262#bib.bib8)\], healthcare\[[33](https://arxiv.org/html/2607.18262#bib.bib66)\], energy systems\[[42](https://arxiv.org/html/2607.18262#bib.bib68)\], and environmental monitoring\[[45](https://arxiv.org/html/2607.18262#bib.bib67)\]\. Most of these domains rely on physical sensors\[[28](https://arxiv.org/html/2607.18262#bib.bib69),[21](https://arxiv.org/html/2607.18262#bib.bib70)\]that produce numeric data inherently subject to uncertainty\[[1](https://arxiv.org/html/2607.18262#bib.bib72),[9](https://arxiv.org/html/2607.18262#bib.bib71)\]\.

This form of uncertainty appears concretely in the circular manufacturing scenarios studied in the Collaborative Research Centre SFB 1574 “Circular Factory”\[[34](https://arxiv.org/html/2607.18262#bib.bib89),[26](https://arxiv.org/html/2607.18262#bib.bib28)\], a 12\-year initiative funded by the German Research Foundation \(DFG\)\. The project aims to enable the perpetual use of technical products by integrating inspection, disassembly, reprocessing, and reassembly into a data\-driven circular production system\. Within this project, more than 60 researchers from engineering, computer science, and production\-systems research across multiple institutions collaborate on a shared knowledge graph infrastructure for inspection and manufacturing data\. The resulting infrastructure is therefore not merely a benchmark context, but a shared integration layer for engineering subprojects that produce, validate, and consume uncertain measurement data\.

As illustrated in Figure[1](https://arxiv.org/html/2607.18262#S1.F1), the inspection of returned industrial products is central to this goal\. In the “Circular Factory” use case, a returned FEIN CG\-15\-125 BL angle grinder is disassembled, and its components undergo geometric inspection using a Zeiss ATOS Q structured\-light 3D scanner\. Extracting geometric characteristics such as individual tooth lengths from the resulting point clouds yields uncertain numeric data rather than exact scalar values: to account for sensor noise and reconstruction uncertainty, the geometric extraction pipeline outputs a probability distribution\. These uncertain measurements must be represented in a common, machine\-interpretable form while remaining queryable together with component identity, measurement context, and units in the shared knowledge graph infrastructure\. Downstream triage, validation, reliability\-modeling, and reassembly\-planning modules require such queryable uncertainty representations\. Analogous needs arise in healthcare and environmental monitoring, where measurement data likewise carry acquisition and device\-specific uncertainty\[[33](https://arxiv.org/html/2607.18262#bib.bib66),[45](https://arxiv.org/html/2607.18262#bib.bib67)\]\.

Yet, the current RDF and SPARQL specifications, comparable query languages \(e\.g\., GQL\[[19](https://arxiv.org/html/2607.18262#bib.bib57)\]\), and specific ontologies such as the Semantic Sensor Network \(SSN\) Ontology\[[22](https://arxiv.org/html/2607.18262#bib.bib4)\], do not provide native support for harmonized querying and analysis of uncertain numeric measurement data\. Such support requires that \(i\) multi\-dimensional sensor measurement data is represented in an efficient manner, such as spatio\-temporal point clouds; \(ii\) measurements are represented as random variables that define distributions over their data domains; \(iii\) essential probabilistic functions are natively supported; and \(iv\) selections, filters, and operations on random variables are integrated with both each other and other knowledge graph representations\.

![Refer to caption](https://arxiv.org/html/2607.18262v1/x1.png)Figure 1:Circular manufacturing inspection workflow\. An extracted crown gear undergoes Zeiss ATOS Q structured\-light 3D scanning, yielding point clouds from which geometric features \(e\.g\., tooth lengthLL\) are modeled as probability distributions\.To address these requirements, this paper makes the following contributions:

1. 1\.Probabilistic representation as random variables\.We model uncertain numeric measurement as random variables defined by a domain and a distribution\. By encoding such distributions as probabilistic RDF literals, we elevate them to first\-class objects that can be accessed, queried, and used in computations\.
2. 2\.Distribution\-aware query algebra\.We extend the SPARQL algebra with a divergence join for distributional matching with a divergence tolerance and decision parameter, and introduce type\-aware expression operators for comparing, transforming, fusing, and combining distributions\.
3. 3\.Extensible probabilistic datatype layer\.We demonstrate the datatype layer with two parametric representations, Gaussian mixture models and Dirichlet distributions, and one non\-parametric representation, histogram\-based empirical distributions, all supported through a shared literal scheme and polymorphic operator interface\.
4. 4\.Implementation, evaluation, and artifacts\.We implement ProbSPARQL on top of Apache Jena and evaluate it through real\-data applicability checks on project\-derived circular\-factory measurement fragments and controlled runtime experiments covering overhead, filter pushdown, divergence\-join strategies, and datatype extensibility\. We report early cross\-institutional uptake within the SFB 1574 consortium, where non\-author partner groups in inspection, materials analysis, and production\-systems research provide consumer scenarios for downstream validation, matching, and planning modules, and release the implementation, generated evaluation data, query workloads, and replication scripts as reproducibility artifacts\.111[https://github\.com/0sidewalkenforcer0/ProbSPARQL](https://github.com/0sidewalkenforcer0/ProbSPARQL)

## 2Use Case Requirements and Consuming Modules

The circular\-factory scenario yields recurring query needs over uncertain measurement data in the shared knowledge graph\. The four use cases below are representative rather than exhaustive and cover the current pilot’s main probability\-aware query capabilities\. They were formulated with engineering partner groups in the cross\-institutional SFB 1574 consortium based on recurring data\-access needs in inspection, materials analysis, and production\-systems research\. These partner groups provide consumer scenarios for downstream validation, matching, and planning modules beyond the ProbSPARQL proposer team\.

R1: Threshold\-based retrieval\(Consumer:Planning and control module\.\) Purpose:Automated triage agents screen returned gears for excessive tooth wear before routing them to reuse, reprocessing, or rejection\. Since tooth\-length measurements are uncertain, the module requires probabilistic rather than deterministic thresholding\. Query type:Probabilistic thresholding \(“Give me all gears which have at least one tooth whose length is below 9\.8 mm with probability at least 0\.9\.”\)

R2: Anomaly detection\(Consumer:Autonomous measurement\-strategy module\.\) Purpose:Measurement\-validation components compare distributions obtained from different sensors or repeated measurements to detect sensor\-specific anomalies, inconsistent reconstructions, or data\-quality problems before the values are consumed by downstream modules\. Query type:Distributional divergence \(“Find all gears where the Jensen–Shannon divergence between the tooth\-length distributions from CT and structured light exceeds 0\.2\.”\)

R3: Motor performance evaluation\(Consumer:Reliability modeling module\.\) Purpose:Diagnostic agents evaluate the output power of returned motors for reliability modeling\. Since power is not directly measured, it is derived from angular speed and torque, both represented as uncertain numeric values\. The consuming module therefore needs a derived power distribution rather than a point estimate\. Query type:Uncertainty propagation \(“Retrieve all motors and compute their probabilistic power distributions derived from uncertain angular\-speed and torque measurements\.”\)

R4: Divergence\-based matching\(Consumer:Reassembly\-planning module\.\) Purpose:Reassembly agents select compatible reclaimed drive and spindle gears for recombination\. Substantially different hardness profiles may indicate contact\-stiffness mismatch and vibration\-relevant incompatibility\. Query type:Divergence join \(“Retrieve all available drive–spindle gear pairs for which the null hypothesisH0:Δ≤ϵH\_\{0\}:\\Delta\\leq\\epsilon, whereΔ\\Deltais the divergence between their hardness distributions, cannot be rejected at decision parameterα=0\.05\\alpha=0\.05\.”\)

## 3Core Concepts for Probabilistic Literals

Computing with probabilistic literals relies on modeling uncertain numeric values as*random variables*\. We classify*operations on random variables*as distribution\-valued transformations or deterministic\-valued evaluations\. A*divergence test*is defined as a statistical hypothesis test\.

###### Definition 1\(Random Variable\[[32](https://arxiv.org/html/2607.18262#bib.bib86)\]\)

LetD⊆ℝdD\\subseteq\\mathbb\{R\}^\{d\}\(d∈ℕ\+d\\in\\mathbb\{N\}^\{\+\}\) be the*domain*\. A*random variable*is a pairX=\(D,P\)X=\(D,P\), wherePPis a probability measure overDD\. Two random variablesX=\(D,P\)X=\(D,P\)andX′=\(D′,P′\)X^\{\\prime\}=\(D^\{\\prime\},P^\{\\prime\}\)are*compatible*if and only ifD=D′D=D^\{\\prime\}\. Compatible random variables are*distributionally equivalent*, writtenX≡X′X\\equiv X^\{\\prime\}, if and only ifP=P′P=P^\{\\prime\}\.

###### Definition 2\(Operations on Random Variables\)

LetXi=\(Di,Pi\)X\_\{i\}=\(D\_\{i\},P\_\{i\}\), fori=1,…,ni=1,\\ldots,n, be random variables with measurable domainsDi⊆ℝdiD\_\{i\}\\subseteq\\mathbb\{R\}^\{d\_\{i\}\}\. An operation on random variables is*distribution\-valued*if its result is a new random variable\. It is*deterministic\-valued*if its result is a deterministic quantity computed from one or more underlying probability measures rather than a random variable\.

