2.2 Alphabet of RIF-BLDThe alphabetSemantics of RIF-BLD consists of a countably infinite set of constant symbols Const , a countably infinite set of variable symbols Var (disjoint from Const ),as a
countably infinite setSpecialization of argument names, ArgNames (disjoint from Const and Var ), connective symbols And and Or , quantifiers Exists and Forall ,RIF-FLD
This normative section defines the symbols = , # , ## , -> , :- , and auxiliary symbols, such as "(" and ")".precise relationship between
the setsemantics of connective symbols, quantifiers, = , etc., is disjoint from ConstRIF-BLD and Var . Variables are written as Unicode strings preceded withthe symbol "?".semantic framework of RIF-FLD.
Specification of the syntax for constant symbolssemantics without reference to RIF-FLD is
given in Section Symbol SpacesDirect Specification of RIF-FLD.RIF-BLD Semantics.
The languagesemantics of RIF-BLDthe RIF Basic Logic Dialect is defined by
specialization from the setsemantics of formulas constructed usingthe above alphabet accordingSemantic Framework for Logic
Dialects of RIF. Section Semantics of a RIF Dialect as a Specialization of the RIF
Framework in that document lists the parameters of the semantic
framework, which we need to specialize for RIF-BLD.
Recall that the rules spelled out below. 2.3 Terms RIF-BLD supports several kindssemantics of terms: constants and variables , positional terms, terms with named arguments , equality , membership , and subclass terms, and framesa dialect is derived
from these notions by specializing the following parameters.
- The effect of the syntax.
-
RIF-BLD does not support negation. This is the word " term " will be used to referonly obvious
simplification with respect to any kind of terms. Formally, terms are definedRIF-FLD as follows: Constants and variables . If t ∈ Const or t ∈ Var then tfar as the semantics is
a simple term . Positional termsconcerned.
- Truth values.
-
IfThe set TV of truth values in RIF-BLD consists of
just two values, t ∈ Constand f such that f
<t 1 , ...,t n are terms then t(t 1 .... Clearly, <t n )is a positional termtotal
order here.
- Data types.
-
Terms with named argumentsRIF-BLD supports all the data types listed in Section Primitive Data
Types of RIF-FLD:
- xsd:long
- xsd:integer
- xsd:decimal
- xsd:string
- xsd:time
- xsd:dateTime
- rdf:XMLLiteral
- rif:text
- Logical entailment.
-
A term with named arguments (a term with named arguments)Recall that logical entailment in RIF-FLD is of the form t(s 1 ->v 1 ... s n ->v n ) , where t ∈ Const , v 1 , ..., v n are terms (positional,defined with
named arguments, frame, etc.),respect to an unspecified set of intended semantic structures and
s 1 , ..., s n are (not necessarily distinct) symbols from thethat dialects of RIF must make this notion concrete. For RIF-BLD,
this set ArgNames .is defined in one of the term t here representstwo following equivalent
ways:
- as a
predicateset of all models; or
-
a function; s 1 , ..., s n represent argument names; and v 1 , ..., v n represent argument values. Terms with named arguments are like positional terms except thatas the argumentsunique minimal model.
These two definitions are named and their orderequivalent for entailment of RIF-BLD
conditions by RIF-BLD sets of formulas, since all rules in RIF-BLD
are Horn -- it is immaterial. Note thata term like f() is both positionalclassical result of Van Emden and with named arguments. Equality terms . If tKowalski
[vEK76].
3 Direct Specification of RIF-BLD
Syntax
This normative section specifies the syntax of RIF-BLD directly,
without referring to RIF-FLD. We define both a membership term if tpresentation
syntax and s are arbitrary terms. Subclass terms . t##san XML syntax. The presentation syntax is not
intended to be a subclass term if t and s are arbitrary terms. Frame terms . t[p 1 ->v 1 ... p n ->v n ]concrete syntax for RIF-BLD. It is a frame term (or simply a frame ) if t , p 1 , ..., p n , v 1 , ..., v n , n ≥ 0, are arbitrary terms. Membership, subclass,defined in
Mathematical English and frame terms are usedis intended to describe objects in object-based logics like F-logic [ KLW95 ]. These terms canbe readily mixed both with positional termsused in the definitions
and terms with named arguments: p(?X q#r[v(1,2)->s] t(d->e f->g)) . 2.4 Well-formednessexamples. This syntax deliberately leaves out details such as
the setdelimiters of all symbols, Const , is partitioned into positional predicate symbols, predicate symbols with named arguments, positional function symbols, function symbols with named arguments, and individuals. Each positional predicate and function symbol has precisely one arity , which is a non-negative integer that tells how many argumentsthe symbol can take. An arity for terms with named arguments (of a symbol with named arguments) is a bag {s 1 ... s k }various syntactic components, escape symbols,
parenthesizing, precedence of argument names ( s i ∈ ArgNames ). Each predicate or function symbol with named arguments has precisely one arity (for terms with named arguments).operators, and the arity of a symbol (or whether itlike. Since RIF is
a predicate, a function, oran individual) is not specified explicitly in RIF-BLD. Instead,interchange format, it is inferreduses XML as follows. Each constant symbol in aits concrete syntax.
