Terminology

Connect to definitions in RDF 1.2 Concepts.

The following definitions from other specifications are used in this document: @@

TODO

RFC 2119 language should be automatically inserted here.

Elements of the Abstract Syntax

Variable

A [=variable=] represents a possible [=RDF term=] in a triple pattern. Variables are also used in expressions.

Expression

An [=expression=] is a function or a functional form. It's arguments are [=RDF terms=]. An expression is evaluated with respect to a [=solution mapping=] to give an [=RDF term=] as the result. Expressions are compatible with SHACL list parameter functions and with SPARQL expressions.

Data block

A [=data block=] is a set of triples. These triples are added to the inference graph as additional facts and are included in the inference process.

Triple template

A [=triple template=] is 3-tuple where each element is either a [=variable=] or an [=RDF term=] (which might be a [=triple term=]). The second element of the tuple must be an [=IRI=] or a [=variable=]. [=Triple templates=] appear in the [=head=] of a [=rule=].

Triple pattern

A [=triple pattern=] is 3-tuple where each element is either a [=variable=] or an [=RDF term=] (which might be a triple term). The second element of the tuple must be an [=IRI=] or a [=variable=]. [=Triple patterns=] can appear as [=triple pattern elements=] as well as inside [=negation elements=].

Condition element

A [=condition element=] is an [=expression=] that evaluates to true or false. [=Condition elements=] appear in the [=body=] of a [=rule=].

Triple pattern element

A [=triple pattern element=] is a [=triple pattern=] used as a [=rule body element=].

Negation element

A [=negation element=] is a [=rule body element=] comprised of a sequence of [=triple pattern elements=] and [=condition elements=].

Assignment element

An [=assignment element=] is a pair consisting of a variable, called the assignment variable, and an expression, called the assignment expression. [=Assignment elements=] appear in the [=body=] of a [=rule=].

Rule body element

A [=rule body element=] (often just "rule element") is any element that can appear in a [=rule body=], a [=triple pattern=], a [=condition element=], a [=negation element=], or an [=assignment element=].

Rule head

A [=rule head=] is a sequence of [=triple templates=].

Rule body

A [=rule body=] is a sequence of [=rule body elements=].

Rule imports

[=Rule imports=] (often just "imports") are a collection of URLs for other rule sets that will be included during evaluation.

Rule

A [=rule=] is a pair of a [=rule head=] (often just "head") and a [=rule body=] (often just "body").

Run-once rule

A [=run-once rule=] is a rule is run once at a particular point in the evaluation of a rule set.

Rule set

A [=rule set=] is a collection of zero or more [=rules=], a collection of zero or more [=data blocks=], and a collection of zero or more [=rule imports=].

Base graph

A [=base graph=] is the [=RDF Graph=] given as input to the evaluation process.

Inference graph

An [=inference graph=] is an [=RDF Graph=] produced by evaluating a [=rule set=]. It contains all triples not present in the [=base graph=] that are inferred by applying the [=rule set=] to the [=base graph=].

Infer

[=Infer=] is the operation that applies a [=rule set=] to a given [=base graph=] and produces an [=inference graph=] containing inferred triples.

Query

[=Query=] is the operation that determines whether a given goal pattern can be derived from a [=base graph=] using the [=rule set=].

In a [=triple pattern=] or a [=triple template=], position 1 of the tuple is informally called the subject, position 2 is informally called the predicate, and position 3 is informally called the object.

Well-formedness Conditions

Well-formedness is a set of conditions on the abstract syntax of SHACL rules. Together, these conditions ensure that a [=variable=] in the [=head=] of a rule has a value defined in the [=body=] of the rule; that each variable in an [=condition element=] or [=assignment expression=] has a value at the point of evaluation; and that each assignment in a rule introduces a new variable, one that has not been used earlier in the rule body.

A [=rule=] is a well-formed rule if all of the following conditions are met:

• For every [=variable=] appearing in a [=triple template=] of the [=head=] of the [=rule=], there is one or more occurrences of a [=variable=] of the same name in the [=triple patterns=] in the [=body=], in an [=assignment element=] in the body, or both.
• For every [=variable=] in an [=expression=] at position i of the [=body=], there is a corresponding [=variable=] of the same name occurring in a [=triple pattern=] at a position j, or occurring as an [=assignment variable=] at a position j, where i > j.
• Each [=assignment variable=] is used in only one [=assignment element=] in the [=body=] of the [=rule=].
• An [=assignment variable=] at position i of a [=rule body=] does not occur in any [=triple pattern=] at position j where i > j.
• For every [=variable=] in an [=assignment expression=] at position i, there is a corresponding variable in a [=triple pattern=] at position j where i > j, or there is an [=assignment variable=] in an [=assignment element=] at position j where i > j.

