RIF Basic Logic Dialect

This specification develops RIF-BLD (the Basic Logic Dialect of the Rule Interchange Format). From a theoretical perspective, RIF-BLD corresponds to the language of definite Horn rules with equality and a standard first-order semantics [CL73]. Syntactically, RIF-BLD has a number of extensions to support features such as objects and frames as in F-logic [KLW95], internationalized resource identifiers (or IRIs, defined by [RFC-3987]) as identifiers for concepts, and XML Schema datatypes [XML-SCHEMA2]. In addition, RIF RDF and OWL Compatibility [RIF-RDF+OWL] defines the syntax and semantics of integrated RIF-BLD/RDF and RIF-BLD/OWL languages. These features make RIF-BLD a Web-aware language. However, it should be kept in mind that RIF is designed to enable interoperability among rule languages in general, and its uses are not limited to the Web.

RIF-BLD is defined in two different ways -- both normative:

• As a direct specification, independently of the RIF Framework for Logic Dialects [RIF-FLD], for the benefit of those who desire a direct path to RIF-BLD, e.g., as prospective implementers, and are not interested in extensibility issues. This version of the RIF-BLD specification is given first.
• As a specialization of the RIF Framework for Logic-based Dialects [RIF-FLD], which is part of the RIF extensibility framework. Building on RIF-FLD, this version of the RIF-BLD specification is comparatively short and is presented in Section RIF-BLD as a Specialization of the RIF Framework at the end of this document. This is intended for the reader who is already familiar with RIF-FLD and does not need to go through the much longer direct specification of RIF-BLD. This section is also useful for dialect designers, as it is a concrete example of how a non-trivial RIF dialect can be derived from the RIF framework for logic dialects.

Logic-based RIF dialects that specialize or extend RIF-BLD in accordance with the RIF Framework for Logic Dialects [RIF-FLD] will be developed in other specifications by the RIF working group.

To give a preview, here is a simple complete RIF-BLD example deriving a ternary relation from its inverse.

Example 1 (An introductory RIF-BLD example).

A rule can be written in English to derive buy relationships (rather than store any of them) from sell relationships (e.g., stored as facts, as exemplified by the second line):

The fact Mary buys LeRif from John can be logically derived by a modus ponens argument. Assuming Web IRIs for the predicates buy and sell, as well as for the individuals John, Mary, and LeRif, the above English text can be represented in RIF-BLD Presentation Syntax as follows.

Document(
  Prefix(cpt http://example.com/concepts#)
  Prefix(ppl http://example.com/people#)
  Prefix(bks http://example.com/books#)

  Group
  (
    Forall ?Buyer ?Item ?Seller (
        cpt:buy(?Buyer ?Item ?Seller) :- cpt:sell(?Seller ?Item ?Buyer)
    )
 
    cpt:sell(ppl:John bks:LeRif ppl:Mary)
  )
)

For the interchange of such rule (and fact) documents, an equivalent RIF-BLD XML Syntax is given in this specification. To formalize their meaning, a RIF-BLD Semantics is specified.

This normative section specifies the syntax of RIF-BLD directly, without relying on [RIF-FLD]. We define both the presentation syntax (below) and an XML syntax in Section XML Serialization Syntax for RIF-BLD. The presentation syntax is normative, but is not intended to be a concrete syntax for RIF-BLD. It is defined in mathematical"mathematical English," a special form of English and is meant to be used in the definitions and examples.for communicating mathematical definitions, examples, etc. This syntax deliberately leaves out details such as the delimiters of the various syntactic components, escape symbols, parenthesizing, precedence of operators, and the like. Since RIF is an interchange format, it uses XML as its concrete syntax and RIF-BLD conformance is described in terms of semantics-preserving transformations.

Note to the reader: this section depends on Section Constants, Symbol Spaces, and Datatypes of [RIF-DTB].

Constants are written as "literal"^^symspace, where literal is a sequence of Unicode characters and symspace is an identifier for a symbol space. Symbol spaces are defined in Section Constants and Symbol Spaces of [RIF-DTB].

The symbols =, #, and ## are used in formulas that define equality, class membership, and subclass relationships. The symbol -> is used in terms that have named arguments and in frame formulas. The symbol External indicates that an atomic formula or a function term is defined externally (e.g., a built-in) and the symbols Prefix and Base are used in abridged representations of IRIs.

The symbol Document is used to specify RIF-BLD documents, Import is an import directive, and the symbol Group is used to organize RIF-BLD formulas into collections.   ☐

The language of RIF-BLD is the set of formulas constructed using the above alphabet according to the rules given below.

1. Constants and variables. If t ∈ Const or t ∈ Var then t is a simple term.
2. Positional terms. If t ∈ Const and t₁, ..., t_n are base terms then t(t₁ ... t_n) is a positional term.
3. Terms with named arguments. A term with named arguments is of the form t(s₁->v₁ ... s_n->v_n), where t ∈ Const and v₁, ..., v_n are base terms and s₁, ..., s_n are pairwise distinct symbols from the set ArgNames.

   The constant t here represents a predicate or a function; s₁, ..., s_n represent argument names; and v₁, ..., v_n represent argument values. The argument names, s₁, ..., s_n, are required to be pairwise distinct. Terms with named arguments are like positional terms except that the arguments are named and their order is immaterial. Note that a term of the form f() isis, trivially, both a positional term and a term with named arguments.

4. Equality terms. t = s is an equality term, if t and s are base terms.
5. Class membership terms (or just membership terms). t#s is a membership term if t and s are base terms.
6. Subclass terms. t##s is a subclass term if t and s are base terms.
7. Frame terms. t[p₁->v₁ ... p_n->v_n] is a frame term (or simply a frame) if t, p₁, ..., p_n, v₁, ..., v_n, n ≥ 0, are base terms.

   Membership, subclass, and frame terms are used to describe objects and class hierarchies.

8. Externally defined terms. If t is a positional, named-argument, or a frame term then External(t) is an externally defined term.

9. Such terms are used for representing built-in functions and predicates as well as "procedurally attached" terms or predicates, which might exist in various rule-based systems, but are not specified by RIF.