###### Definition 3\(Divergence Test\[[27](https://arxiv.org/html/2607.18262#bib.bib87)\]\)

LetX=\(D,P\)X=\(D,P\)andX′=\(D,P′\)X^\{\\prime\}=\(D,P^\{\\prime\}\)be compatible random variables, and letΔ\\Deltabe a statistical divergence measure satisfyingΔ​\(X,X′\)=0\\Delta\(X,X^\{\\prime\}\)=0if and only ifX≡X′X\\equiv X^\{\\prime\}\. Given a toleranceϵ≥0\\epsilon\\geq 0and a parameterα∈\(0,1\)\\alpha\\in\(0,1\), the*divergence test*formulates compatibility as the hypothesis\-testing problemH0:Δ​\(X,X′\)≤ϵH\_\{0\}:\\Delta\(X,X^\{\\prime\}\)\\leq\\epsilonagainstH1:Δ​\(X,X′\)\>ϵH\_\{1\}:\\Delta\(X,X^\{\\prime\}\)\>\\epsilon\. The test regardsXXandX′X^\{\\prime\}as compatible whenH0H\_\{0\}is not rejected by the configured decision strategy\.

BecauseΔ\\Deltamay lack a closed form, ProbSPARQL makes the decision strategy configurable and evaluates its latency–accuracy trade\-off in Section[8](https://arxiv.org/html/2607.18262#S8)\.

## 4ProbSPARQL Syntax and Datatype Interpretation

We adopt standard RDF/SPARQL notation\[[43](https://arxiv.org/html/2607.18262#bib.bib26),[44](https://arxiv.org/html/2607.18262#bib.bib27)\]\. Let𝒱\\mathcal\{V\},ℐ\\mathcal\{I\},ℬ\\mathcal\{B\}, andℒ\\mathcal\{L\}be pairwise disjoint sets of variables, IRIs, blank nodes, and literals, respectively, and let𝒯=ℐ∪ℬ∪ℒ\\mathcal\{T\}=\\mathcal\{I\}\\cup\\mathcal\{B\}\\cup\\mathcal\{L\}\(as in\[[29](https://arxiv.org/html/2607.18262#bib.bib65)\]\)\. We partitionℒ\\mathcal\{L\}into two disjoint subsets,ℒ𝑟𝑑𝑓\\mathcal\{L\}\_\{\\mathit\{rdf\}\}andℒ𝑑𝑖𝑠𝑡\\mathcal\{L\}\_\{\\mathit\{dist\}\}, whereℒ𝑟𝑑𝑓\\mathcal\{L\}\_\{\\mathit\{rdf\}\}are ordinary RDF literals andℒ𝑑𝑖𝑠𝑡\\mathcal\{L\}\_\{\\mathit\{dist\}\}are RDF literals whose datatypes encode probability distributions\.

### 4\.1Representing Random Variables in RDF

We model a random variable as an RDF node linked to its domain and to a probabilistic distribution literal inℒ𝑑𝑖𝑠𝑡\\mathcal\{L\}\_\{\\mathit\{dist\}\}\(green in Figure[3](https://arxiv.org/html/2607.18262#S6.F3)\)\. This keeps the random variable addressable in the RDF graph while delegating distribution parameters to a typed literal\.

As a concrete instantiation, we introduce the*Gaussian Mixture Model literal*\(GMM literal\) with the custom datatype IRIuq:gmmLiteral\. Its value space is the set of Gaussian mixture models\. Gaussian mixtures are widely used as a flexible approximation for continuous probability densities while still supporting tractable computations\. Its lexical spaceLuq:gmmLiteralL\_\{\\texttt\{uq:gmmLiteral\}\}consists of JSON\-encoded lexical forms\[[4](https://arxiv.org/html/2607.18262#bib.bib38)\]\. Each lexical form must contain six mandatory fields:n\_components,dimensions,covariance\_type\("full","diag", or"spherical"\),weights,means, andcovariances\. Figure[2](https://arxiv.org/html/2607.18262#S4.F2)illustrates a concrete two\-dimensional GMM literal together with its JSON lexical form, corresponding density visualization, and mathematical form of the mixture\.

![Refer to caption](https://arxiv.org/html/2607.18262v1/x2.png)Figure 2:A 2D Gaussian mixture model instantiation\. The left pane shows the JSON literal encoding the distribution parameters, while the right pane visualizes the corresponding probability density together with the mathematical form of the mixture\.#### Interpretation\.

Each probabilistic datatype is associated with a lexical space, a value space, and a partial interpretation from its lexical space to its value space, defined only for datatype\-specific well\-formed lexical forms\. For the datatypeuq:gmmLiteral, let𝖦𝖬𝖬\\mathsf\{GMM\}denote the value space of Gaussian mixture models\. The partial interpretation⟦⋅⟧𝑑𝑖𝑠𝑡:Luq:gmmLiteral⇀𝖦𝖬𝖬\\llbracket\\cdot\\rrbracket\_\{\\mathit\{dist\}\}:L\_\{\\texttt\{uq:gmmLiteral\}\}\\rightharpoonup\\mathsf\{GMM\}maps a well\-formed JSON lexical form to the corresponding GMM\. If the JSON lexical form is malformed or violates the required field\-level constraints, the interpretation is undefined\. The formal GMM definition and well\-formedness constraints are provided in Section S\.3\.2 of the supplementary material\[[46](https://arxiv.org/html/2607.18262#bib.bib91)\]\.

*Extensible probabilistic datatypes\.*The same datatype\-interpretation scheme applies to other probabilistic datatypes: each datatype provides its own lexical syntax, well\-formedness constraints, value space, and partial interpretation, while the RDF modeling pattern remains unchanged\. Compositional data can be encoded usinguq:dirichletLiteral, for example\{"alphas": \[5\.0, 2\.0, 3\.0\]\}\. Histogram literals can be encoded usinguq:histogramLiteral, for example\{"dimensions": 1, "edges": \[0, 10, 20\], "weights": \[0\.3, 0\.7\]\}\.

### 4\.2Query Grammar and Expressions

ProbSPARQL follows the standard SPARQL graph\-pattern syntax\[[29](https://arxiv.org/html/2607.18262#bib.bib65)\]and extends it with probabilistic filter functions and aθ\\theta\-join withDIVJOINsyntax\. Non\-standard graph\-pattern and filter constructs are highlighted in blue\. A ProbSPARQL query is a graph pattern𝒫\\mathcal\{P\}generated by

𝒫\\displaystyle\\mathcal\{P\}\\:=B​G​P​∣Filterℱ​\(𝒫\)∣​ProjW​\(𝒫\)​∣Union​\(𝒫,𝒫\)∣​Join​\(𝒫,𝒫\)\\displaystyle:=BGP\\mid\\textsc\{Filter\}\_\{\\,\\mathcal\{F\}\}\(\\mathcal\{P\}\)\\mid\\textsc\{Proj\}\_\{\\,W\}\(\\mathcal\{P\}\)\\mid\\textsc\{Union\}\(\\mathcal\{P\},\\mathcal\{P\}\)\\mid\\textsc\{Join\}\(\\mathcal\{P\},\\mathcal\{P\}\)∣JoinΘ​\(𝒫,𝒫\)∣​Minus​\(𝒫,𝒫\)​∣Diffℱ​\(𝒫,𝒫\)∣​Optℱ​\(𝒫,𝒫\),\\displaystyle\\ \\mid\{\\color\[rgb\]\{0,0\.4453125,0\.69921875\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0\.4453125,0\.69921875\}\\textsc\{Join\}\_\{\\Theta\}\(\\mathcal\{P\},\\mathcal\{P\}\)\}\\mid\\textsc\{Minus\}\(\\mathcal\{P\},\\mathcal\{P\}\)\\mid\\textsc\{Diff\}\_\{\\,\\mathcal\{F\}\}\(\\mathcal\{P\},\\mathcal\{P\}\)\\mid\\textsc\{Opt\}\_\{\\,\\mathcal\{F\}\}\(\\mathcal\{P\},\\mathcal\{P\}\),whereW⊆𝒱W\\subseteq\\mathcal\{V\}is the finite set of projected variables\. Here,JoinΘ​\(𝒫,𝒫\)\{\\color\[rgb\]\{0,0\.4453125,0\.69921875\}\\definecolor\[named\]\{pgfstrokecolor\}\{rgb\}\{0,0\.4453125,0\.69921875\}\\textsc\{Join\}\_\{\\Theta\}\(\\mathcal\{P\},\\mathcal\{P\}\)\}denotes a parameterized join pattern with a finite setΘ\\Thetaof atomic divergence constraints\. Each constraint is written abstractly as?​x≅ϵ,α?​y?x\\cong\_\{\\epsilon,\\alpha\}?y, where?​x?xand?​y?yrange over variables from the left and rightΘ\\Theta\-join operands, respectively,ϵ\\epsilonis the divergence tolerance, andα\\alphais the decision parameter of the corresponding test\.

Filters inherit the standard SPARQL filter language and are extended with concrete probabilistic filter functions\. We useprob:divergenceTest\(E,E,ϵ,α\)\(E,E,\\epsilon,\\alpha\)for divergence testing andprob:sameDistribution\(E,E\)\(E,E\)for exact distributional equality\. For example,FILTER\(prob:divergenceTest\(?d1, ?d2, 0\.1, 0\.05\)\)invokes the divergence test withϵ=0\.1\\epsilon=0\.1andα=0\.05\\alpha=0\.05\. For the formal fragment considered, we restrict filters to safe expressions: for everyFilterℱ​\(𝒫\)\\textsc\{Filter\}\_\{\\,\\mathcal\{F\}\}\(\\mathcal\{P\}\), we requirevar⁡\(ℱ\)⊆var⁡\(𝒫\)\\operatorname\{var\}\(\\mathcal\{F\}\)\\subseteq\\operatorname\{var\}\(\\mathcal\{P\}\)\.