3.1 Alphabet of RIF-BLD
formula (orDefinition
(Alphabet). The alphabet of RIF-BLD consists
of
- a countably infinite set of
formulas) is expected to occur in at most one context: as an individual,constant symbols
Const
- a
function symbolcountably infinite set of variable symbols
Var (disjoint from Const)
- a
particular arity, a predicate symbolcountably infinite set of a particular arity,argument names, ArgNames
(disjoint from Const and Var)
- connective symbols And, Or
an individual., and
:-
- quantifiers Exists and Forall
- the
arity ofsymbols =, #, ##,
->, and External
- the grouping symbol Group
- auxiliary symbols, such as "(" and
its type")"
The set of connective symbols, quantifiers, =, etc., is
then determined by its context. If a symboldisjoint from Const occurs in more than one context,and Var. The formula (or a set of formulas) is not considered to be well-formedargument names
in RIF-BLD. 2.5 Formulas Any term (positional orArgNames are written as unicode strings that must not
start with a question mark, "?". Variables are written as
Unicode strings preceded with named arguments) ofthe form p(...)symbol "?".
Constants are written as "literal"^^symspace, where
pliteral is a predicate symbol,sequence of Unicode characters and
symspace is alsoan atomic formulaidentifier for a symbol space. Symbol
spaces are defined in Section Symbol
Spaces of the RIF-FLD document.
|
The definition of symbol spaces will eventually be also given in
the document Data Types
and Builtins, so the above reference will be to that document
instead of RIF-FLD.
|
The symbols =, #, and ## are used in
formulas that define equality, class membership, subclass,and frame terms are also atomic formula. Simplesubclass
relationships. The symbol -> is used in terms (constantsthat have
named arguments and variables) are notin frame formulas. Not all atomic formulas are well-formed -- see Section Well-formedness . A well-formed atomic formula isThe symbol External
indicates that an atomic formula thator a function term is alsodefined
externally (e.g., a well-formed term. More general formulas are constructed out ofbuiltin).
The atomic formulassymbol Group is used to organize RIF-BLD rules into
collections and annotate them with metadata. ☐
The helplanguage of logical connectives. A formulaRIF-BLD is a statement that can have one ofthe following forms: Atomic : If φ is a well-formed atomic formula then it is also a formula. Conjunction : If φ 1 , ..., φ n , n ≥ 0, areset of formulas then so is And( φ 1 ... φ n ) . As a special case, And() is allowedconstructed using
the above alphabet according to the rules given below.
3.2 Terms
RIF-BLD supports several kinds of terms: constants and
is treated as a tautology, i.e., a formula that is always true. Disjunction : If φ 1variables, ..., φ npositional terms, terms with named
arguments, n ≥ 0, are formulas then so is Or( φ 1 ... φ n ) . When n=0equality, we get Or() as a special case; it is treated as a formula that is always false. Existentials :membership, and
subclass atomic formulas, and frame formulas. The
word "term" will be used to refer to any kind of these
constructs.
Definition
(Term).
- Constants and variables. If
φt ∈ Const
or t ∈ Var then t is a formulasimple
term.
- Positional terms. If t ∈ Const and
?Vt1, ..., ?Vtn are
variablessimple, positional, or named-argument terms
then Exists ?Vt(t1 ... ?Vtn ( φ) is a
formula. Formulas constructed using the above definitions are called RIF-BLD conditionspositional term.
- Terms with named arguments.
RIF-BLD rules are defined as follows: Rule : If φ is an atomic formula and ψ isA RIF-BLD condition then φ :- ψterm with named
arguments is a formula, provided that φ does not haveof the signature bi_atomic (i.e., is not a builtin predicate). Universals : If φ is a ruleform
t(s1->v1 ...
sn->vn), where t ∈
Const and ?Vv1, ...,
?Vvn are variables then Forall ?V 1 ... ?V n ( φ ) is a formula, called an explicitly quantified rule . 2.6 EBNF Grammar forsimple, positional, or
named-argument terms and s1, ...,
sn are pairwise distinct symbols from the Presentation Syntax of RIF-BLD So far,set
ArgNames.