   Note that frame terms are allowed to be externally defined. Therefore, externally defined objects can be accessed using the more natural frame-based interface. For instance, External("http://example.com/acme"^^rif:iri["http://example.com/mycompany/president"^^rif:iri(?Year) -> ?Pres]) could be an interface provided to access an externally defined method "http://example.com/mycompany/president"^^rif:iri of an external object "http://example.com/acme"^^rif:iri.   ☐

   Feature At Risk #1: External frames

   Note: This feature is "at risk" and may be removed from this specification based on feedback. Please send feedback to public-rif-comments@w3.org.

Observe that the argument names of frame terms, p₁, ..., p_n, are base terms and, as a special case, can be variables. In contrast, terms with named arguments can use only the symbols from ArgNames to represent their argument names. They cannot be constants from Const or variables from Var. (The reason for this restriction has to do with the complexity of unification, which is used by several inference mechanisms of first-order logic.)

1. Atomic: If φ is an atomic formula then it is also a formula.
2. Condition formula: A condition formula is either an atomic formula or a formula that has one of the following forms:
   • Conjunction: If φ₁, ..., φ_n, n ≥ 0, are condition formulas then so is And(φ₁ ... φ_n), called a conjunctive formula. As a special case, And() is allowed and is treated as a tautology, i.e., a formula that is always true.
   • Disjunction: If φ₁, ..., φ_n, n ≥ 0, are condition formulas then so is Or(φ₁ ... φ_n), called a disjunctive formula. As a special case, Or() is permitted and is treated as a contradiction, i.e., a formula that is always false.
   • Existentials: If φ is a condition formula and ?V₁, ..., ?V_n are variables then Exists ?V₁ ... ?V_n(φ) is an existential formula.

   Condition formulas are intended to be used inside the premises of rules. Next we define the notion of RIF-BLD rules, sets of rules, and RIF documents.

3. Rule implication: φ :- ψ is a formula, called rule implication, if:
   • φ is an atomic formula or a conjunction of atomic formulas,
   • ψ is a condition formula, and
   • none of the atomic formulas in φ is an externally defined term (i.e., a term of the form External(...)).

   Feature At Risk #2: Equality in the rule conclusion (φ in the above)

   Note: This feature is "at risk" and may be removed from this specification based on feedback. Please send feedback to public-rif-comments@w3.org.

4. Universal rule: If φ is a rule implication and ?V₁, ..., ?V_n are variables then Forall ?V₁ ... ?V_n(φ) is a formula, called a universal rule. It is required that all the free variables in φ occur among variables ?V₁ ... ?V_n in the quantification part. A variable ?v is free in φ if it does not occur in a subformula of φ of the form Q ?v (ψ), where Q is a quantifier (Forall or Exists). Universal rules will also be referred to as RIF-BLD rules.
5. Universal fact: If φ is an atomic formula then Forall ?V₁ ... ?V_n(φ) is a formula, called a universal fact, provided that all the free variables in φ occur among the variables ?V₁ ... ?V_n.

   Universal facts are often considered to be rules without premises (or having true as their premises).

6. Group: If φ₁, ..., φ_n are RIF-BLD rules, universal facts, variable-free rule implications, variable-free atomic formulas, or group formulas then Group(φ₁ ... φ_n) is a group formula.

   Group formulas are used to represent sets of rules and facts. Note that some of the φ_i's can be group formulas themselves, which means that groups can be nested.

7. Document: An expression of the form Document(directive₁ ... directive_n Γ) is a RIF-BLD document formula (or simply a document formula), if
   • Γ is aan optional group formula that makesencompasses the actuallogical content of the document.
   • directive₁, ..., directive_n areis an optional sequence of directives. A directive can be an import directive, a prefixbase directive, or a baseprefix directive.
   • • A base directive has the form Base(iri), where iri is a unicode string in the form of an IRI.

       Like prefix directives,The Base directives dodirective does not affect the semantics. They are used asIt defines a syntactic shortcutsshortcut for expanding relative IRIs into full IRIs, as described in Section Constants and Symbol Spaces of [RIF-DTB].

     • A prefix directive has the form Prefix(p v), where p is an alphanumeric string that serves as the prefix name and v is a macro-expansion for p -- a string that forms an IRI.

       Like the Base directive, the Prefix directives do not affect the semantics of RIF documents. Instead, they are used asdefine shorthands to allow more concise representation of IRI constants. This mechanism is explained in [RIF-DTB], Section Constants and Symbol Spaces.

     • An import directive can have one of these two forms: Import(t) or Import(t p). Here t is an IRI constant and p is a term. The constant t indicates the location of another document to be imported and p is called the profile of import.

       Section Direct Specification of RIF-BLD Semantics of this document defines the semantics for the directive Import(t) only. The semantics of the directive Import(t p) is given in [RIF-RDF+OWL]. It is used for importing non-RIF-BLD logical entities, such as RDF data and OWL ontologies. The profile specifies what kind of entity is being imported and under what semantics (for instance, the various RDF entailment regimes).

     ThereA document formula can becontain at most one Base directive in the sequence of directives in a document formula. It must be the first directive indirective. The sequence,Base directive, if present, must be first, followed by a sequenceany number of Prefix directives (again, if present),directives, followed by a sequenceany number of Import directives.

All parts of a document formula -- the directives and the group formula -- are optional and can be omitted.In this definition, the component formulas φ, φ_i, ψ_i, and Γ are said to be subformulas of the respective formulas (condition, rule, group, etc.) that are built with the help ofusing these components.   ☐

The above definitions endow RIF-BLD with a wide variety of syntactic forms for terms and formulas, which creates infrastructure for exchanging syntactically diverse rule languages. Systems that do not support some of the syntax directly can still support it through syntactic transformations. For instance, disjunctions in the rule body can be eliminated through a standard transformation, such as replacing p :- Or(q r) with a pair of rules p :- q,   p :- r. Terms with named arguments can be reduced to positional terms by ordering the arguments by their names and incorporating themthe ordered argument names into the predicate name. For instance, p(bb->1 aa->2) can be represented as p_aa_bb(2,1).