A [=rule set=] is "well-formed" if and only if all of the [=rules=] of the rule set are "well-formed".

Rule Dependency

Triple pattern dependency

A [=triple pattern=] in the rule body of rule `R1` depends on a rule `R2` if the head of `R2` can possibly generate triples that match the [=triple pattern=] in `R1`. The dependency is positive if the [=triple pattern=] is a [=rule body element=] and the dependency is negative if the [=triple pattern=] appears in a [=negation element=].

Rule dependency

A rule `R1` depends on rule `R2` if any rule element of `R1` depends on `R2`. The dependency is negative if any of the rule elements have a negative dependency on `R2`; otherwise the dependency is positive.

Dependency graph

A [=dependency graph=] of a [=rule set=] is a graph where the vertices are rules and an edge exists where one rule R1 depends on rule R2. An edge is labeled either positive or negative according to the [=rule dependency=].

Transitive rule dependency

A rule `R1` has a [=transitive dependency=] on rule `R2` if there is a path in the [=dependency graph=] from `R1` to `R2`.

Recursive rule dependency

A rule `R` has a [=recursive dependency=] if there is a cyclic path in the [=dependency graph=] involving `R`.

## output -- Dependency graph with rule vertices and labeled edges.

define buildDependencyGraph(ruleSet):

    ## edgeLabelMap maps (R1, R2) to "positive" or "negative"
    let edgeLabelMap be a map from (rule, rule) to label

    define mergeLabel(oldLabel, newLabel):
        ## Negative dependency dominates positive dependency.
        if oldLabel == "negative" or newLabel == "negative":
            return "negative"
        else:
            return "positive"
        endif
        enddefine

    ## Classify all triple patterns in a rule, looking inside negation elements.
    foreach rule R1 in ruleSet:

        let bodyDependencies = {}
        foreach rule body element RBE in the body of R1:
            if RBE is a negation element:
                foreach triple pattern element TPE in RBE:
                    let item be a pair (TPE, "negative")
                    add item to bodyDependencies
                endfor
            else if RBE is a triple pattern element:
                let item be a pair (RBE, "postive")
                add item to bodyDependencies
            else if RBE is a condition element:
                ## Do nothing
            else if RBE is an assignment element:
                ## Do nothing
            endif
        endfor

        foreach pair ( triple pattern TP, depLabel) in bodyDependencies:

            foreach rule R2 in ruleSet:

                foreach triple template TT in head of R2:
                    ## "possibly generate" / matching is defined in
                    ## 3.3 Rule Dependency.
                    if TT can possibly generate a triple pattern TP:
                        let key = (R1, R2)
                        if edgeLabelMap contains key:
                            let oldLabel = edgeLabelMap.get(key)
                            let merged = mergeLabel(oldLabel, depLabel)
                            edgeLabelMap.set(key, merged)
                        else:
                            edgeLabelMap.set(key, depLabel)
                        endif
                    endif
                endfor

            endfor

        endfor
    endfor

    let DP = { }
    foreach entry ((R1, R2), label) in edgeLabelMap:
        DP = { edge (R1 -> R2) labeled label } ∪︀ DP
    endfor

    the result is DP
    enddefine
            

Notes:

• A [=triple template=] with components `ts`, `tp`, `to` can possibly generate a triple with component RDF terms `s`, `p`, `o` if `ts` is a variable or `ts` is the same RDF term as `s`, `tp` is a variable or `tp` is the same RDF term as `p`, and `to` is a variable or `to` is the same RDF term as `o`.

  In addition, if any pair of `ts`, `tp`, and `to` are the same variable, then the corresponding pair of `s`, `p`, and `o` must be the same.

  Revise

• The dependency graph is not affected by the data graph.

Examples:

            @@ Examples of triple patttern dependencies.
          

            @@ Examples of rule dependencies.
          

Stratification

[=Stratification=] is the process of partitioning a [=rule set=] into an ordered sequence of [=stratification layers=] (also known as "strata", singular "stratum"), forming a [=stratification=]. Rules in lower [=strata=] are evaluated before rules in higher [=strata=].

[=Stratification=] imposes constraints on dependencies between [=rules=] to ensure that [=negation elements=] depend only on results computed using earlier (lower) [=strata=]. This guarantees a single well-defined outcome from the evaluation of a [=rule set=] over a given [=base graph=].