The syntax of RIF-BLD was specified in Mathematical English. Tool developers, however, preferterm t here represents a predicate or a function;
s1, ..., sn represent
argument names; and v1, ...,
vn represent argument values. The more formal EBNF notation, which we will give next. Several points shouldargument
names, s1, ..., sn, are
required to be kept in mind regarding this notation.pairwise distinct. Terms with named arguments are
like positional terms except that the syntax of first-order logicarguments are named and their
order is not context-free, so EBNF cannot capture the syntax of RIF-BLD precisely. For instance, it cannot capture the well-formedness conditions , i.e., the requirementimmaterial. Note that each symbol in RIF-BLD can occur in at most one context. As a result, the grammar, below, defines onlya supersetterm of RIF-BLD.the EBNF syntaxform f() is
not a concrete syntax: it does not address the details of how constantsboth positional and variables are represented,with named arguments.
- Equality terms. If t and
its are
simple, positional, or named-argument terms
then t = s is not sufficiently precise about the delimiters and escape symbols. Instead, white spacean equality
term.
- Class membership terms (or just membership
terms). t#s is
informally used asa delimiter,membership term if
t and white space is implied in productions that use Kleene star. For instance, TERM* is to be understood as TERM TERM ... TERMs are simple, positional, where each ' ' abstracts from oneor
more blanks, tabs, newlines, etc. This is done on purpose, since RIF's presentation syntaxnamed-argument terms.
- Subclass terms. t##s is
intended asa tool for specifying the semanticssubclass
term if t and for illustration of the main RIF concepts through examples. Its are simple,
positional, or named-argument terms.
- Frame terms. t[p1->v1 ...
pn->vn] is
not intended asa concrete syntax forframe term
(or simply a rule language. RIF definesframe) if t,
p1, ..., pn,
v1, ..., vn, n ≥
0, are simple, positional, or
named-argument terms.
- Externally defined terms. If t is a
concrete syntax onlyterm then
External(t) is an externally defined
term.
-
Such terms are used for exchanging rules,representing builtin functions and
that syntax is XML-based, obtainedpredicates as a refinementwell as "procedurally attached" terms or predicates,
which might exist in various rule-based systems, but are not
specified by RIF. ☐
Membership, subclass, and serialization of the EBNF syntax. 2.6.1 EBNF for RIF-BLD Condition Language The Condition Language represents formulas that can beframe terms are used in the body of the RIF-BLD rules. It is supposedto be a common part of a numberdescribe
objects and class hierarchies.
3.3 Well-formedness of RIF dialects, including RIF PRD .Terms
The EBNF grammar for a supersetset of the RIF-BLD condition languageall symbols, Const, is as follows. CONDITION ::= 'And' '(' CONDITION* ')' | 'Or' '(' CONDITION* ')' | 'Exists' Var+ '(' CONDITION ')' | COMPOUND COMPOUND ::= Uniterm | Equal | Member | Subclass | Frame Uniterm ::= Const '(' (TERM* | (Const '->' TERM)*) ')' Equal ::= TERM '=' TERM Member ::= TERM '#' TERM Subclass ::= TERM '##' TERM Frame ::= TERM '[' (TERM '->' TERM)* ']' TERM ::= Const | Var | COMPOUND Const ::= LITERAL '^^' SYMSPACE Var ::= '?' VARNAMEpartitioned into
Each predicate and disjunctions of conditions, respectively. Exists introduces existentially quantified variables. Here Var+ standsfunction symbol has precisely one
arity.
- For positional symbols, an arity is a non-negative integer that
tells how many arguments the
list of variablessymbol can take.
- For symbols that take named arguments, an arity is a set
{s1 ... sk} of argument names
(si ∈ ArgNames), which are
freeallowed for
that symbol.
The arity of a symbol (or whether it is a predicate, a function,
or an individual) is not specified in CONDITION .RIF-BLD conditions permit only existential variables, but RIF-FLD syntax allows arbitrary quantification, which can be usedexplicitly. Instead,
it is inferred as follows. Each constant symbol in some dialects.a CONDITION can also beRIF-BLD
formula (or a COMPOUND term, i.e.set of formulas) may occur in at most one context: as
an individual, a Uniterm , Equal , Member , Subclass ,function symbol of a particular arity, or Frame .a
predicate symbol of a particular arity. The production forarity of the non-terminal TERM defines RIF-BLD terms -- constants, variables, or COMPOUND terms.symbol and
its type is then determined by its context. If a symbol from
Const occurs in more than one context in a set of
formulas, the RIF-BLD presentation syntax doesset is not commit to any particular vocabularywell-formed in RIF-BLD.