RIF-BLD allows every term and formula (including terms and formulas that occur inside other terms and formulas) to be optionally preceded by an annotation of the form (* id φ *), where id is a rif:iri constant and φ is a frame formula or a conjunction of frame formulas. Both items inside the annotation are optional. The id part represents the identifier of the term/formula to which the annotation is attached and φ is the metadata part of the annotation. RIF-BLD does not impose any restrictions on φ apart from what is stated above. In particular, it may include variables, function symbols, rif:local constants, and so on.

Document formulas with and without annotations will be referred to as RIF-BLD documents.

A convention is used to avoid a syntactic ambiguity in the above definition. For instance, in (* id φ *) t[w -> v] the metadata annotation could be attributed to the term t or to the entire frame t[w -> v]. The convention in RIF-BLD is that the above annotation is considered to be syntactically attached to the entire frame. Yet, since φ iscan be a conjunction, some conjuncts can be used to provide metadata targeted to the object part, t, of the frame. Generally, the convention associates each annotation to the largest term or formula it precedes.

It is suggestedWe suggest to use Dublin Core, RDFS, and OWL properties for metadata, along the lines of Section 7.1 of [OWL-Reference]-- specifically owl:versionInfo, rdfs:label, rdfs:comment, rdfs:seeAlso, rdfs:isDefinedBy, dc:creator, dc:description, dc:date, and foaf:maker.

• If s occurs as an atomic subformula of the form s(...) with arity α (the arity can correspond to a positional occurrence of s or to an occurrence with named arguments)then s occurs in the context of a predicate symbol with arity α.
• If s occurs as a term (not subformula) of the form s(...) with arity α then s occurs in the context of a function symbol with arity α.
• If s occurs as an atomic subformula External(s(...)) with arity α then s occurs in the context of an external predicate symbol with arity α.
• If s occurs as a term (not subformula) External(s(...)) with arity α then s occurs in the context of an external function symbol with arity α.   ☐

Definition (Imported document). Let Δ be a document formula and Import(t) be one of its import directives, where t is an IRI constant that identifies another document formula, Δ'. We say that Δ' is directly imported into Δ.

A document formula Δ' is said to be imported into Δ if it is either directly imported into Δ or it is imported (directly or not) into some other formula that is directly imported into Δ.     ☐

Definition (Well-formed formula). A formula φ is well-formed iff:

• every constant symbol (whether rif:local or not) mentioned in φ occurs in exactly one context.
• if φ is a document formula and Δ'₁, ..., Δ'_k are all of its imported documents, then every non-rif:local constant symbol mentioned in φ or any of the imported Δ'_is must occur in exactly one context (in all of the Δ'_is).
• Whenever a formula contains a term or a subformula of the form External(t), t must be an instance of the coherent set of external schemas (Section Schemas for Externally Defined Terms of [RIF-DTB]) associated with the language of RIF-BLD.   ☐
• If t is an instance of the coherent set of external schemas associated with the language then t can occur only as External(t), i.e., as an external term or atomic formula.

Definition (Language of RIF-BLD). The language of RIF-BLD consists of the set of all well-formed formulas and is determined by:

• the alphabet of the language and
• a set of coherent external schemas, which determine the available built-ins and other externally defined predicates and functions.   ☐

• The syntax of first-order logic is not context-free, so EBNF cannot capture the syntax of RIF-BLD precisely. For instance, it cannot capture the section on well-formedness conditions, i.e., the requirement that each symbol in RIF-BLD can occur in at most one context. As a result, the EBNF grammar defines a strict superset of RIF-BLD (not all rulesformulas that are derivable using the EBNF grammar are well-formed rulesformulas in RIF-BLD).
• The EBNF grammar does not address theall details of how constants (defined in [RIF-DTB]) and variables are represented, and it is not sufficiently precise about the delimiters and escape symbols. Instead,White space is informally used as thea delimiter, and white spaceis implied in productions that use Kleene star. For instance, TERM* is to be understood as TERM TERM ... TERM, where each ' ' abstracts from one or more blanks, tabs, newlines, etc. This is so because RIF's presentation syntax is a tool for specifying the semantics and for illustration of the main RIF concepts through examples. It is not intended as a concrete syntax for a rule language. RIF defines a concrete syntax only for exchanging rules, and that syntax is XML-based, obtained as a refinement and serialization of the presentation syntax.
• For all the above reasons, the EBNF grammar is not normative. (The non-normative status of the EBNF grammer should not be confused with the normative status ofRecall, however, that the RIF-BLD presentation syntax.)syntax, as specified in mathematical English, is normative.

  FORMULA        ::= IRIMETA? 'And' '(' FORMULA* ')' |
                     IRIMETA? 'Or' '(' FORMULA* ')' |
                     IRIMETA? 'Exists' Var+ '(' FORMULA ')' |
                     ATOMIC |
                     IRIMETA? 'External' '(' Atom | Frame ')'
  ATOMIC         ::= IRIMETA? (Atom | Equal | Member | Subclass | Frame)
  Atom           ::= UNITERM
  UNITERM        ::= Const '(' (TERM* | (Name '->' TERM)*) ')'
  Equal          ::= TERM '=' TERM
  Member         ::= TERM '#' TERM
  Subclass       ::= TERM '##' TERM
  Frame          ::= TERM '[' (TERM '->' TERM)* ']'
  TERM           ::= IRIMETA? (Const | Var | Expr | 'External' '(' Expr ')')
  Expr           ::= UNITERM
  Const          ::= '"' UNICODESTRING '"^^' SYMSPACE | CONSTSHORT
   IRICONST ::= '"' IRI '"^^' 'rif:iri'Name           ::= UNICODESTRING
  Var            ::= '?' UNICODESTRING
  SYMSPACE       ::= ANGLEBRACKIRI | CURIE
 
  IRIMETA        ::= '(*' IRICONST? (Frame | 'And' '(' Frame* ')')? '*)'

The RIF-BLD presentation syntax does not commit to any particular vocabulary and permits arbitrary Unicode strings in constant symbols, argument names, and variables. Constant symbols can have this form: "UNICODESTRING"^^SYMSPACE, where SYMSPACE is a ANGLEBRACKIRI or CURIE that represents an identifier of the symbol space of the constant, and UNICODESTRING is a Unicode string from the lexical space of that symbol space. ANGLEBRACKIRI and CURIE are defined in Section Shortcuts for Constants in RIF's Presentation Syntax of [RIF-DTB]. Constant symbols can also have several shortcut forms, which are represented by the non-terminal CONSTSHORT. These shortcuts are also defined in the same section of [RIF-DTB]. One of them is the CURIE shortcut, which is extensively used in the examples in this document. Names are Unicode character sequences. Variables are composed of UNICODESTRING symbols prefixed with a ?-sign.a ?-sign.