Stratification

A [=stratification=] of a [=rule set=] is a sequence of sets of rules. Each rule in a [=rule set=] appears in exactly one [=stratification layer=].

Stratification layer

Each set of rules in a [=stratification=] is called a [=stratification layer=], also called a "stratum".

## output -- Map: Integer -> Set of rules.

define stratification(ruleSet):

    let DP = Dependency graph for the rule set.
    let stratumMap be a map from rule to integer

    ## The dependency graph should satisfy the stratification condition.
    ## Check unbounded work due to a violation of the stratification condition.
    let limit = num rules  + 1
    let maxStratum = 0

    ## initialize stratumMap
    foreach rule in ruleSet:
        stratumMap.set(rule, 0)
        endfor

    boolean changed = true;
    while changed:
        changed = false;
        foreach edge E in DP:
            ## Edge from pRule to qRule with a label
            let pRule = source of edge
            let qRule = destination of the edge
            let label = edge label

            if label == "positive" :
                if stratumMap.get(pRule) < stratumMap.get(qRule) :
                    stratumMap.set(pRule, stratumMap.get(qRule))
                    changed = true;
                    endif
                endif
            if label == "negative" :
                if stratumMap.get(pRule) <= stratumMap.get(qRule) :
                    let xStratum = 1 + stratumMap.get(qRule)
                    if ( xStratum > limit )
                        ## Stratification requirement violated
                        error "Stratification error"
                        endif
                    stratumMap.put(pRule, xStratum)
                    maxStratum = max(maxStratum, xStratum)
                    changed = true;
                    endif
                endif
            endfor
        endwhile

    ## Initialize the result map.
    let stratumLevels be a map from integer to a set of rules
    for i = 0 to maxStratum
        stratumLevels.set(i, {})
        endfor

    ## Gather rules in stratumMap with the same level number
    for rule R in map stratumMap:
        let stratumNum = stratumMap.get(R)
        let S = {R} ∪︀ stratumLevel.get(stratumNum)
        stratumLevel.set(stratumNum, S)
        endfor

    the result is stratumLevels
    endfunction
            

Shape Rules Language syntax

The grammar is given below.

Mapping the AST to the abstract syntax.

RDF Rules Syntax

Vocabulary: shacl-rules.ttl.

Evaluation Definitions

Solution mapping

A Solution mapping, μ, is a partial function μ : V → T, where V is the set of all variables and T is the set of all [=RDF terms=]. The domain of μ is denoted by dom(μ), and it is the subset of V for which μ is defined. We use the term [=solution=] where it is clear that a [=solution mapping=] is meant. Write μ₀ for the solution mapping, such that dom(μ0) is the empty set.

Substitution function

A substitution function, or just a substitution, is a function subst(μ, [=triple pattern=]) that returns a [=triple pattern=] where each occurrence in the [=triple pattern=] of a variable that is in the dom(μ) is replaced by the [=RDF term=] given by the [=solution mapping=] for var. If the triple pattern result has no variables, then it is an [=RDF Triple=].

Evaluation graph

A [=evaluation graph=] is an [=RDF Graph=] that combines the [=base graph=] and all triples produced during the evaluation of a rule set.

Graph match

A [=graph match=] finds the ways to map a triple pattern onto triples in an [=RDF Graph=].

Let G be an [=RDF graph=] and TP be a triple pattern. The function `graphMatch(G, TP)` returns a set of all possible solutions that, when applied to the triple pattern, produce a triple that is in the [=evaluation graph=]

Let G be an [=RDF graph=] and TP be a triple pattern.

graphMatch(G, TP) = { μ | subst(μ, TP) is a triple in G }

Solution compatible

Two solutions S1 and S2 are [=compatible=] if they agree on the variables in common.

Let S1 and S2 be solutions.

compatible(μ1, μ2) = true
                      if forall v in dom(μ1) intersection dom(μ2)
                          μ1(v) = μ2(v)
compatible(μ1, μ2) = false otherwise
              

Solution sequence

A [=solution sequence=] is a multi-set of solutions. There is no defined order to the sequence. It is equivalent to an unordered list and it can contain duplicates.

Solution merge

If two solutions are compatible, the merge of two solutions is the solution that maps variables of each solution to the [=RDF term=] from one or other of the solutions.

Let μ₁, μ₂ be solution mappings, and S1 and S2 be solution sequences.

merge(μ1, μ2) = { μ |
                    μ(v) = μ1(v) if v in dom(μ1)
                    μ(v) = μ2(v) otherwise }

merge(S1, S2) = { μ |
                    μ1 in S1, μ2 in S2
                    and compatible(μ1, μ2)
                    μ(v) = merge(μ1, μ2)

Say the domain is `dom(S1) ∪︀ dom(S2)`.