For the namesa term of variables or forthe literals used in constant symbols.form External(t) to be well-formed,
t must be an instance of an external
schema, i.e., a schema of an externally specified term, as
defined in Section Schemas for Externally Defined Terms of RIF-FLD.
Also, if a term of the examples, variables are denoted by Unicode character sequences beginning with a ?-sign. Constant symbols have the form: LITERAL^^SYMSPACE , where SYMSPACE isform External(p(...)) occurs as
an IRI string that identifiesatomic formula then the symbol spaceoccurrence of the constant and LITERALp is considered
to be a Unicode string frompredicate occurrence.
|
The lexical spacedefinition of external schemas will eventually also appear
in the document Data
Types and Builtins, so the above reference will be to that
symbol space.document instead of RIF-FLD.
|
3.4 Formulas
Any term (positional or with named arguments) of the form
p(...) (or External(p(...)), where p is
a predicate symbol, is also an atomic formula.
Equality, membership, subclass, and subclassframe terms are self-explanatory. Uniterms ( Universalalso atomic
formulas. A formula of the form External(p(...)) is also
called an externally defined atomic formula.
Simple terms )(constants and variables) are terms that can be either positional or with named arguments.not formulas.
Not all atomic formulas are well-formed. A frame termwell-formed atomic
formula is an atomic formula that is also a well-formed
term composed(see Section Well-formedness of an object Id and a collection of attribute-value pairs. Example 1 shows conditions thatTerms). More general formulas are
composedconstructed out of uniterms, frames, and existentials.the examples ofatomic formulas with the frames showhelp of logical
connectives.
Definition
(Well-formed formula). A well-formed formula is a
statement that variables can occur in the syntactic positionshas one of object Ids, object properties, or property values. Example 1 (RIF-BLD conditions) We use the prefix bks to abbreviate http://example.com/books# and the prefix auth for http://example.com/authors#. Positional terms: book^^rif:local(auth:rifwg^^rif:iri bks:LeRif^^rif:iri) Exists ?X (book^^rif:local(?X LeRif^^rif:local)) Terms with named arguments: book^^rif:local(author^^rif:local->auth:rifwg^^rif:iri title^^rif:local->bks:LeRif^^rif:iri) Exists ?X (book^^rif:local(author^^rif:local->?X title^^rif:local->bks:LeRif^^rif:iri)) Frames: wd1^^rif:local[author^^rif:local->auth:rifwg^^rif:iri title^^rif:local->bks:LeRif^^rif:iri ] Exists ?X (wd2^^rif:local[author^^rif:local->?X title^^rif:local->bks:LeRif^^rif:iri ]) Exists ?X (wd2^^rif:local#book^^rif:local[author^^rif:local->?X title^^rif:local->bks:LeRif^^rif:iri]) Exists ?I ?X (?I[author^^rif:local->?X title^^rif:local->bks:LeRif^^rif:iri]) Exists ?I ?X (?I#book^^rif:local[author^^rif:local->?X title^^rif:local->bks:LeRif^^rif:iri]) Exists ?S (wd2^^rif:local[author^^rif:local->auth:rifwg^^rif:iri ?S->bks:LeRif^^rif:iri]) Exists ?X ?S (wd2^^rif:local[author^^rif:local->?X ?S->bks:LeRif^^rif:iri]) Exists ?I ?X ?S (?I#book^^rif:local[author->?X ?S->bks:LeRif^^rif:iri]) 2.6.2 EBNF for RIF-BLD Rule Language The presentation syntax for Horn rules extends the syntax in Section EBNF for RIF-BLD Condition Language withthe following productions. Ruleset ::= RULE* RULE ::= 'Forall' Var+ '(' RULE ')' | Implies | COMPOUND Implies ::= COMPOUND ':-' CONDITIONforms:
- Atomic: If φ is a
Rulesetwell-formed atomic formula
then it is also a set of RIF rules. Rules are generated by the Implies production, with optional Forall -quantification. Varwell-formed formula.