Following Definition[2](https://arxiv.org/html/2607.18262#Thmdefinition2), ProbSPARQL distinguishes two classes of probabilistic expressions:*distribution\-valued*expressionsEdistE^\{\\mathrm\{dist\}\}, which return random variables, and*deterministic\-valued*expressionsEdetE^\{\\mathrm\{det\}\}, which return deterministic values\. ProbSPARQL does not overload SPARQL arithmetic symbols such as\+\+and∗\*: arithmetic over scalar RDF literals remains unchanged, while arithmetic\-like operations over distribution\-valued literals are exposed through explicit distribution\-valued operators with datatype\-specific semantics\.

###### Definition 4\(Distribution\-Valued Expression\)

A distribution\-valued expressionEdistE^\{\\mathrm\{dist\}\}is generated by

Edist::=\\displaystyle E^\{\\mathrm\{dist\}\}:=?​x​∣c∣​scale​\(Edist,λ\)​∣shift​\(Edist,v\)∣​linearTransform​\(Edist,A,b\)\\displaystyle?x\\mid c\\mid\\mathrm\{scale\}\(E^\{\\mathrm\{dist\}\},\\lambda\)\\mid\\mathrm\{shift\}\(E^\{\\mathrm\{dist\}\},v\)\\mid\\mathrm\{linearTransform\}\(E^\{\\mathrm\{dist\}\},A,b\)∣marginal​\(Edist,S\)∣​joint​\(Edist,Edist\)∣mix​\(Edist,Edist,w\)\\displaystyle\\mid\\mathrm\{marginal\}\(E^\{\\mathrm\{dist\}\},S\)\\mid\\mathrm\{joint\}\(E^\{\\mathrm\{dist\}\},E^\{\\mathrm\{dist\}\}\)\\mid\\mathrm\{mix\}\(E^\{\\mathrm\{dist\}\},E^\{\\mathrm\{dist\}\},w\)∣multiply​\(Edist,Edist\)∣​fuse​\(Edist,Edist\)∣convolve​\(Edist,Edist\),\\displaystyle\\mid\\mathrm\{multiply\}\(E^\{\\mathrm\{dist\}\},E^\{\\mathrm\{dist\}\}\)\\mid\\mathrm\{fuse\}\(E^\{\\mathrm\{dist\}\},E^\{\\mathrm\{dist\}\}\)\\mid\\mathrm\{convolve\}\(E^\{\\mathrm\{dist\}\},E^\{\\mathrm\{dist\}\}\),where?​x∈𝒱?x\\in\\mathcal\{V\}must be bound to probabilistic literals, andc∈ℒ𝑑𝑖𝑠𝑡c\\in\\mathcal\{L\}\_\{\\mathit\{dist\}\}\.

###### Definition 5\(Deterministic\-Valued Expression\)

A deterministic\-valued expressionEdetE^\{\\mathrm\{det\}\}is generated by

Edet::=\\displaystyle E^\{\\mathrm\{det\}\}:=?​x​∣c∣​pdf​\(Edist,t\)​∣logpdf​\(Edist,t\)∣​cdf​\(Edist,t\)∣mean​\(Edist\)\\displaystyle?x\\mid c\\mid\\mathrm\{pdf\}\(E^\{\\mathrm\{dist\}\},t\)\\mid\\mathrm\{logpdf\}\(E^\{\\mathrm\{dist\}\},t\)\\mid\\mathrm\{cdf\}\(E^\{\\mathrm\{dist\}\},t\)\\mid\\mathrm\{mean\}\(E^\{\\mathrm\{dist\}\}\)∣std​\(Edist\)∣​quantile​\(Edist,p\)​∣map​\(Edist\)∣​modeCount​\(Edist\)\\displaystyle\\mid\\mathrm\{std\}\(E^\{\\mathrm\{dist\}\}\)\\mid\\mathrm\{quantile\}\(E^\{\\mathrm\{dist\}\},p\)\\mid\\mathrm\{map\}\(E^\{\\mathrm\{dist\}\}\)\\mid\\mathrm\{modeCount\}\(E^\{\\mathrm\{dist\}\}\)∣jsd\(Edist,Edist\),\\displaystyle\\mid\\mathrm\{jsd\}\(E^\{\\mathrm\{dist\}\},E^\{\\mathrm\{dist\}\}\),where?​x∈𝒱?x\\in\\mathcal\{V\}must be bound to ordinary RDF literals, andc∈ℒ𝑟𝑑𝑓c\\in\\mathcal\{L\}\_\{\\mathit\{rdf\}\}\.

The remaining symbols denote datatype\-specific operator parameters\. Their signatures, result types, dimensionality requirements, and compatibility conditions are given in Section S\.4 of the supplementary material\[[46](https://arxiv.org/html/2607.18262#bib.bib91)\]\.

## 5ProbSPARQL Query Semantics

We adopt the standard SPARQL algebra and evaluation framework\[[43](https://arxiv.org/html/2607.18262#bib.bib26)\]and specify below only the modifications relevant to probabilistic literals and divergence joins\.

### 5\.1Distributional Compatibility

A*solution mapping*is a partial functionμ:𝒱→𝒯\\mu:\\mathcal\{V\}\\to\\mathcal\{T\}with domaindom⁡\(μ\)\\operatorname\{dom\}\(\\mu\)\. The key modification to standard SPARQL lies in the treatment of equality on probabilistic literals\. We define a binary relation≡𝑑𝑖𝑠𝑡\\equiv\_\{\\mathit\{dist\}\}on𝒯\\mathcal\{T\}by settingt≡𝑑𝑖𝑠𝑡t′t\\equiv\_\{\\mathit\{dist\}\}t^\{\\prime\}iff either \(i\)t,t′∈𝒯∖ℒ𝑑𝑖𝑠𝑡t,t^\{\\prime\}\\in\\mathcal\{T\}\\setminus\\mathcal\{L\}\_\{\\mathit\{dist\}\}andt=t′t=t^\{\\prime\}, or \(ii\)t,t′∈ℒ𝑑𝑖𝑠𝑡t,t^\{\\prime\}\\in\\mathcal\{L\}\_\{\\mathit\{dist\}\}and⟦t⟧𝑑𝑖𝑠𝑡=⟦t′⟧𝑑𝑖𝑠𝑡\\llbracket t\\rrbracket\_\{\\mathit\{dist\}\}=\\llbracket t^\{\\prime\}\\rrbracket\_\{\\mathit\{dist\}\}\. In all other cases,t≢𝑑𝑖𝑠𝑡t′t\\not\\equiv\_\{\\mathit\{dist\}\}t^\{\\prime\}\. Two solution mappingsμ\\muandμ′\\mu^\{\\prime\}are compatible, writtenμ∼μ′\\mu\\sim\\mu^\{\\prime\}, iff∀?​x∈dom⁡\(μ\)∩dom⁡\(μ′\),μ​\(?​x\)≡𝑑𝑖𝑠𝑡μ′​\(?​x\)\.\\forall?x\\in\\operatorname\{dom\}\(\\mu\)\\cap\\operatorname\{dom\}\(\\mu^\{\\prime\}\),\\mu\(?x\)\\equiv\_\{\\mathit\{dist\}\}\\mu^\{\\prime\}\(?x\)\.

###### Definition 6

Ifμ∼μ′\\mu\\sim\\mu^\{\\prime\}, then their union, writtenμ⊎μ′\\mu\\uplus\\mu^\{\\prime\}, is the partial mapping overdom⁡\(μ\)∪dom⁡\(μ′\)\\operatorname\{dom\}\(\\mu\)\\cup\\operatorname\{dom\}\(\\mu^\{\\prime\}\)defined by\(μ⊎μ′\)​\(?​x\)=μ​\(?​x\)\(\\mu\\uplus\\mu^\{\\prime\}\)\(?x\)=\\mu\(?x\)for all?​x∈dom⁡\(μ\)?x\\in\\operatorname\{dom\}\(\\mu\)and\(μ⊎μ′\)​\(?​x\)=μ′​\(?​x\)\(\\mu\\uplus\\mu^\{\\prime\}\)\(?x\)=\\mu^\{\\prime\}\(?x\)for all?​x∈dom⁡\(μ′\)?x\\in\\operatorname\{dom\}\(\\mu^\{\\prime\}\)\.

Accordingly, ordinary join in ProbSPARQL follows the standard SPARQL join structure, while compatibility on shared probabilistic literals is interpreted via≡𝑑𝑖𝑠𝑡\\equiv\_\{\\mathit\{dist\}\}rather than by lexical identity of their encoded forms\.222For example, in a Gaussian mixture model withkkcomponents, the syntactic ordering of the components in the JSON encoding is irrelevant to the represented distribution\.

### 5\.2Expression Evaluation

The evaluation of an expressionEEunder a solution mappingμ\\mu, denoted⟦E⟧μ\\llbracket E\\rrbracket\_\{\\mu\}, is defined recursively\. Base cases are⟦?x⟧μ=μ\(?x\)\\llbracket?x\\rrbracket\_\{\\mu\}=\\mu\(?x\)for variables?​x∈𝒱?x\\in\\mathcal\{V\}and⟦c⟧μ=c\\llbracket c\\rrbracket\_\{\\mu\}=cfor literalsc∈ℒc\\in\\mathcal\{L\}\. For anynn\-ary built\-in operator𝗈𝗉\\mathsf\{op\},

⟦𝗈𝗉\(E1,…,En\)⟧μ=Op\(⟦E1⟧μ,…,⟦En⟧μ\),\\llbracket\\mathsf\{op\}\(E\_\{1\},\\dots,E\_\{n\}\)\\rrbracket\_\{\\mu\}=\\textsc\{Op\}\(\\llbracket E\_\{1\}\\rrbracket\_\{\\mu\},\\dots,\\llbracket E\_\{n\}\\rrbracket\_\{\\mu\}\),whereOpis the semantic function corresponding to𝗈𝗉\\mathsf\{op\}\. Detailed datatype\-specific operator definitions are provided in Section S\.4 of the supplementary material\[[46](https://arxiv.org/html/2607.18262#bib.bib91)\]\. Probabilistic operators are partial and yield𝑒𝑟𝑟𝑜𝑟\\mathit\{error\}whenever their datatype\-specific typing, lexical well\-formedness, dimensionality, or compatibility requirements are not satisfied\.