Say that two solutions that have no variables in common are compatible.

Preparation for Evaluation

The first step in evaluating a [=rule set=] is to prepare a single, valid rule set. This involves gathering all imported rule sets, building a single, combined rule set, and then calculating the stratification for the combined rule set.

                MR.rules = RS1.rules ∪︀ RS2.rules
                MR.data = merge(RS1.data, RS2.data)
                MR.imports = {}
            

define imports(rule set RS, set of URLs V), returning rule set
    let I = the set of import URLs declared for the rule set RS
    let RS2 = RS
    foreach URL x in I:
       if x ∉ V:
           V = V ∪︀ { x }
           read rule set RS3 from URL x
           RS2 = rulesetMerge(RS2, imports(RS3, V))
        endif
    endfor
    result is RS2
enddefine

let RS be a ruleset
let V = {}
if RS has a location, V = { location of RS }
result is imports(RS, V)

Evaluation of an Expression

Sketch

@@ Reference SPARQL expression evaluation Expression Evaluation
@@ Reference SPARQL EBV Effective Boolean Value (EBV)

define evalFunction(F, μ):
    let [x/μ] be
        if x is an RDF term, then [x/row] is x
        if x is a variable, then [x/row] is μ(x)
        ## By well-formedness, it is an error if x is not in the row.
    return evalFunction(F(expr1, expr2 ...), row) = F(evalFunction(expr1, row), evalFunction(expr2, row), ...)
        if an error is returned by evalFunction: return error
    return evalFunction(FF(expr1, expr2) , row) = ... things that are not functions like `IF`

Evaluation of a Rule

let R be a well-formed rule.

let rule R = (H, B) where
             H is the sequence of triple templates in the head
             B is the sequence of triple pattern elements,
                condition elements, negation elements,
                and assignment elements in the body

# Solution sequence of one solution that does not map any variables.
let SEQ0: Solution sequence = { μ0 }

let G = evaluation graph

# Evaluate rule body
# This function returns a sequence of solutions
define evalRuleElements(B, SEQ, G):

    for each rule element rElt in B:

        if rElt is a triple pattern TP:
            X = graphMatch(G, TP)
            SEQ1 = {}
            for each μ1 in X:
                for each μ2 in SEQ:
                  if compatible(μ1, μ2)
                    μ3 = merge(μ1, μ2)
                    add μ3 to SEQ1
                    endif
                endfor
             endfor

        if rElt is a condition element with expression F:
            SEQ1 = {}
            for each solution μ in SEQ:
                let x = evalFunction(F, μ)
                if x is true:
                    add μ to SEQ1
                    endif
                endfor
            endif

        if rElt is a negation expression with body elements N:
            SEQ1 = {}
            for each solution μ in SEQ:
                S = sequence{ μ }
                NEG = evalRuleElements(N, S, G)
                if NEG is empty
                    add μ to SEQ1
                    endif
                endfor
            endif

        if rElt is an assignment with variable V and expression expr
            SEQ1 = {}
            for each solution S in SEQ:
                let x = evalFunction(expr, S)
                if x is not an error:
                    add(V, x) to S
                    add S to SEQ1
                else
                    # Error: drop solution S
                    endif
                endfor
            endif

        if SEQ1 is empty
            SEQ = {}
            return SEQ
            endif

        SEQ = SEQ1
        endfor

     return SEQ
     enddefine

let SEQ = evalRuleElements(B, SEQ0, G)

# Evaluate rule head
let H = empty set
for each μ in SEQ:
    let S = {}
    for each triple template TT in head
        let triple = subst(μ, TT)
        Add triple to S
        endfor
    H = H union S
    endfor

result eval(R, G) is H

Note that `H` may contain triples that are also in the data graph.

Evaluation of a Rule Set

let G0 be the input base graph
let RS be the rule set
let D be the graph of all DATA triples in RS

Apply stratification to RS

let L be the sequence of layers after stratification

# Inference graph
let GI = { t ∈ D  | t ∉ in G0 }

# Evaluation graph.
let GE = G0 ∪︀ D

for each layer in L:
    let finished = false
    while !finished:
        finished = true
        for each rule in layer:
            let X = eval(rule, GE)
            let Y = { t ∈ X | t ∉ in GE }
            if Y is not empty:
                finished = false
                GI = Y ∪︀ GI
                GE = Y ∪︀ GE
                endif
            endfor
        endwhile
    endfor
the result is GI