Equality, membership, and subclass terms are self-explanatory. An Atom and Expr (expression) can either be positional or with named arguments. A frame term is a term composed of an object Id and a collection of attribute-value pairs. An External(Atom) is a call to an externally defined predicate; External(Frame) is a call to an externally defined frame. Likewise, External(Expr) is a call to an externally defined function.

As explained in Section RIF-BLD Annotations in the Presentation Syntax, RIF-BLD formulas and terms can be prefixed with optional annotations, IRIMETA, for identification and metadata. IRIMETA is represented using (*...*)-brackets that contain an optional IRI constant, IRICONST, as identifier followed by an optional Frame or conjunction of Frames as metadata. An IRICONST is the special case of a Const with the symbol space rif:iri, again permitting the shortcut forms defined in [RIF-DTB]. One such specialization is '"' IRI '"^^' 'rif:iri' from the Const production, where IRI is a sequence of Unicode characters that forms an internationalized resource identifier as defined by [RFC-3987].

Example 2 (RIF-BLD conditions).

This example shows conditions that are composed of atoms, expressions, frames, and existentials. In frame formulas variables are shown in the positions of object Ids, object properties, and property values. For brevity, we use the CURIE shortcut notation prefix:suffix for constant symbols, which is understood as a macro that expands into an IRI obtained by concatenation of the prefix definition and suffix. Thus, if bks is a prefix that expands into http://example.com/books# then bks:LeRif is an abbreviation for "http://example.com/books#LeRif"^^rif:iri. This and other shortcuts are defined in [RIF-DTB]. Assume that the following prefix directives appear in the preamble to the document:

Prefix(bks  http://example.com/books#)
Prefix(auth http://example.com/authors#)
Prefix(cpt  http://example.com/concepts#)

Positional terms:
  
  cpt:book(auth:rifwg bks:LeRif)
  Exists ?X (cpt:book(?X bks:LeRif))

Terms with named arguments:

  cpt:book(cpt:author->auth:rifwg  cpt:title->bks:LeRif)
  Exists ?X (cpt:book(cpt:author->?X cpt:title->bks:LeRif))

Frames:

  bks:wd1[cpt:author->auth:rifwg cpt:title->bks:LeRif]
  Exists ?X (bks:wd2[cpt:author->?X  cpt:title->bks:LeRif])
  Exists ?X (And (bks:wd2#cpt:book  bks:wd2[cpt:author->?X  cpt:title->bks:LeRif]))
  Exists ?I ?X (?I[cpt:author->?X  cpt:title->bks:LeRif])
  Exists ?I ?X (And (?I#cpt:book ?I[cpt:author->?X  cpt:title->bks:LeRif]))
  Exists ?S (bks:wd2[cpt:author->auth:rifwg ?S->bks:LeRif])
  Exists ?X ?S (bks:wd2[cpt:author->?X ?S->bks:LeRif])
  Exists ?I ?X ?S (And (?I#cpt:book  ?I[author->?X ?S->bks:LeRif]))

The presentation syntax for RIF-BLD rules extends the syntax in Section EBNF for RIF-BLD Condition Language with the following productions.

 Document  ::= IRIMETA? 'Document' '(' Base? Prefix* Import* Group? ')'
 Base      ::= 'Base' '(' IRI ')'
 Prefix    ::= 'Prefix' '(' Name IRI ')'
 Import    ::= IRIMETA? 'Import' '(' IRICONST PROFILE? ')'
 Group     ::= IRIMETA? 'Group' '(' (RULE | Group)* ')'
 RULE      ::= (IRIMETA? 'Forall' Var+ '(' CLAUSE ')') | CLAUSE
 CLAUSE    ::= Implies | ATOMIC
 Implies   ::= IRIMETA? (ATOMIC | 'And' '(' ATOMIC* ')') ':-' FORMULA
 PROFILE   ::= TERM

For convenience, we reproduce the condition language part of the EBNF below.

  FORMULA        ::= IRIMETA? 'And' '(' FORMULA* ')' |
                     IRIMETA? 'Or' '(' FORMULA* ')' |
                     IRIMETA? 'Exists' Var+ '(' FORMULA ')' |
                     ATOMIC |
                     IRIMETA? 'External' '(' Atom | Frame ')'
  ATOMIC         ::= IRIMETA? (Atom | Equal | Member | Subclass | Frame)
  Atom           ::= UNITERM
  UNITERM        ::= Const '(' (TERM* | (Name '->' TERM)*) ')'
  Equal          ::= TERM '=' TERM
  Member         ::= TERM '#' TERM
  Subclass       ::= TERM '##' TERM
  Frame          ::= TERM '[' (TERM '->' TERM)* ']'
  TERM           ::= IRIMETA? (Const | Var | Expr | 'External' '(' Expr ')')
  Expr           ::= UNITERM
  Const          ::= '"' UNICODESTRING '"^^' SYMSPACE | CONSTSHORT
   IRICONST ::= '"' IRI '"^^' 'rif:iri'Name           ::= UNICODESTRING
  Var            ::= '?' UNICODESTRING
  SYMSPACE       ::= ANGLEBRACKIRI | CURIE
 
  IRIMETA        ::= '(*' IRICONST? (Frame | 'And' '(' Frame* ')')? '*)'

Recall that an IRI has the form of an internationalized resource identifier as defined by [RFC-3987].

A RIF-BLD Document consists of an optional Base, followed by any number of Prefixes, followed by any number of Imports, followed by an optional Group. Base and Prefix justserve as shortcut mechanisms for (long)IRIs. An Import indicates the location of a document to be imported and an optional profile. A RIF-BLD Group is a nestedcollection of any number of RULE elements along with any number of nested Groups.