### 5\.3Probabilistic Filter Conditions

Standard SPARQL filter semantics are inherited, except that equality over probabilistic literals is interpreted via≡dist\\equiv\_\{\\mathrm\{dist\}\}\. Order comparisons such as<<,≤\\leq,\>\>, and≥\\geqremain scalar: they are defined only for orderable RDF literals and yield𝑒𝑟𝑟𝑜𝑟\\mathit\{error\}on probabilistic literals\. We specify below the additional probabilistic filter functions\.

For a filterℱ\\mathcal\{F\}and a solution mappingμ\\mu, let⟦ℱ⟧μ∈\{⊤,⊥,𝑒𝑟𝑟𝑜𝑟\}\\llbracket\\mathcal\{F\}\\rrbracket\_\{\\mu\}\\in\\\{\\top,\\bot,\\mathit\{error\}\\\}denote the truth\-value ofℱ\\mathcal\{F\}underμ\\mu\. Letvi=⟦Ei⟧μv\_\{i\}=\\llbracket E\_\{i\}\\rrbracket\_\{\\mu\}fori=1,2i=1,2\. When defined, writexi=⟦vi⟧𝑑𝑖𝑠𝑡x\_\{i\}=\\llbracket v\_\{i\}\\rrbracket\_\{\\mathit\{dist\}\}\. LetCμC\_\{\\mu\}hold iffv1v\_\{1\}andv2v\_\{2\}are well\-formed probabilistic literals andx1x\_\{1\}andx2x\_\{2\}are compatible\. For compatiblex1x\_\{1\}andx2x\_\{2\}, writeAccϵ,α​\(x1,x2\)\\mathrm\{Acc\}\_\{\\epsilon,\\alpha\}\(x\_\{1\},x\_\{2\}\)iff the null hypothesisH0:Δ​\(x1,x2\)≤ϵH\_\{0\}:\\Delta\(x\_\{1\},x\_\{2\}\)\\leq\\epsiloncannot be rejected under the configured decision strategy with parameterα\\alpha\. Then

⟦𝗉𝗋𝗈𝖻:𝖽𝗂𝗏𝖾𝗋𝗀𝖾𝗇𝖼𝖾𝖳𝖾𝗌𝗍\(E1,E2,ϵ,α\)⟧μ=\{⊤,if​Cμ​and​Accϵ,α​\(x1,x2\),⊥,if​Cμ​and​¬Accϵ,α​\(x1,x2\),𝑒𝑟𝑟𝑜𝑟,otherwise\.\\llbracket\\mathsf\{prob\{:\}divergenceTest\}\(E\_\{1\},E\_\{2\},\\epsilon,\\alpha\)\\rrbracket\_\{\\mu\}=\\begin\{cases\}\\top,&\\text\{if \}C\_\{\\mu\}\\text\{ and \}\\mathrm\{Acc\}\_\{\\epsilon,\\alpha\}\(x\_\{1\},x\_\{2\}\),\\\\ \\bot,&\\text\{if \}C\_\{\\mu\}\\text\{ and \}\\neg\\mathrm\{Acc\}\_\{\\epsilon,\\alpha\}\(x\_\{1\},x\_\{2\}\),\\\\ \\mathit\{error\},&\\text\{otherwise\}\.\\end\{cases\}The filter function𝗉𝗋𝗈𝖻:𝗌𝖺𝗆𝖾𝖣𝗂𝗌𝗍𝗋𝗂𝖻𝗎𝗍𝗂𝗈𝗇\\mathsf\{prob\{:\}sameDistribution\}provides an explicit form of distributional equality:

⟦𝗉𝗋𝗈𝖻:𝗌𝖺𝗆𝖾𝖣𝗂𝗌𝗍𝗋𝗂𝖻𝗎𝗍𝗂𝗈𝗇\(E1,E2\)⟧μ=\{⊤,if​Cμ​and​v1≡𝑑𝑖𝑠𝑡v2,⊥,if​Cμ​and​v1≢𝑑𝑖𝑠𝑡v2,𝑒𝑟𝑟𝑜𝑟,otherwise\.\\llbracket\\mathsf\{prob\{:\}sameDistribution\}\(E\_\{1\},E\_\{2\}\)\\rrbracket\_\{\\mu\}=\\begin\{cases\}\\top,&\\text\{if \}C\_\{\\mu\}\\text\{ and \}v\_\{1\}\\equiv\_\{\\mathit\{dist\}\}v\_\{2\},\\\\ \\bot,&\\text\{if \}C\_\{\\mu\}\\text\{ and \}v\_\{1\}\\not\\equiv\_\{\\mathit\{dist\}\}v\_\{2\},\\\\ \\mathit\{error\},&\\text\{otherwise\}\.\\end\{cases\}

### 5\.4Θ\\Theta\-Join

We next formalize the parameterized join⨝Θ\\Join\_\{\\Theta\}\. LetΘ\\Thetabe a finite set of atomic divergence constraints of the form?​x≅ϵ,α?​y?x\\cong\_\{\\epsilon,\\alpha\}?y, where?​x?xand?​y?yare variables from the two join operands,ϵ\\epsilonis a divergence tolerance, andα\\alphais the decision parameter of the associated divergence test\. For two solution mappingsμ\\muandμ′\\mu^\{\\prime\}, we writeμ,μ′⊧\(?​x≅ϵ,α?​y\)\\mu,\\mu^\{\\prime\}\\models\(?x\\cong\_\{\\epsilon,\\alpha\}?y\)iff?​x∈dom⁡\(μ\)?x\\in\\operatorname\{dom\}\(\\mu\),?​y∈dom⁡\(μ′\)?y\\in\\operatorname\{dom\}\(\\mu^\{\\prime\}\),μ​\(?​x\),μ′​\(?​y\)∈ℒ𝑑𝑖𝑠𝑡\\mu\(?x\),\\mu^\{\\prime\}\(?y\)\\in\\mathcal\{L\}\_\{\\mathit\{dist\}\}, the interpreted probabilistic values⟦μ\(?x\)⟧𝑑𝑖𝑠𝑡\\llbracket\\mu\(?x\)\\rrbracket\_\{\\mathit\{dist\}\}and⟦μ′\(?y\)⟧𝑑𝑖𝑠𝑡\\llbracket\\mu^\{\\prime\}\(?y\)\\rrbracket\_\{\\mathit\{dist\}\}are compatible, and the null hypothesisH0:Δ\(⟦μ\(?x\)⟧𝑑𝑖𝑠𝑡,⟦μ′\(?y\)⟧𝑑𝑖𝑠𝑡\)≤ϵH\_\{0\}:\\Delta\(\\llbracket\\mu\(?x\)\\rrbracket\_\{\\mathit\{dist\}\},\\llbracket\\mu^\{\\prime\}\(?y\)\\rrbracket\_\{\\mathit\{dist\}\}\)\\leq\\epsiloncannot be rejected at decision parameterα\\alpha\. We writeμ,μ′⊧Θ\\mu,\\mu^\{\\prime\}\\models\\Thetaiff every atomic constraint inΘ\\Thetais satisfied\. TheΘ\\Theta\-join of two sets of solution mappingsΩ1\\Omega\_\{1\}andΩ2\\Omega\_\{2\}is then defined by

Ω1⨝ΘΩ2=\{μ1⊎μ2∣μ1∈Ω1,μ2∈Ω2,μ1∼μ2,μ1,μ2⊧Θ\}\.\\Omega\_\{1\}\\Join\_\{\\Theta\}\\Omega\_\{2\}=\\left\\\{\\mu\_\{1\}\\uplus\\mu\_\{2\}\\mid\\mu\_\{1\}\\in\\Omega\_\{1\},\\ \\mu\_\{2\}\\in\\Omega\_\{2\},\\ \\mu\_\{1\}\\sim\\mu\_\{2\},\\ \\mu\_\{1\},\\mu\_\{2\}\\models\\Theta\\right\\\}\.

### 5\.5Evaluation Rules

The evaluation of a graph pattern𝒫\\mathcal\{P\}over an RDF graphGG, denoted⟦𝒫⟧G\\llbracket\\mathcal\{P\}\\rrbracket\_\{G\}, is defined as in standard SPARQL\[[43](https://arxiv.org/html/2607.18262#bib.bib26)\], except for the additional clause introduced by ProbSPARQL:⟦JoinΘ\(𝒫1,𝒫2\)⟧G=⟦𝒫1⟧G⨝Θ⟦𝒫2⟧G\.\\llbracket\\textsc\{Join\}\_\{\\Theta\}\(\\mathcal\{P\}\_\{1\},\\mathcal\{P\}\_\{2\}\)\\rrbracket\_\{G\}=\\llbracket\\mathcal\{P\}\_\{1\}\\rrbracket\_\{G\}\\Join\_\{\\Theta\}\\llbracket\\mathcal\{P\}\_\{2\}\\rrbracket\_\{G\}\.The remaining recursive evaluation clauses, together with the algebraic characterization of theΘ\\Theta\-join and its non\-trivial optimization rules are given in Section S\.9 of the supplement\[[46](https://arxiv.org/html/2607.18262#bib.bib91)\]\.

