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| <!DOCTYPE html> | ||
| <html lang="en"> | ||
| <head> | ||
| <meta charset="utf-8"/> | ||
| <meta name="viewport" content="width=device-width"/> | ||
| <title>Algorithms for Static Analysis and Validation of the Shape Expressions Language</title> | ||
| <link rel="stylesheet" href="local.css"/> | ||
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| <script src='https://www.w3.org/Tools/respec/respec-w3c-common' | ||
| defer class='remove'></script> | ||
| <!-- script src='../primer/respec-w3c-common.js' async class='remove'></script --> | ||
| <script src="jquery-2.1.4.min.js" type="text/javascript"></script> | ||
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| <script class='remove'> | ||
| var respecConfig = { | ||
| localBiblio: { | ||
| "shex-spec" : { | ||
| "authors": ["Eric Prud'hommeaux", "Iovka Boneva", "Jose Emilio Labra Gayo", "Gregg Kellogg"], | ||
| "title": "Shape Expressions Language 2.next", | ||
| "href": "index.html" | ||
| }, | ||
| "shex-vocab": { | ||
| "authors": ["Gregg Kellogg"], | ||
| "title": "Shape Expression Vocabulary", | ||
| "href": "http://www.w3.org/ns/shex#" | ||
| }, | ||
| "shape-map": { | ||
| "authors": ["Eric Prud'hommeaux", "Thomas Baker"], | ||
| "title": "ShapeMap Structure and Language", | ||
| "href": "http://shex.io/shape-map/" | ||
| } | ||
| }, | ||
| specStatus: "CG-DRAFT", | ||
| shortName: "shex-algorithms", | ||
| latestVersion: "", | ||
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| previousPublishDate: "", | ||
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| editors: [ | ||
| { name: "Iovka Boneva", | ||
| url: "http://cristal.univ-lille.fr/~boneva/", | ||
| company: "University of Lille", | ||
| companyURL: "http://www.univ-lille1.fr/" }, | ||
| { name: "Jérémie Dusart", | ||
| company: "Inria", | ||
| companyURL: "http://www.inria.fr/" }, | ||
| ], | ||
| wg: "Shape Expressions Community Group", | ||
| wgURI: "https://www.w3.org/community/shex/", | ||
| wgPublicList: "public-shex", | ||
| //wgPatentURI: "https://www.w3.org/2004/01/pp-impl/73865/status", | ||
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| bugTracker: { | ||
| open: "https://github.com/shexSpec/shex/issues", | ||
| new: "https://github.com/shexSpec/shex/issues/new" | ||
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| otherLinks: [{ | ||
| key: "Version control", | ||
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| value: "Github Repository", | ||
| href: "" | ||
| }] | ||
| }], | ||
| }; | ||
| </script> | ||
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| <script type="text/javascript" src="scripts.js"></script> | ||
| </head> | ||
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| <body> | ||
| <section id="abstract"> | ||
| <p> | ||
| The <a href="index.html">Shape Expressions (ShEx) language</a> describes <a>RDF nodes</a> and <a>graph</a> structures, organized in a shapes schema. | ||
| A shapes schema must comply to a series of <a href="index.html#schema-requirements">requirements</a>. | ||
| </p> | ||
| <p> | ||
| This document presents the algorithms that allow to test whether a syntactically correct shapes schema satisfies the requirements. | ||
| It also presents algorithms for validating an RDF graph against a ShEx schema that satisfies the requirements. | ||
| </p> | ||
| </section> | ||
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| <section id="sotd"> | ||
| <p>This document is being developed by the | ||
| <a href="https://www.w3.org/community/shex/">Shape Expressions Community Group</a>. | ||
| <p> | ||
| This version is an editor's draft. | ||
| </p> | ||
| </section> | ||
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| <section id="introduction"> | ||
| <h2>Introduction</h2> | ||
| </section> | ||
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| <section id="schema-requirements"> | ||
| <h2>Algorithms for Checking the Schema Requirements</h2> | ||
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| <section id="shape-expr-ref"> | ||
| <h3>Shape Expression Reference Requirement</h3> | ||
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| <p> | ||
| The <a href="index.html#shapeExprRef-requirement">definition of this requirement</a> is in the shape expressions language specification. | ||
| </p> | ||
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| <p>For a ShEx schema <span class="math">sch</span>, let <span class="math">ShapeExprIds(sch)</span> be the set of shape expression identifiers in <span class="math">sch</span>.</p> | ||