Rules are generated using CLAUSE elements. The RULE production has two alternatives:

• In the first, a CLAUSE is in the scope of the Forall quantifier. In that case, all variables mentioned in CLAUSE are required to also appear among the variables in the Var+ sequence.
• In the second alternative, CLAUSE appears on its own. In that case, CLAUSE cannot have variables.

Frame, Var, ATOMIC, and FORMULA were defined as part of the syntax for positive conditions in Section EBNF for RIF-BLD Condition Language. In the CLAUSE production, an ATOMIC is what is usually called a fact. An Implies rule can have an ATOMIC or a conjunction of ATOMIC elements as its conclusion; it has a FORMULA as its premise. Note that, by a definition in Section Formulas, formulas that query externally defined atoms (i.e., formulas of the form External(Atom(...))) are not allowed in the conclusion part of a rule (ATOMIC does not expand to External).

Example 3 (RIF-BLD rules).

This example shows a business rule borrowed from the document RIF Use Cases and Requirements:

As before, for better readability we use the compact URI notation defined in [[[RIF-DTB], Section Constants and Symbol Spaces. Again, prefix directives are assumed in the preamble to the document. Then, two versions of the main part of the document are given.

Prefix(ppl  http://example.com/people#)
Prefix(cpt  http://example.com/concepts#)
Prefix(func http://www.w3.org/2007/rif-builtin-function#)
Prefix(pred http://www.w3.org/2007/rif-builtin-predicate#)

a. Universal form:

   Forall ?item ?deliverydate ?scheduledate ?diffduration ?diffdays (
        cpt:reject(ppl:John ?item) :-
            And(cpt:perishable(?item)
                cpt:delivered(?item ?deliverydate ppl:John)
                cpt:scheduled(?item ?scheduledate)
                ?diffduration = External(func:subtract-dateTimes(?deliverydate ?scheduledate))
                ?diffdays = External(func:days-from-duration(?diffduration))
                External(pred:numeric-greater-than(?diffdays 10)))
   )

b. Universal-existential form:

   Forall ?item (
        cpt:reject(ppl:John ?item ) :-
            Exists ?deliverydate ?scheduledate ?diffduration ?diffdays (
                 And(cpt:perishable(?item)
                     cpt:delivered(?item ?deliverydate ppl:John)
                     cpt:scheduled(?item ?scheduledate)
                     ?diffduration = External(func:subtract-dateTimes(?deliverydate ?scheduledate))
                     ?diffdays = External(func:days-from-duration(?diffduration))
                     External(pred:numeric-greater-than(?diffdays 10)))
            )
   )

Example 4 (A RIF-BLD document containing an annotated group).

This example shows a complete document containing a group formula that consists of two RIF-BLD rules. The first of these rules is copied from Example 3a. The group is annotated with an IRI identifier and frame-represented Dublin Core metadata.

Document(
  Prefix(ppl  http://example.com/people#)
  Prefix(cpt  http://example.com/concepts#)
  Prefix(dc   http://purl.org/dc/terms/)
  Prefix(func http://www.w3.org/2007/rif-builtin-function#)
  Prefix(pred http://www.w3.org/2007/rif-builtin-predicate#)
  Prefix(xs   http://www.w3.org/2001/XMLSchema#)
  
  (* "http://sample.org"^^rif:iri pd[dc:publisher -> http://www.w3.org/
                                     dc:date -> "2008-04-04"^^xs:date] *)
  Group
  (
    Forall ?item ?deliverydate ?scheduledate ?diffduration ?diffdays (
        cpt:reject(ppl:John ?item) :-
            And(cpt:perishable(?item)
                cpt:delivered(?item ?deliverydate ppl:John)
                cpt:scheduled(?item ?scheduledate)
                ?diffduration = External(func:subtract-dateTimes(?deliverydate ?scheduledate))
                ?diffdays = External(func:days-from-duration(?diffduration))
                External(pred:numeric-greater-than(?diffdays 10)))
    )
 
    Forall ?item (
        cpt:reject(ppl:Fred ?item) :- cpt:unsolicited(?item)
    )
  )
)

This normative section specifies the semantics of RIF-BLD directly, without relying on [RIF-FLD].

Recall that the presentation syntax of RIF-BLD allows the use of macros, which are specified via the Prefix and Base directives.directives, and various shortcuts for integers, strings, and rif:local symbols. The semantics, below, is described using the full syntax, i.e., the description assumeswe assume that all shortcuts and macros have already been expanded as explaineddefined in [RIF-DTB], Section Constants and Symbol Spaces.

The key concept in a model-theoretic semantics of a logic language is the notion of a semantic structure. The definition, below, is a little bit more general than necessary. This is done in order to better see the connection with the semantics of the RIF framework described in [RIF-FLD].

Definition (Semantic structure). A semantic structure, I, is a tuple of the form <TV, DTS, D, D_ind, D_func, I_C, I_V, I_F, I_frame, I _SFNF, I_sub, I_isa, I₌, I_external, I_truth>. Here D is a non-empty set of elements called the domain of I, and D_ind, D_func are nonempty subsets of D. D_ind is used to interpret the elements of Const that are individuals and D_func is used to interpret the elements of Const that are function symbols. As before, Const denotes the set of all constant symbols and Var the set of all variable symbols. TV denotes the set of truth values that the semantic structure uses and DTS is a set of identifiers for primitive datatypes (please refer to Section Datatypes of [RIF-DTB] for the semantics of datatypes).

The other components of I are total mappings defined as follows:

1. I_C maps Const to D.

   This mapping interprets constant symbols. In addition:

2. • If a constant, c ∈ Const, is an individual then it is required that I_C(c) ∈ D_ind.
   • If c ∈ Const, is a function symbol (positional or with named arguments) then it is required that I_C(c) ∈ D_func.
3. I_V maps Var to D_ind.

   This mapping interprets variable symbols.

4. I_F maps D to functions D*_ind → D (here D*_ind is a set of all sequences of any finite length over the domain D_ind).

   This mapping interprets positional terms. In addition:

5. • If d ∈ D_func then I_F(d) must be a function D*_ind → D_ind.
   • This means that when a function symbol is applied to arguments that are individual objects then the result is also an individual object.
6. I _SFNF maps D to the set of total functions of the form SetOfFiniteSets(ArgNames × D_ind) → D.