## 6Circular Manufacturing Vocabulary

Figure[3](https://arxiv.org/html/2607.18262#S6.F3)shows the vocabulary pattern for the circular\-factory KG\. It combines theAngle Grinder Subgraph, theInspection Measurement Subgraph, and theRandom Variable Representationwith four ontology modules: theProduct Ontology333[https://w3id\.org/circularfactory/PerpetualProduct](https://w3id.org/circularfactory/PerpetualProduct), theAngle Grinder Ontology, theInspection Measurement Ontology, and theUnits of Measurement Ontology\. TheProduct Ontologyprovides generic classes such ascfc:Product,cfc:Part, andcfc:Component, while theAngle Grinder Ontologyspecializes them throughag:AngleGrinderandag:CrownGear\. TheInspection Measurement Ontologycaptures concepts such asim:MeasurableCharacteristic,im:PointCloud, andim:StructuredLightMeasurement\. TheUnits of Measurement Ontology\[[30](https://arxiv.org/html/2607.18262#bib.bib88)\]supplies physical units such asom:millimetre\.

The instanceim:structuredLightMeasurement1in theInspection Measurement Subgraphis linked viauq:representedByto the random\-variable nodeuq:randomVariable1in theRandom Variable Representation, rather than being reduced to a detached uncertainty annotation\. This node records a non\-negative numeric domain viauq:hasDomainand a distribution encoded as auq:gmmLiteralviauq:hasDistribution\. This design keeps product structure, inspection provenance, scalar summaries, units, and uncertainty representation modular, while allowing graph patterns to retrieve the relevant distribution\-valued literal for probabilistic evaluation\.

![Refer to caption](https://arxiv.org/html/2607.18262v1/x3.png)Figure 3:Circular manufacturing instantiation of the ProbSPARQL modeling pattern, connecting theInspection Measurement Subgraphto theRandom Variable Representationwhile preserving product structure, point\-cloud input, scalar value, and unit information\.
## 7Implementation

Our open\-source prototype extends the Apache Jena\[[39](https://arxiv.org/html/2607.18262#bib.bib53)\]ARQ query processor to evaluate probabilistic literals and ProbSPARQL operators during query execution\.

The implementation introduces a polymorphic probabilistic datatype layer\. Each supported datatype is associated with a parser from its lexical form to an internal value representation, a serializer back to RDF literals, and datatype\-specific operator backends\. Foruq:gmmLiteral, JSON lexical forms are parsed into Gaussian mixture value objects used by backend routines for distribution evaluation, uncertainty propagation, and divergence estimation\. Additional datatypes are supported by registering their datatype\-specific parsers, serializers, value representations, and operator backends, including divergence or distance routines for comparison\-oriented operators\.

These extensions allow probabilistic expressions to be evaluated inside the SPARQL engine, rather than exporting distribution literals to an external application for post\-processing\. This in\-engine evaluation reduces serialization and data\-transfer overhead\. The extended processor is deployed as an Apache Jena Fuseki endpoint and serves as the execution platform for the experimental evaluation in Section[8](https://arxiv.org/html/2607.18262#S8)\.

## 8Evaluation

We evaluate ProbSPARQL through four questions derived from the circular\-factory query\-layer requirements: \(EQ1\) Can project\-derived circular\-factory measurement data be represented and queried through the proposed random\-variable and probabilistic\-literal pattern? \(EQ2\) What overhead is introduced by probabilistic literals and operators inside the SPARQL execution layer? \(EQ3\) Does in\-engine probabilistic filtering provide an integration benefit over application\-layer post\-processing? \(EQ4\) Which latency–accuracy trade\-offs arise for divergence\-based matching in component\-pairing workloads?

We answer EQ1 on a real circular\-factory KG fragment derived from project measurement data\. We answer EQ2–EQ4 on controlled ontology\-conformant benchmarks\. For mixture\-complexity and operator\-overhead experiments, we use a fixed graph with 1,000 angle\-grinder instances and approximately 300k triples, varying the number of Gaussian mixture componentsK∈\{1,3,5,10\}K\\in\\\{1,3,5,10\\\}\. For graph\-size scalability, we fixK=3K=3and vary the generated graph from 10 angle\-grinder instances with 3,078 triples to 5,000 instances with 1,535,008 triples\. The schema mirrors our circular\-factory KG infrastructure\[[26](https://arxiv.org/html/2607.18262#bib.bib28)\], while the generated distributions provide controlled workloads for comparing probabilistic query operators\. All experiments were run on a consumer\-grade laptop444Apple MacBook Pro with M4 Pro chip, 48 GB memory, macOS, Java 21, Jena 6\.0\.0\.\. We report the median of 10 runs after 3 warm\-up iterations; standard deviations across runs were below 5% of the median for all measurements\. Extended protocols, full result tables, complete query templates, workload\-construction details, divergence\-decision pseudocode, graph\-size scalability, dimensionality scaling, and datatype\-extensibility experiments are provided in Sections S\.5–S\.8 of the supplement\[[46](https://arxiv.org/html/2607.18262#bib.bib91)\]\.

We compare against baseline SPARQL execution whenever a deterministic counterpart is available\. We do not use existing probabilistic database systems as runtime baselines because their data models and execution architectures are not directly comparable to RDF graphs with distribution\-valued probabilistic literals\. Such a comparison would conflate probabilistic\-literal overhead with differences in data model, encoding, and query execution\. Moreover, these systems do not provide the SPARQL\-compatible divergence join required by R4\.

### 8\.1Pilot Validation on Real Circular\-Factory Measurement Data

To answer EQ1, we validate ProbSPARQL on a project\-derived circular\-factory KG fragment\. The graph stores material\-analysis data for drive and spindle gears, including Barkhausen\-noise root\-mean\-square \(RMS\) signal measurements, residual\-stress measurements, hardness measurements, and process conditions\. Uncertain material\-state measurements are represented as random variables with GMM\-encoded probabilistic RDF literals, while empirical roughness distributions are represented through histogram literals\. The graph contains 4,037 triples across project\-derived measurement subsets, with 97 sample resources, 20 process\-condition resources, 280 measurement resources, and 163 random\-variable resources\. The main text shows R4 because it exercises the dedicatedDIVJOINoperator\. The query intent and representative bindings were reviewed with material\-analysis and reassembly\-planning partners outside the ProbSPARQL implementation team and matched the intended hardness\-based scenario\.

1SELECT?driveGear?spindleGearWHERE\{

2\{?driveGearaag:DriveGear;im:hasHardCharacteristic?hardcharA\.

3?hardcharAim:isMeasuredBy?measA\.

4?measAom:hasUnitom:HardnessVickers;uq:representedBy?rvA\.

5?rvAuq:hasDomainuq:nonNegativeDouble;uq:hasDistribution?distA\.\}

6DIVJOIN\(?distA,?distB,0\.3,0\.05\)

7\{?spindleGearaag:SpindleGear;im:hasHardCharacteristic?hardcharB\.

8?hardcharBim:isMeasuredBy?measB\.

9?measBom:hasUnitom:HardnessVickers;uq:representedBy?rvB\.

10?rvBuq:hasDomainuq:nonNegativeDouble;uq:hasDistribution?distB\.\}

11\}

Listing 1:R4: Divergence join for component matchingListing[1](https://arxiv.org/html/2607.18262#LST1)retrieves candidate drive–spindle gear pairs, follows the measurement links to their hardness random variables, and extracts the corresponding probabilistic hardness distributions\. The callDIVJOIN\(?distA, ?distB, 0\.3, 0\.05\)evaluates whetherH0:Δ≤ϵH\_\{0\}:\\Delta\\leq\\epsiloncannot be rejected for the two distributions, usingϵ=0\.3\\epsilon=0\.3and configured decision parameterα=0\.05\\alpha=0\.05\. Matching is based on distributional compatibility rather than scalar equality\. This real\-data execution demonstrates that project\-derived hardness distributions can be retrieved and matched directly by the probabilistic join operator\. Runtime behavior and scalability are evaluated next on the generated ontology\-conformant graphs\.

### 8\.2System Overhead and Filter Pushdown

ProbSPARQL adds probabilistic literals and operators without changing the surrounding graph\-pattern structure\. We therefore isolate its latency overhead using deterministic counterparts for retrieval, threshold\-based filtering \(R1\), and uncertainty\-propagating arithmetic \(R3\)\. Distribution comparison \(R2\) is reported separately because it is dominated by Monte Carlo estimation and has no direct deterministic counterpart\.

Execution overhead across mixture complexities\.Figure[5](https://arxiv.org/html/2607.18262#S8.F5)shows that retrieval incurs a1\.251\.25–1\.42×1\.42\{\\times\}overhead across allKKvalues, driven primarily by JSON literal deserialization\. CDF\-based filtering adds per\-literalO​\(K\)O\(K\)CDF evaluations, raising the overhead slightly to1\.431\.43–1\.60×1\.60\{\\times\}\. Uncertainty\-propagating arithmetic reaches1\.701\.70–1\.95×1\.95\{\\times\}due to theO​\(K2\)O\(K^\{2\}\)closed\-form moment computations inprob:multiply\. Across all three query types, the weak dependence onKKindicates that parsing cost dominates the lightweight mathematical operations\.

![Refer to caption](https://arxiv.org/html/2607.18262v1/Fig/latency_overhead.png)Figure 4:Execution latency for deterministic \(Det\. Literal\) and probabilistic \(GMM Literal\) queries across mixture complexitiesK∈\{1,3,5,10\}K\\in\\\{1,3,5,10\\\}\. Distribution comparison is omitted due to its distinct sampling\-based scaling behavior\.
![Refer to caption](https://arxiv.org/html/2607.18262v1/Fig/inEngine.png)Figure 5:Latency comparison for CDF\-based probabilistic filtering\. In\-engine evaluation avoids serialization and application\-layer data transfer, achieving2\.0×2\.0\{\\times\}–3\.5×3\.5\{\\times\}speedups depending on filter selectivity\.