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| <p>Algorithm checkShapeExpressionReferenceRequirement</p> | ||
| <pre><code> | ||
| Input: sch: a schema | ||
| Output: true sch satisfies the shape expression reference requirement, false otherwise | ||
| whenever the latter is true, produces also D: the shape expressions dependency graph induced by sch | ||
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| let ShapeExprIds be the set of all shape expression ids in sch | ||
| let ShapeExprRefs be the set of all shape expression references in sch | ||
| if shapeExprRefs is not included in shapeExprIds then return false | ||
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| initiate a directed edge labelled graph D which nodes are all Shape, NodeConstraint, ShapeNot, ShapeOr, ShapeAnd and shapeExprRef elements that appear in sch | ||
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| foreach ShapeExpr se in the vertex set of D | ||
| if se is a ShapeOr or a ShapeAnd then | ||
| foreach ShapeExpr sub in se.shapeExprs | ||
| add to D an edge from se to sub labelled with pos | ||
| else if se is a ShapeNot then | ||
| add to D an edge from se to se.shapeExpr labelled with neg | ||
| else if se is a shapeExprRef then | ||
| add to D an edge from se to shapeExprWithId(se) labelled with pos | ||
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| if D contains a cycle then | ||
| return false | ||
| else | ||
| return (true, D) | ||
| </code></pre> | ||
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| </section> | ||
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| <section id="triple-expr-ref"> | ||
| <h3>Triple Expression Reference Requirement</h3> | ||
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| <p> | ||
| The <a href="index.html#tripleExprRef-requirement">definition of this requirement</a> is in the shape expressions language specification. | ||
| </p> | ||
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| </section> | ||
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| <section id="negation"> | ||
| <h3>Negation Requirement</h3> | ||
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| <p> | ||
| The <a href="index.html#negation-requirement">definition of this requirement</a> is in the shape expressions language specification. | ||
| </p> | ||
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| <p>Algorithm for computing the dependency graph of a schema.</p> | ||
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| <pre><code> | ||
| Input: schema a ShexSchema | ||
| Output: D the shape expressions dependency graph | ||
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| let DSE be the dependency graph as computed by checkShapeExpressionReferenceRequirement | ||
| let DTE be the dependency graph as computed by checkTripleExpressionReferenceRequirement | ||
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| let atomicTripleConstraints be the map that with every TripleExpression s.expression that appears in schema (for some Shape s) associates the set of TripleConstrants tc s.t. there is a path in DTE between s.expression and tc; | ||
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| let atomicShapes be the map that with every ShapeExpression tc.valueExpr that appears in schema (for some TripleConstraint tc) associates the set of Shapes s s.t. there is path in DSE from tc.valueExpr to s | ||
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| initiate a directed edge-labelled graph D with vertex set that contains all the Shapes in schema | ||
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| foreach Shape s vertex of D s.t. s.expression is in the key set of atomicTripleConstraints | ||
| foreach TripleConstraint tc in atomicTripleConstraints(s.expression) | ||
| if tc.valueExpr is present then | ||
| foreach Shape s2 that belongs to atomicShapes(tc.valueExpr) | ||
| if there is at least one path in DSE from tc.valueExpr to s2 that traverses an odd number of neg-labelled edges then | ||
| add to D a neg-labelled edge from s to s2 | ||
| else | ||
| add to D a pos-labeled edge from s to s2 | ||
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| return D | ||
| </code></pre> | ||
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| <p>Algorithm for computing the stratification of a schema.</p> | ||
| <pre><code> | ||
| Input: schema a ShexSchema | ||
| Output: a map stratum from the set of Shapes in schema to natural numbers, or ERROR if the schema cannot be stratified | ||
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| let D be the dependency graph of schema | ||