   This mapping interprets function symbols with named arguments. In addition:

7. • If d ∈ D_func then I _SFNF(d) must be a function SetOfFiniteSets(ArgNames × D_ind) → D_ind.
   • This is analogous to the interpretation of positional terms with two differences:
   • • Each pair <s,v> ∈ ArgNames × D_ind represents an argument/value pair instead of just a value in the case of a positional term.
     • The arguments of a term with named arguments constitute a finite set of argument/value pairs rather than a finite ordered sequence of simple elements. So, the order of the arguments does not matter.
8. I_frame maps D_ind to total functions of the form SetOfFiniteBags(D_ind × D_ind) → D.

   This mapping interprets frame terms. An argument, d ∈ D_ind, to I_frame representrepresents an object and the finite bag {<a1,v1>, ..., <ak,vk>} represents a bag of attribute-value pairs for d. We will see shortly how I_frame is used to determine the truth valuation of frame terms.

   Bags (multi-sets) are used here because the order of the attribute/value pairs in a frame is immaterial and pairs may repeat: o[a->b a->b]. Such repetitions arise naturally when variables are instantiated with constants. For instance, o[?A->?B ?C->?D] becomes o[a->b a->b] if variables ?A and ?C are instantiated with the symbol a and ?B, ?D with b.

9. I_sub gives meaning to the subclass relationship. It is a mapping of the form D_ind × D_ind → D.

   The operator ## is required to be transitive, i.e., c1 ## c2 and c2 ## c3 must imply c1 ## c3. This is ensured by a restriction in Section Interpretation of Formulas.

10. I_isa gives meaning to class membership. It is a mapping of the form D_ind × D_ind → D.

    The relationships # and ## are required to have the usual property that all members of a subclass are also members of the superclass, i.e., o # cl and cl ## scl must imply o # scl. This is ensured by a restriction in Section Interpretation of Formulas.

11. I₌ is a mapping of the form D_ind × D_ind → D.

    It gives meaning to the equality operator.

12. I_truth is a mapping of the form D → TV.

    It is used to define truth valuation for formulas.

13. I_external is a mapping from the coherent set of schemas for externally defined functions to total functions D* → D. For each external schema σ = (?X₁ ... ?X_n; τ) in the [[DTB#def-external-schema-set|coherentcoherent set of external schemas]schemas associated with the language, I_external(σ) is a function of the form Dⁿ → D.

    For every external schema, σ, associated with the language, I_external(σ) is assumed to be specified externally in some document (hence the name external schema). In particular, if σ is a schema of a RIF built-in predicate or function, I_external(σ) is specified in [RIF-DTB] so that:

    • If σ is a schema of a built-in function then I_external(σ) must be the function defined in the aforesaid document.
    • If σ is a schema of a built-in predicate then I_truth ο (I_external(σ)) (the composition of I_truth and I_external(σ), a truth-valued function) must be as specified in [RIF-DTB].

For convenience, we also define the following mapping I from terms to D:

• I(k) = I_C(k), if k is a symbol in Const
• I(?v) = I_V(?v), if ?v is a variable in Var
• I(f(t₁ ... t_n)) = I_F(I(f))(I(t₁),...,I(t_n))
• I(f(s₁->v₁ ... s_n->v_n)) = I _SFNF(I(f))({<s₁,I(v₁)>,...,<s_n,I(v_n)>})

• Here we use {...} to denote a set of argument/value pairs.

• I(o[a₁->v₁ ... a_k->v_k]) = I_frame(I(o))({<I(a₁),I(v₁)>, ..., <I(a_n),I(v_n)>})

• Here {...} denotes a bag of attribute/value pairs.

• I(c1##c2) = I_sub(I(c1), I(c2))
• I(o#c) = I_isa(I(o), I(c))
• I(x=y) = I₌(I(x), I(y))
• I(External(t)) = I_external(σ)(I(s₁), ..., I(s_n)), if t is an instance of the external schema σ = (?X₁ ... ?X_n; τ) by substitution ?X₁/s₁ ... ?X_n/s₁.

  Note that, by definition, External(t) is well formed only if t is an instance of an external schema. Furthermore, by the definition of coherent sets of external schemas, t can be an instance of at most one such schema, so I(External(t)) is well-defined.

The effect of datatypes. The set DTS must include the datatypes described in Section Primitive Datatypes of [RIF-DTB].

The datatype identifiers in DTS impose the following restrictions. Given dt ∈ DTS, let LS_dt denote the lexical space of dt, VS_dt denote its value space, and L_dt: LS_dt → VS_dt the lexical-to-value-space mapping (for the definitions of these concepts, see Section Primitive Datatypes of [RIF-DTB]. Then the following must hold:

• VS_dt ⊆ D_ind; and
• For each constant "lit"^^dt such that lit ∈ LS_dt, I_C("lit"^^dt) = L_dt(lit).

That is, I_C must map the constants of a datatype dt in accordance with L_dt.

RIF-BLD does not impose restrictions on I_C for constants in symbol spaces that are not datatypes mentioned in DTS.   ☐

1. Positional atomic formulas: TVal_I(r(t₁ ... t_n)) = I_truth(I(r(t₁ ... t_n)))
2. Atomic formulas with named arguments: TVal_I(p(s₁->v₁ ... s_k->v_k)) = I_truth(I(p(s₁->v₁ ... s_k->v_k))).
3. Equality: TVal_I(x = y) = I_truth(I(x = y)).
   • To ensure that equality has precisely the expected properties, it is required that:
     I_truth(I(x = y)) = t if and only if I(x) = I(y) and that I_truth(I(x = y)) = f otherwise.
   • This is tantamount to saying that TVal_I(x = y) = t if I(x) = I(y).
4. Subclass: TVal_I(sc ## cl) = I_truth(I(sc ## cl)).

   To ensure that the operator ## is transitive, i.e., c1 ## c2 and c2 ## c3 imply c1 ## c3, the following is required:

   For all c1, c2, c3 ∈ D,   if TVal_I(c1 ## c2) = TVal_I(c2 ## c3) = t   then TVal_I(c1 ## c3) = t.
5. Membership: TVal_I(o # cl) = I_truth(I(o # cl)).