In\-engine vs\. post\-processing performance\.After quantifying the basic overhead of probabilistic literals and operators, we next measure the serialization and transfer costs avoided by in\-engine probabilistic filtering\. The*in\-engine*strategy appliesFILTER\(prob:cdf\(?d, 9\.8\) \>= 0\.9\)before joining with downstream patterns\. The*post\-processing*baseline retrieves all distributions and evaluates the CDF filter using Python’s SciPy\. Figure[5](https://arxiv.org/html/2607.18262#S8.F5)reports median latencies across four filter selectivities spanning the range observed in our dataset, from highly selective \(1% of components flagged\) to moderately selective \(30%\)\. Post\-processing latency remains at approximately 1 800 ms regardless of selectivity, as the entire set of GMM literals must be serialized and transferred over the network before filtering begins\. In\-engine evaluation applies the filter before serialization and transfer, achieving a3\.41×3\.41\{\\times\}speedup at a selectivity of 0\.01, and maintaining a1\.96×1\.96\{\\times\}advantage even at 0\.30\.

The standalone distribution\-comparison workload, driven by Monte Carlo sampling, scales from 1 853 ms \(K=1K\{=\}1\) to 10 744 ms \(K=10K\{=\}10\)\. This cost becomes critical in use case R4, where divergence\-test joins evaluate candidate pairs generated by the join operands\. Section[8\.3](https://arxiv.org/html/2607.18262#S8.SS3)evaluates decision strategies that reduce the per\-candidate\-pair cost of divergence joins\.

### 8\.3Divergence Join Optimization

The divergence join evaluates theΘ\\Theta\-join condition over candidate pairs generated from the two join operands\. In the prototype, this operator is exposed through theDIVJOINsyntax shown in Listing[1](https://arxiv.org/html/2607.18262#LST1)\. While the operator retains a quadratic worst\-case pair enumeration, we exploit a key algorithmic insight: evaluatingH0:Δ≤ϵH\_\{0\}:\\Delta\\leq\\epsilonis a binary decision\. This avoids computing high\-precision divergence estimates for every candidate pair and enables substantial reductions in candidate\-pair decision time via bounding\[[5](https://arxiv.org/html/2607.18262#bib.bib76)\]and early\-stopping mechanisms\[[41](https://arxiv.org/html/2607.18262#bib.bib75)\]\.

To evaluate these optimizations, we compare five decision strategies across controlled candidate\-pair workloads with high\-precision reference JSD values\[[10](https://arxiv.org/html/2607.18262#bib.bib37)\], grouped by decision difficulty\. The*Hard*workload comprises near\-threshold pairs whose reference JSD lies withinϵ±0\.05\\epsilon\\pm 0\.05, where estimation variance is likely to affect the binary decision\. The*Mixed*workload provides a balanced baseline by combining pairs from all difficulty levels in equal proportions\. Table[1](https://arxiv.org/html/2607.18262#S8.T1)reports F1 and MAE over each workload, together with the median decision time per candidate pair\.

Sampling strategies\.The fixed\-budget Monte Carlo baselines useN=10,000N=10\{,\}000samples per candidate pair\.*MC\-Naive*applies standard Monte Carlo estimation\.*MC\-Strat*uses variance\-reduced stratified sampling\[[31](https://arxiv.org/html/2607.18262#bib.bib73)\]\.*SPRT*applies Wald’s sequential probability ratio test\[[41](https://arxiv.org/html/2607.18262#bib.bib75)\]for early termination in hypothesis testing, configured with significance levelα=0\.05\\alpha=0\.05\.*Det\-Bound*uses an analytic lower bound on JSD derived from the Data Processing Inequality\[[5](https://arxiv.org/html/2607.18262#bib.bib76)\]as a deterministic rejection test\. When used standalone, unresolved cases are treated as non\-rejections ofH0H\_\{0\}\.*Adaptive\-Cascade*first applies Det\-Bound to rejectH0H\_\{0\}when possible, then triggers SPRT for sequential evaluation, and finally falls back to MC\-Strat for ambiguous boundary cases that exhaust the sample budget\.

Table 1:Divergence\-test strategies on Hard and Mixed workloads\. Results report F1 \(%\), MAE, and median per\-candidate\-pair latency \(ms\)\.Results\.Table[1](https://arxiv.org/html/2607.18262#S8.T1)compares five strategies across the two workloads\. Since the framework is strategy\-agnostic, users may select a strategy according to their latency and accuracy requirements\. We use Adaptive\-Cascade as the default strategy because it offers the most balanced performance across decision difficulties\.

On the*Hard*workload, MC\-Strat achieves the highest F1 score \(97\.7%\) but requires 23\.34 ms per candidate\-pair decision\. Adaptive\-Cascade lowers the decision latency to 3\.27 ms while maintaining a 95\.9% F1 score\. This yields a7\.9×7\.9\{\\times\}speedup over MC\-Naive and improves over standalone SPRT in F1 score \(94\.3%\)\. On the*Mixed*workload, Adaptive\-Cascade reaches 99\.0% F1 with a 1\.04 ms per\-candidate\-pair latency, yielding a24\.1×24\.1\{\\times\}speedup over MC\-Naive\.

Although Det\-Bound is nearly cost\-free, its conservative lower bound misses many near\-threshold decisions on the*Hard*workload\. Adaptive\-Cascade therefore uses it only as an initial rejection step and falls back to SPRT and MC\-Strat for unresolved cases, improving F1 to 95\.9% while retaining low latency\.

## 9Related Work

*Similarity\-Aware Extensions of SPARQL\.*Several works extend SPARQL with similarity mechanisms over strings\[[23](https://arxiv.org/html/2607.18262#bib.bib58)\], scalar attributes\[[13](https://arxiv.org/html/2607.18262#bib.bib59),[14](https://arxiv.org/html/2607.18262#bib.bib60)\], nearest\-neighbour joins\[[3](https://arxiv.org/html/2607.18262#bib.bib61)\], or embedding vectors\[[25](https://arxiv.org/html/2607.18262#bib.bib62)\]\. These approaches make approximate matching available for deterministic objects, whereas ProbSPARQL targets uncertain numeric measurements whose values are probability distributions\. The resulting query problem is therefore distributional compatibility between uncertain measurements, rather than approximate matching between deterministic values\.

*Probabilistic Relational Databases\.*A foundational line of probabilistic database work reasons about tuple\-level uncertainty and query answers through possible\-world or lineage\-based semantics\[[8](https://arxiv.org/html/2607.18262#bib.bib16),[2](https://arxiv.org/html/2607.18262#bib.bib63),[18](https://arxiv.org/html/2607.18262#bib.bib50)\]\. Other systems support attribute\-level uncertainty or simulation\-based query evaluation\[[35](https://arxiv.org/html/2607.18262#bib.bib51),[20](https://arxiv.org/html/2607.18262#bib.bib52)\], and interval\- or bound\-based approaches improve tractability for complex queries\[[12](https://arxiv.org/html/2607.18262#bib.bib77)\]\. These systems provide foundations for uncertain data management in relational schemas, whereas ProbSPARQL targets RDF graphs where uncertain numeric measurements must remain connected to component identity, units, provenance, and other KG context\.

*Uncertainty in RDF Knowledge Graphs\.*A major line of work on uncertainty in RDF and knowledge graphs attaches uncertainty to triples or annotations\[[15](https://arxiv.org/html/2607.18262#bib.bib40),[40](https://arxiv.org/html/2607.18262#bib.bib43),[11](https://arxiv.org/html/2607.18262#bib.bib20),[47](https://arxiv.org/html/2607.18262#bib.bib54),[36](https://arxiv.org/html/2607.18262#bib.bib47)\], while probabilistic ontology formalisms attach uncertainty to axioms or derived entailments\[[7](https://arxiv.org/html/2607.18262#bib.bib44),[24](https://arxiv.org/html/2607.18262#bib.bib45),[16](https://arxiv.org/html/2607.18262#bib.bib46)\]\. These frameworks primarily model uncertainty about the truth, confidence, or degree of graph statements\. ProbSPARQL addresses a different layer in which graph statements remain deterministic, while uncertain numeric measurement values are queried as probability distributions within SPARQL query evaluation\.

*Ontologies for Sensor and Numeric Data\.*Vocabularies such as ProbOnto\[[37](https://arxiv.org/html/2607.18262#bib.bib30)\], SSN\[[22](https://arxiv.org/html/2607.18262#bib.bib4)\], and the Data Cube\[[6](https://arxiv.org/html/2607.18262#bib.bib15)\]describe probability distributions, sensor observations, provenance, and statistical datasets in RDF\. These vocabularies provide a descriptive layer, but not executable SPARQL semantics for querying and combining distribution\-valued measurements\. ProbSPARQL complements them with query\-layer operations over distribution\-valued probabilistic literals\.

## 10Discussion, Impact, and Conclusion

*Practical gap\.*ProbSPARQL addresses the gap between uncertain numeric measurements produced by circular\-factory inspection and manufacturing workflows and the deterministic literal model of current RDF/SPARQL systems\. It keeps measurement uncertainty connected to the surrounding product, component, unit, and measurement context, while making probabilistic filtering, propagation, comparison, and matching executable inside the SPARQL query layer\.

*Integration impact\.*The main integration benefit is that uncertainty\-aware query operations remain inside the KG workflow rather than a separate application\-layer pipeline\. An ordinary SPARQL pipeline with external numerical processing must retrieve candidate distributions before filtering or matching can be applied\. Expanding distribution parameters into RDF triples keeps the data inside the graph, but it makes query patterns verbose and shifts datatype\-specific operations into query logic or external code\. A separate probabilistic data store can support numerical processing, but it separates uncertainty handling from the product, component, unit, and measurement context\. ProbSPARQL keeps these operations within the same KG interface for downstream uncertainty\-aware access\.