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| if D contains a strongly connected component stc that contains a neg-labelled edge then | ||
| raise an error | ||
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| let S be a directed graph which set of vertices is the set of strongly connected components of D | ||
| let L be a topological sorting of the vertices of S | ||
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| foreach s node of D | ||
| let stratum(s) be the index in L of the unique strongly connected component stc s.t. s belongs to stc | ||
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| return stratum | ||
| </code></pre> | ||
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| </section> | ||
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| </section> | ||
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| <section id="validation"> | ||
| <h2>Algorithms for Validation</h2> | ||
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| <section id="refine-validation"> | ||
| <h3>Refine Validation</h3> | ||
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| <p>Algorithm for refine validation.</p> | ||
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| <pre><code> | ||
| Input: an RDF graph G, a ShexSchema schema which set of shapes is S | ||
| Output: the shape map completeTyping(G,sch) | ||
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| let stratum=stratification(sch) | ||
| let k be the greatest number in the range of stratum | ||
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| let typing be the empty set | ||
| for i in 0 .. k do | ||
| let shapes-i be the set of Shapes s in S s.t. stratum(s) = i | ||
| let typing-i = nodes(G) x shapes-i | ||
| do | ||
| changing = false | ||
| foreach (n,s) in typing-i do | ||
| if not matches(n,s,G,Sch,typing-i union typing) | ||
| remove (n,s) from typing-i | ||
| changing = true | ||
| while (changing) | ||
| add typing-i to typing | ||
| end for | ||
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| return typing | ||
| </code></pre> | ||
| </section> | ||
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| <section id="recursive-validation"> | ||
| <h3>Recursive Validation</h3> | ||
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| <p>Algorithm isValid(G,sch,n,se) for checking a unique node s against a unique shape expression s.</p> | ||
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| <pre><code> | ||
| Input: RDF graph G, ShexSchema sch with set of Shapes S, node n in G, Shape s in sch, hyp: a stack over Nodes(G)xS | ||
| Output: true if satisfies(G,sch,n,s,completeTyping(G,sch)), false otherwise | ||
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| push (n,s) on hyp | ||
| let sm be an empty shape map | ||
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| let atomicTripleConstranits be the set of atomic triple constraints of s.expression | ||
| let neighbourhood be the set of (predicate,n2) s.t. either (n, predicate, n2) in G and predicate appears in a triple constraint in allTC, or (n2, predicate, n) in G and ^predicate appears in a triple constraint in allTC | ||
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| for (pred,n2) in neighbourhood do | ||
| foreach tc in allTC | ||
| if tc.predicate == pred then | ||
| if tc.valueExpr is present then | ||
| foreach s2 in atomicShapes(tc.valueExpr) | ||
| if isValid(G,sch,n2,s2) then | ||
| add (n2,s2) to sm | ||
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| add all elements of hyp to sm | ||
| result = satisfies(n,s,G,sm) | ||
| pop from hyp | ||
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| return result | ||
| </code></pre> | ||
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| <p>Recursive validation algorithm.</p> | ||
| <pre><code> | ||
| Input: an RDF graph G, a ShexSchema schema which set of shapes is S, an initial shapeMap map | ||
| Output: true if satisfies(n,s,G,sch,completeTyping(G,sch)) for every (n,s) in map, false otherwise | ||
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| let sm be an empty shape map | ||
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| foreach (n,se) in map | ||
| foreach Shape s in atomicShapes of se | ||
| if isValid(s,n,G,sch) then | ||
| add (n,s) to sm | ||
| if not satisfies(n,se,G,sch,sm) then | ||
| return false | ||
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| return true | ||
| </code></pre> | ||
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| </section> | ||
| </section> | ||
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| </body> | ||
| </html> |
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