   To ensure that all members of a subclass are also members of the superclass, i.e., o # cl and cl ## scl implies o # scl, the following is required:

   For all o, cl, scl ∈ D,   if TVal_I(o # cl) = TVal_I(cl ## scl) = t   then   TVal_I(o # scl) = t.
6. Frame: TVal_I(o[a₁->v₁ ... a_k->v_k]) = I_truth(I(o[a₁->v₁ ... a_k->v_k])).

   Since the bag of attribute/value pairs represents the conjunctions of all the pairs, the following is required, if k > 0:

   TVal_I(o[a₁->v₁ ... a_k->v_k]) = t if and only if TVal_I(o[a₁->v₁]) = ... = TVal_I(o[a_k->v_k]) = t.
7. Externally defined atomic formula: TVal_I(External(t)) = I_truth(I_external(σ)(I(s₁), ..., I(s_n))), if t is an atomic formula that is an instance of the external schema σ = (?X₁ ... ?X_n; τ) by substitution ?X₁/s₁ ... ?X_n/s₁.

   Note that, by definition, External(t) is well-formed only if t is an instance of an external schema. Furthermore, by the definition of coherent sets of external schemas, t can be an instance of at most one such schema, so I(External(t)) is well-defined.

8. Conjunction: TVal_I(And(c₁ ... c_n)) = t if and only if TVal_I(c₁) = ... = TVal_I(c_n) = t. Otherwise, TVal_I(And(c₁ ... c_n)) = f.

9. The empty conjunction is treated as a tautology, so TVal_I(And()) = t.

10. Disjunction: TVal_I(Or(c₁ ... c_n)) = f if and only if TVal_I(c₁) = ... = TVal_I(c_n) = f. Otherwise, TVal_I(Or(c₁ ... c_n)) = t.

11. The empty disjunction is treated as a contradiction, so TVal_I(Or()) = f.

12. Quantification:
    • TVal_I(Exists ?v₁ ... ?v_n (φ)) = t if and only if for some I*, described below, TVal_I*(φ) = t.
    • TVal_I(Forall ?v₁ ... ?v_n (φ)) = t if and only if for every I*, described below, TVal_I*(φ) = t.

    Here I* is a semantic structure of the form <TV, DTS, D, D_ind, D_func, I_C, I*_V, I_F, I_frame, I _SFNF, I_sub, I_isa, I₌, I_external, I_truth>, which is exactly like I, except that the mapping I*_V, is used instead of I_V.   I*_V is defined to coincide with I_V on all variables except, possibly, on ?v₁,...,?v_n.

13. Rule implication:
    • TVal_I(conclusion :- condition) = t, if either TVal_I(conclusion)=t or TVal_I(condition)=f.
    • TVal_I(conclusion :- condition) = f   otherwise.
14. Groups of rules:

    If Γ is a group formula of the form Group(φ₁ ... φ_n) then

    • TVal_I(Γ) = t if and only if TVal_I(φ₁) = t, ..., TVal_I(φ_n) = t.
    • TVal_I(Γ) = f   otherwise.

    This means that a group of rules is treated as a conjunction.   ☐

Definition (Semantic multi-structure). A semantic multi-structure is a set {I^Δ₁, ..., I^Δ_n}, n>0, where I^Δ₁, ..., I^Δ_n are semantic structures labeledadorned with document formulas. These structures must be identical in all respects except that the mappings I_C^Δ₁, ..., I_C^Δ_n might differ on the constants in Const that belong to the rif:local symbol space. The above set is allowed to have at most one semantic structure with the same label.adornment.     ☐

With the help of semantic multi-structures we can now explain the semantics of RIF documents.

Definition (Truth valuation of a document formula). Let Δ be a document formula and let Δ₁, ..., Δ_k be all the RIF-BLD document formulas that are imported (directly or indirectly, according to Definition Imported document) into Δ. Let Γ, Γ₁, ..., Γ_k denote the respective group formulas associated with these documents. If any of these Γ_i is missing (which is a possibility, since every part of a document is optional), assume that it is a tautology, such as a = a, so that every TVal function maps such a Γ_i to the truth value t. Let I = {I^Δ, I^Δ₁, ..., I^Δ_k, ...} be a semantic multi-structure, whichmulti-structure that contains semantic structures labeledadorned with at least the documents Δ, Δ₁, ..., Δ_k. Then we define:

Note that this definition considers only those document formulas that are reachable via the one-argument import directives. Two argument import directives are ignorednot covered here. Their semantics is defined by the document RIF RDF and OWL Compatibility [RIF-RDF+OWL].         ☐

The above definitions make the intent behind the rif:local constants clear: occurrences of such constants in different documents can be interpreted differently even if they have the same name. Therefore, each document can choose the names for the rif:local constants freely and without regard to the names of such constants used in the imported documents.

• a normative mapping from the RIF-BLD presentation syntax to XML (Section Translation BetweenMapping from the RIF-BLDPresentation andSyntax to the XML SyntaxesSyntax), and
• a normative XML Schema for the XML syntax (Appendix XML Schema for BLD).

Recall that the syntax of RIF-BLD is not context-free and thus cannot be fully captured by EBNF and XML Schema. Still, validity with respect to XML Schema can be a useful test. To reflect this state of affairs, we define two notions of syntactic correctness. The weaker notion checks correctness only with respect to XML Schema, while the stricter notion represents "true" syntactic correctness.

Definition (Valid BLD document in XML syntax). A valid BLD document in the XML syntax is an XML document that is valid w.r.t.with respect to the XML schema in Appendix XML Schema for BLD.   ☐

Definition (Conformant BLD document in XML syntax). A conformant BLD document in the XML syntax is a valid BLD document in the XML syntax that is the image of a well-formed RIF-BLD document in the presentation syntax (see Definition Well-formed formula in Section Formulas) under the presentation-to-XML syntax mapping χ_bld defined in Section Translation BetweenMapping from the RIF-BLDPresentation andSyntax to the XML SyntaxesSyntax.   ☐

The XML serialization for RIF-BLD is alternating or fully striped [ANF01]. A fully striped serialization views XML documents as objects and divides all XML tags into class descriptors, called type tags, and property descriptors, called role tags [TRT03]. We follow the tradition of using capitalized names for type tags and lowercase names for role tags.