*Pilot status\.*The prototype is an active query\-layer pilot being integrated into the SFB 1574 circular\-factory infrastructure, not yet a production service\. Current and scheduled take\-up spans the data\-infrastructure, inspection, material\-analysis, planning/control, reliability\-modeling, and reassembly\-planning teams\. The pilot validates the modeling pattern and partner\-reviewed query templates on project\-derived measurement fragments\. The evaluation demonstrates feasible in\-engine execution, effective filter pushdown, configurable divergence\-join strategies, and datatype extensibility across three probabilistic representations\.

*Lessons and limitations\.*The pilot yields three lessons\. Distribution\-valued literals need explicit random\-variable nodes to preserve domain, unit, and measurement context\. Selective probabilistic filters benefit from in\-engine evaluation by avoiding candidate\-distribution transfer\. Divergence\-based matching needs configurable decision strategies for module\-specific latency and accuracy requirements\. Remaining challenges include worst\-case quadratic candidate enumeration for divergence joins, compatible\-domain mappings for cross\-datatype comparison, and validation with downstream engineering modules\. Future work will investigate index\-supported pruning, richer unit and domain checks, and feedback from broader pilot deployment in circular\-factory workflows\.

*Supplemental Material Statement:*The supplement\[[46](https://arxiv.org/html/2607.18262#bib.bib91)\]provides detailed formal semantics, complete query workloads, extended evaluation, and algebraic proofs\.

## Declaration of use of Generative AI

ChatGPT was used to assist with language editing of this Work, including grammar, wording, clarity, and stylistic refinement, and to provide suggestions for improving figure readability\. All AI\-assisted edits and suggestions were reviewed and verified by the authors, who remain fully responsible for the content of this Work\.