The all-uppercase classes in the presentation syntax, such as FORMULA, become XML Schema groups in Appendix XML Schema for BLD. They act like macros and are not visible in instance markup. The other classes as well as non-terminals and symbols (such as Exists or =) become XML elements with optional attributes, as shown below.

RIF-BLD uses [XML1.0] for its XML syntax.

XML serialization of RIF-BLD in Section EBNF for RIF-BLD Condition Language uses the following elements.

- And       (conjunction)
- Or        (disjunction)
- Exists    (quantified formula for 'Exists', containing declare and formula roles)
- declare   (declare role, containing a Var)
- formula   (formula role, containing a FORMULA)
- Atom      (atom formula, positional or with named arguments)
- External  (external call, containing a content role)
- content   (content role, containing an Atom, for predicates, or Expr, for functions)
- Member    (member formula)
- Subclass  (subclass formula)
- Frame     (Frame formula)
- object    (Member/Frame role, containing a TERM or an object description)
- op        (Atom/Expr role for predicates/functions as operations)
- args      (Atom/Expr positional arguments role, with fixed 'ordered' attribute, containing n TERMs)
- instance  (Member instance role)
- class     (Member class role)
- sub       (Subclass sub-class role)
- super     (Subclass super-class role)
- slot      (Atom/Expr or Frame slot role, with fixed 'ordered' attribute, containing a Name or TERM followed by a TERM)
- Equal     (prefix version of term equation '=')
- Expr      (expression formula, positional or with named arguments)
- left      (Equal left-hand side role)
- right     (Equal right-hand side role)
- Const     (individual, function, or predicate symbol, with optional 'type' attribute)
- Name      (name of named argument)
- Var       (logic variable)
   
- id        (identifier role, containing IRICONST)
- meta      (meta role, containing metadata as a Frame or Frame conjunction)

The id and meta elements, which are expansions of the IRIMETA element, can occur optionally as the initial children of any Class element.

For the XML Schema definition of the RIF-BLD condition language see Appendix XML Schema for BLD.

The XML syntax for symbol spaces utilizesuses the type attribute associated with the XML term elements such aselement Const. For instance, a literal in the xs:dateTime datatype can beis represented as <Const type="&xs;dateTime">2007-11-23T03:55:44-02:30</Const>. RIF-BLD also utilizesuses the ordered attribute to indicate that the orderedness ofchildren of the elementsargs and slot it is associated with.elements are ordered.

Example 5 (A RIF condition and its XML serialization).

This example illustrates XML serialization for RIF conditions. As before, the compact URI notation is used for better readability. Assume that the following prefix directives are found in the preamble to the document:

Prefix(bks    http://example.com/books#)
Prefix(cpt    http://example.com/concepts#)
Prefix(curr   http://example.com/currencies#)
Prefix(rif    http://www.w3.org/2007/rif#)
Prefix(xs     http://www.w3.org/2001/XMLSchema#)

RIF condition

   And (Exists ?Buyer (cpt:purchase(?Buyer ?Seller
                                    cpt:book(?Author bks:LeRif)
                                    curr:USD(49)))
        ?Seller=?Author )

XML serialization

   <And>
     <formula>
       <Exists>
         <declare><Var>Buyer</Var></declare>
         <formula>
           <Atom>
             <op><Const type="&rif;iri">&cpt;purchase</Const></op>
             <args ordered="yes">
               <Var>Buyer</Var>
               <Var>Seller</Var>
               <Expr>
                 <op><Const type="&rif;iri">&cpt;book</Const></op>
                 <args ordered="yes">
                   <Var>Author</Var>
                   <Const type="&rif;iri">&bks;LeRif</Const>
                 </args>
               </Expr>
               <Expr>
                 <op><Const type="&rif;iri">&curr;USD</Const></op>
                 <args ordered="yes"><Const type="&xs;integer">49</Const></args>
               </Expr>
             </args>
           </Atom>
         </formula>
       </Exists>
     </formula>
     <formula>
       <Equal>
         <left><Var>Seller</Var></left>
         <right><Var>Author</Var></right>
       </Equal>
     </formula>
   </And>

Example 6 (A RIF condition with named arguments and its XML serialization).

This example illustrates XML serialization of RIF conditions that involve terms with named arguments. As in Example 5, we assume the following prefix directives:

Prefix(bks    http://example.com/books#)
Prefix(cpt    http://example.com/concepts#)
Prefix(curr   http://example.com/currencies#)
Prefix(rif    http://www.w3.org/2007/rif#)
Prefix(xs     http://www.w3.org/2001/XMLSchema#)

RIF condition:

   And (Exists ?Buyer ?P (
                 And (?P#cpt:purchase
                      ?P[cpt:buyer->?Buyer
                         cpt:seller->?Seller
                         cpt:item->cpt:book(cpt:author->?Author cpt:title->bks:LeRif)
                         cpt:price->49
                         cpt:currency->curr:USD]))
        ?Seller=?Author)


XML serialization:

   <And>
     <formula>
       <Exists>
         <declare><Var>Buyer</Var></declare>
         <declare><Var>P</Var></declare>
         <formula>
           <And>
             <formula>
               <Member>
                 <instance><Var>P</Var></instance>
                 <class><Const type="&rif;iri">&cpt;purchase</Const></class>
               </Member>
             </formula>
             <formula>
               <Frame>
                 <object>
                   <Var>P</Var>
                 </object>
                 <slot ordered="yes">
                   <Const type="&rif;iri">&cpt;buyer</Const>
                   <Var>Buyer</Var>
                 </slot>
                 <slot ordered="yes">
                   <Const type="&rif;iri">&cpt;seller</Const>
                   <Var>Seller</Var>
                 </slot>
                 <slot ordered="yes">
                   <Const type="&rif;iri">&cpt;item</Const>
                   <Expr>
                     <op><Const type="&rif;iri">&cpt;book</Const></op>
                     <slot ordered="yes">
                       <Name>&cpt;author</Name>
                       <Var>Author</Var>
                     </slot>
                     <slot ordered="yes">
                       <Name>&cpt;title</Name>
                       <Const type="&rif;iri">&bks;LeRif</Const>
                     </slot>
                   </Expr>
                 </slot>
                 <slot ordered="yes&