## References

- \[1\]C\. C\. Aggarwal and P\. S\. Yu\(2009\)A survey of uncertain data algorithms and applications\.IEEE Trans\. Knowl\. Data Eng\.21\(5\),pp\. 609–623\.External Links:[Document](https://dx.doi.org/10.1109/TKDE.2008.190)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1)\.
- \[2\]P\. Agrawal, O\. Benjelloun, A\. D\. Sarma, C\. Hayworth, S\. U\. Nabar, T\. Sugihara, and J\. Widom\(2006\)Trio: A system for data, uncertainty, and lineage\.InVLDB,pp\. 1151–1154\.External Links:[Link](http://dl.acm.org/citation.cfm?id=1164231)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p2.1)\.
- \[3\]D\. Arroyuelo, B\. Bustos, A\. Gómez\-Brandón, A\. Hogan, G\. Navarro, and J\. L\. Reutter\(2024\)Worst\-case\-optimal similarity joins on graph databases\.Proc\. ACM Manag\. Data2\(1\),pp\. 39:1–39:26\.External Links:[Document](https://dx.doi.org/10.1145/3639294)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p1.1)\.
- \[4\]T\. Bray\(2017\-12\)The JavaScript Object Notation \(JSON\) Data Interchange Format\.Request for Comments,RFC Editor\.Note:RFC 8259External Links:[Document](https://dx.doi.org/10.17487/RFC8259)Cited by:[§4\.1](https://arxiv.org/html/2607.18262#S4.SS1.p2.1)\.
- \[5\]T\. M\. Cover and J\. A\. Thomas\(2006\)Elements of information theory \(2\. ed\.\)\.Wiley\.External Links:[Document](https://dx.doi.org/10.1002/047174882X)Cited by:[§8\.3](https://arxiv.org/html/2607.18262#S8.SS3.p1.2),[§8\.3](https://arxiv.org/html/2607.18262#S8.SS3.p3.4)\.
- \[6\]R\. Cyganiak and D\. Reynolds\(2014\-01\)The RDF Data Cube Vocabulary\.Note:W3C RecommendationExternal Links:[Link](https://www.w3.org/TR/vocab-data-cube/)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p4.1)\.
- \[7\]P\. C\. G\. da Costa, K\. B\. Laskey, and K\. J\. Laskey\(2008\)PR\-OWL: A bayesian ontology language for the semantic web\.InURSW \(LNCS Vol\.\),Lecture Notes in Computer Science, Vol\.5327,pp\. 88–107\.External Links:[Document](https://dx.doi.org/10.1007/978-3-540-89765-1%5F6)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p3.1)\.
- \[8\]N\. N\. Dalvi, C\. Ré, and D\. Suciu\(2009\)Probabilistic databases: diamonds in the dirt\.Commun\. ACM52\(7\),pp\. 86–94\.External Links:[Document](https://dx.doi.org/10.1145/1538788.1538810)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p2.1)\.
- \[9\]A\. Deshpande, C\. Guestrin, S\. Madden, J\. M\. Hellerstein, and W\. Hong\(2004\)Model\-driven data acquisition in sensor networks\.InVLDB,pp\. 588–599\.External Links:[Document](https://dx.doi.org/10.1016/B978-012088469-8.50053-X)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1)\.
- \[10\]D\. M\. Endres and J\. E\. Schindelin\(2003\)A new metric for probability distributions\.IEEE Trans\. Inf\. Theory49\(7\),pp\. 1858–1860\.External Links:[Document](https://dx.doi.org/10.1109/TIT.2003.813506)Cited by:[§8\.3](https://arxiv.org/html/2607.18262#S8.SS3.p2.1)\.
- \[11\]H\. Fang\(2019\)PSPARQL: A querying language for probabilistic RDF data\.Complex\.2019,pp\. 8258197:1–8258197:7\.External Links:[Document](https://dx.doi.org/10.1155/2019/8258197)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p3.1)\.
- \[12\]S\. Feng, B\. Glavic, A\. Huber, and O\. A\. Kennedy\(2021\)Efficient uncertainty tracking for complex queries with attribute\-level bounds\.InSIGMOD Conference,pp\. 528–540\.External Links:[Document](https://dx.doi.org/10.1145/3448016.3452791)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p2.1)\.
- \[13\]S\. Ferrada, B\. Bustos, and A\. Hogan\(2020\)Extending SPARQL with similarity joins\.InISWC \(1\),Lecture Notes in Computer Science, Vol\.12506,pp\. 201–217\.External Links:[Document](https://dx.doi.org/10.1007/978-3-030-62419-4%5F12)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p1.1)\.
- \[14\]S\. Ferrada, B\. Bustos, and A\. Hogan\(2024\)Similarity joins and clustering for SPARQL\.Semantic Web15\(5\),pp\. 1701–1732\.External Links:[Document](https://dx.doi.org/10.3233/SW-243540)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p1.1)\.
- \[15\]Y\. Fukushige\(2005\)Representing probabilistic relations in RDF\.InISWC\-URSW,pp\. 106–107\.External Links:[Link](https://ceur-ws.org/Vol-173/pos_paper5.pdf)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p3.1)\.
- \[16\]R\. Giugno and T\. Lukasiewicz\(2002\)P\-SHOQ\(D\): A probabilistic extension of SHOQ\(D\) for probabilistic ontologies in the semantic web\.InJELIA,Lecture Notes in Computer Science, Vol\.2424,pp\. 86–97\.External Links:[Document](https://dx.doi.org/10.1007/3-540-45757-7%5F8)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p3.1)\.
- \[17\]A\. Hogan, E\. Blomqvist, M\. Cochez, C\. d’Amato, G\. de Melo, C\. Gutierrez, S\. Kirrane, J\. E\. L\. Gayo, R\. Navigli, S\. Neumaier, A\. N\. Ngomo, A\. Polleres, S\. M\. Rashid, A\. Rula, L\. Schmelzeisen, J\. Sequeda, S\. Staab, and A\. Zimmermann\(2021\)Knowledge graphs\.Synthesis Lectures on Data, Semantics, and Knowledge,Morgan & Claypool Publishers\.External Links:[Document](https://dx.doi.org/10.2200/S01125ED1V01Y202109DSK022)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1)\.
- \[18\]J\. Huang, L\. Antova, C\. Koch, and D\. Olteanu\(2009\)MayBMS: a probabilistic database management system\.InSIGMOD Conference,pp\. 1071–1074\.External Links:[Document](https://dx.doi.org/10.1145/1559845.1559984)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p2.1)\.
- \[19\]International Organization for Standardization\(2024\)ISO/iec 39075 — information technology — database languages — graph query language \(gql\)\.Note:[https://www\.iso\.org/standard/76120\.html](https://www.iso.org/standard/76120.html)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p4.1)\.
- \[20\]R\. Jampani, F\. Xu, M\. Wu, L\. L\. Perez, C\. M\. Jermaine, and P\. J\. Haas\(2008\)MCDB: a monte carlo approach to managing uncertain data\.InSIGMOD Conference,pp\. 687–700\.External Links:[Document](https://dx.doi.org/10.1145/1376616.1376686)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p2.1)\.
- \[21\]K\. Janowicz, A\. Haller, S\. J\. D\. Cox, D\. L\. Phuoc, and M\. Lefrançois\(2019\)SOSA: A lightweight ontology for sensors, observations, samples, and actuators\.J\. Web Semant\.56,pp\. 1–10\.External Links:[Document](https://dx.doi.org/10.1016/J.WEBSEM.2018.06.003)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1)\.
- \[22\]K\. Janowicz, A\. Haller, S\. Cox, D\. L\. Phuoc, and K\. Taylor\(2017\)Semantic sensor network ontology\.Note:W3C Recommendation[https://www\.w3\.org/TR/vocab\-ssn/](https://www.w3.org/TR/vocab-ssn/)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p4.1),[§9](https://arxiv.org/html/2607.18262#S9.p4.1)\.
- \[23\]C\. Kiefer, A\. Bernstein, and M\. Stocker\(2007\)The fundamentals of isparql: A virtual triple approach for similarity\-based semantic web tasks\.InISWC/ASWC,Lecture Notes in Computer Science, Vol\.4825,pp\. 295–309\.External Links:[Document](https://dx.doi.org/10.1007/978-3-540-76298-0%5F22)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p1.1)\.
- \[24\]D\. Koller, A\. Y\. Levy, and A\. Pfeffer\(1997\)P\-CLASSIC: A tractable probablistic description logic\.InAAAI/IAAI,pp\. 390–397\.External Links:[Link](http://www.aaai.org/Library/AAAI/1997/aaai97-060.php)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p3.1)\.
- \[25\]M\. Kulmanov, S\. Kafkas, A\. Karwath, A\. Malic, G\. V\. Gkoutos, M\. Dumontier, and R\. Hoehndorf\(2018\)Vec2SPARQL: integrating SPARQL queries and knowledge graph embeddings\.InSWAT4LS,CEUR Workshop Proceedings, Vol\.2275\.External Links:[Link](https://ceur-ws.org/Vol-2275/paper12.pdf)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p1.1)\.
- \[26\]G\. Lanza, B\. Deml, S\. Matthiesen, M\. Martin, O\. Brützel, and R\. Hörsting\(2024\)The vision of the circular factory for the perpetual innovative product\.at – Automatisierungstechnik72\(9\),pp\. 774–788\.External Links:[Document](https://dx.doi.org/10.1515/auto-2024-0012)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p2.1),[§8](https://arxiv.org/html/2607.18262#S8.p2.2)\.
- \[27\]E\. L\. Lehmann and J\. P\. Romano\(2005\)Testing statistical hypotheses\.3rd edition,Springer Texts in Statistics,Springer,New York, NY\.External Links:ISBN 978\-0\-387\-27605\-2,[Document](https://dx.doi.org/10.1007/0-387-27605-X)Cited by:[Definition 3](https://arxiv.org/html/2607.18262#Thmdefinition3)\.
- \[28\]H\. Patni, C\. A\. Henson, and A\. P\. Sheth\(2010\)Linked sensor data\.InCTS,pp\. 362–370\.External Links:[Document](https://dx.doi.org/10.1109/CTS.2010.5478492)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1)\.
- \[29\]J\. Pérez, M\. Arenas, and C\. Gutierrez\(2009\)Semantics and complexity of SPARQL\.ACM Trans\. Database Syst\.34\(3\),pp\. 16:1–16:45\.External Links:[Document](https://dx.doi.org/10.1145/1567274.1567278)Cited by:[§4\.2](https://arxiv.org/html/2607.18262#S4.SS2.p1.2),[§4](https://arxiv.org/html/2607.18262#S4.p1.10)\.
- \[30\]H\. Rijgersberg, M\. van Assem, and J\. Top\(2013\-01\)Ontology of units of measure and related concepts\.Semant\. Web4\(1\),pp\. 3–13\.External Links:ISSN 1570\-0844,[Document](https://dx.doi.org/10.3233/SW-2012-0069)Cited by:[§6](https://arxiv.org/html/2607.18262#S6.p1.1)\.
- \[31\]C\. P\. Robert and G\. Casella\(2004\)Monte carlo statistical methods\.Springer Texts in Statistics,Springer\.External Links:[Document](https://dx.doi.org/10.1007/978-1-4757-4145-2)Cited by:[§8\.3](https://arxiv.org/html/2607.18262#S8.SS3.p3.4)\.
- \[32\]V\. K\. Rohatgi and A\. K\. Md\. E\. Saleh\(2015\)An introduction to probability and statistics\.3rd edition,John Wiley & Sons\.External Links:ISBN 978\-1\-118\-79964\-2,[Document](https://dx.doi.org/10.1002/9781118799635)Cited by:[Definition 1](https://arxiv.org/html/2607.18262#Thmdefinition1)\.
- \[33\]M\. Rotmensch, Y\. Halpern, A\. Tlimat,et al\.\(2017\)Learning a health knowledge graph from electronic medical records\.Scientific Reports7,pp\. 5994\.External Links:[Document](https://dx.doi.org/10.1038/s41598-017-05778-z)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1),[§1](https://arxiv.org/html/2607.18262#S1.p3.1)\.
- \[34\]\(2024\)SFB 1574: Circular Factory for the Perpetual Product\.Note:[https://www\.sfb1574\.kit\.edu/english/index\.php](https://www.sfb1574.kit.edu/english/index.php)Funded by the German Research Foundation \(DFG\)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p2.1)\.
- \[35\]S\. Singh, C\. Mayfield, S\. Mittal, S\. Prabhakar, S\. E\. Hambrusch, and R\. Shah\(2008\)Orion 2\.0: native support for uncertain data\.InSIGMOD Conference,pp\. 1239–1242\.External Links:[Document](https://dx.doi.org/10.1145/1376616.1376744)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p2.1)\.
- \[36\]U\. Straccia\(2009\)A minimal deductive system for general fuzzy RDF\.InRR,Lecture Notes in Computer Science, Vol\.5837,pp\. 166–181\.External Links:[Document](https://dx.doi.org/10.1007/978-3-642-05082-4%5F12)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p3.1)\.
- \[37\]M\. J\. Swat, P\. Grenon, and S\. M\. Wimalaratne\(2016\)ProbOnto: ontology and knowledge base of probability distributions\.Bioinform\.32\(17\),pp\. 2719–2721\.External Links:[Document](https://dx.doi.org/10.1093/BIOINFORMATICS/BTW170)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p4.1)\.
- \[38\]R\. B\. Thapa, D\. Hernández, N\. Brandt, J\. Klein, E\. Hoffmann, S\. Staab, M\. Selzer, and G\. Lanza\(2025\)A roadmap to create a knowledge graph for the circular factory for the perpetual product\.InKG4S@ESWC,CEUR Workshop Proceedings, Vol\.4002,pp\. 46–52\.Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1)\.
- \[39\]The Apache Software Foundation\(2025\)Apache jena\.External Links:[Link](https://jena.apache.org/)Cited by:[§7](https://arxiv.org/html/2607.18262#S7.p1.1)\.
- \[40\]O\. Udrea, V\. S\. Subrahmanian, and Z\. Majkic\(2006\)Probabilistic RDF\.InIRI,pp\. 172–177\.External Links:[Document](https://dx.doi.org/10.1109/IRI.2006.252408)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p3.1)\.
- \[41\]A\. Wald\(1945\)Sequential tests of statistical hypotheses\.The Annals of Mathematical Statistics16\(2\),pp\. 117–186\.External Links:[Document](https://dx.doi.org/10.1214/aoms/1177731118)Cited by:[§8\.3](https://arxiv.org/html/2607.18262#S8.SS3.p1.2),[§8\.3](https://arxiv.org/html/2607.18262#S8.SS3.p3.4)\.
- \[42\]X\. Wang, H\. Han, and B\. Gao\(2025\)Enhancing power transformer fault diagnosis through dynamic knowledge graph reasoning\.IEEE Trans\. Instrum\. Meas\.74,pp\. 1–9\.External Links:[Document](https://dx.doi.org/10.1109/TIM.2024.3504557)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1)\.
- \[43\]World Wide Web Consortium\(2013\)SPARQL 1\.1 query language\.Note:W3C Recommendation, 21 March 2013External Links:[Link](https://www.w3.org/TR/sparql11-query/)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1),[§4](https://arxiv.org/html/2607.18262#S4.p1.10),[§5\.5](https://arxiv.org/html/2607.18262#S5.SS5.p1.5),[§5](https://arxiv.org/html/2607.18262#S5.p1.1)\.
- \[44\]World Wide Web Consortium\(2014\)RDF 1\.1 primer\.Note:W3C RecommendationExternal Links:[Link](https://www.w3.org/TR/rdf11-primer/)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1),[§4](https://arxiv.org/html/2607.18262#S4.p1.10)\.
- \[45\]J\. Wu, F\. Orlandi, D\. O’Sullivan, and S\. Dev\(2024\)On the use of virtual knowledge graphs to improve environmental sensor data accessibility\.IEEE J\. Sel\. Top\. Appl\. Earth Obs\. Remote\. Sens\.17,pp\. 6671–6682\.External Links:[Document](https://dx.doi.org/10.1109/JSTARS.2024.3370389)Cited by:[§1](https://arxiv.org/html/2607.18262#S1.p1.1),[§1](https://arxiv.org/html/2607.18262#S1.p3.1)\.
- \[46\]J\. Wu and et alSupplementary material of ProbSPARQL\.Note:[https://0sidewalkenforcer0\.github\.io/ProbSPARQL/ISWC\_SupplementaryMaterial\.pdf](https://0sidewalkenforcer0.github.io/ProbSPARQL/ISWC_SupplementaryMaterial.pdf)Cited by:[§10](https://arxiv.org/html/2607.18262#S10.p5.1),[§4\.1](https://arxiv.org/html/2607.18262#S4.SS1.SSS0.Px1.p1.2),[§4\.2](https://arxiv.org/html/2607.18262#S4.SS2.p4.1),[§5\.2](https://arxiv.org/html/2607.18262#S5.SS2.p1.12),[§5\.5](https://arxiv.org/html/2607.18262#S5.SS5.p1.5),[§8](https://arxiv.org/html/2607.18262#S8.p2.2)\.
- \[47\]A\. Zimmermann, N\. Lopes, A\. Polleres, and U\. Straccia\(2012\)A general framework for representing, reasoning and querying with annotated semantic web data\.J\. Web Semant\.11,pp\. 72–95\.External Links:[Document](https://dx.doi.org/10.1016/J.WEBSEM.2011.08.006)Cited by:[§9](https://arxiv.org/html/2607.18262#S9.p3.1)\.