- Conjunction: If φ1,
COMPOUND...,
φn, n ≥ 0, and CONDITION were defined as part of the syntax for positive conditions in Section EBNF for RIF-BLD Condition Language . Note that COMPOUND termsare well-formed formulas
then so is And(φ1 ... φn), called a
conjunctive formula. As a special case, And() is
allowed and is treated as rules with an empty condition part -- they are usually called facts . Note that, bya definition in Section Formulas , atomic formulastautology, i.e., a formula that correspond to builtin predicates (i.e.,is
always true.
- Disjunction: If φ1, ...,
φn, n ≥ 0, are well-formed formulas
with signature bi_atomicthen so is Or(φ1 ... φn), called a
disjunctive formula. When n=0, we get Or()
as a special case; it is treated as a contradiction, i.e., a
formula that is always false.
- Existentials: If φ is a well-formed formula and
?V1, ..., ?Vn are
not allowed invariables
then Exists ?V1 ... ?Vn(φ)
is an existential formula.
Formulas constructed using the conclusion partabove definitions are called
RIF-BLD
conditions. The following formulas lead to the
notion of a RIF-BLD rule.
-
This restrictionRule implication: If φ is not reflected in the EBNF syntax. The document RIF Use Casesan well-formed atomic
formula and Requirements includesψ is a use case "Negotiating eBusiness Contracts Across Rule Platforms", which discussesRIF-BLD condition then φ :-
ψ is a businesswell-formed formula, called rule implication,
provided that φ is not externally defined (i.e.,
does not have the form External(...)).
- Quantified rule
slightly modified here:: If an itemφ is perishablea rule implication and
?V1, ..., ?Vn are variables
then Forall ?V1 ... ?Vn(φ)
is a well-formed formula, called quantified rule. It is
delivered to John more than 10 days after the scheduled delivery date thenrequired that all the item will be rejected by him.free (i.e., non-quantified) variables in
the Presentation EBNF Syntax used throughout this document, this rule can be writtenφ occur in one of these two equivalent ways: Example 2 (RIF-BLD rules) Here we use the prefix ppl as an abbreviation for http://example.com/people#.the prefix op is used for a yet-to-be-determined IRI, whichForall ?V1
... ?Vn. Quantified rules will also be
used for RIF builtin predicates. a. Universal form: Forall ?item ?deliverydate ?scheduledate ?diffduration ?diffdays ( reject^^rif:local(ppl:John^^rif:iri ?item) :- And(perishable^^rif:local(?item) delivered^^rif:local(?item ?deliverydate ppl:John^^rif:iri) scheduled^^rif:local(?item ?scheduledate) fn:subtract-dateTimes-yielding-dayTimeDuration(?deliverydate ?scheduledate ?diffduration) fn:get-days-from-dayTimeDuration(?diffduration ?diffdays) op:numeric-greater-than(?diffdays 10)) ) b. Universal-existential form: Forall ?item ( reject^^rif:local(ppl#John^^rif:iri ?item ) :- Exists ?deliverydate ?scheduledate ?diffduration ?diffdays ( And(perishable^^rif:local(?item) delivered^^rif:local(?item ?deliverydate ppl:John^^rif:iri) scheduled^^rif:local(?item ?scheduledate) fn:subtract-dateTimes-yielding-dayTimeDuration(?deliverydate ?scheduledate ?diffduration) fn:get-days-from-dayTimeDuration(?diffduration ?diffdays) op:numeric-greater-than(?diffdays 10)) ) ) 2.7 XML Serialization for the Interchange of RIF-BLD The XML serialization forreferred to as RIF-BLD presentation syntax given in this sectionrules.
- Group: If φ is
alternatinga frame term and
ρ1, ..., ρn are RIF-BLD
rules or fully striped (e.g., Alternating Normal Form ). Positional information is optionally exploited only for the arg role elements. For example, role elements ( declaregroup formulas (they can be mixed) then Group φ
(ρ1 ... ρn) and formulaGroup (ρ1
... ρn) are explicit withingroup formulas.
Group formulas are intended to represent sets of rules annotated
with metadata. This metadata is specified using an optional frame
term φ. Note that some of the Exists element. Followingρi's can
be group formulas themselves, which means that groups can be
nested. This allows one to attach metadata to various subsets of
rules, which may be inside larger rule sets, which in turn can be
annotated. ☐
It can be seen from the examplesdefinitions that RIF-BLD has a wide
variety of Java and RDF, we use capitalized namessyntactic forms for class elementsterms and namesformulas. This provides
the infrastructure for exchanging rule languages that start with lowercasesupport rich
collections of syntactic forms. Systems that do not support some of
that syntax directly can still support it through syntactic
transformations. For role elements. The all-uppercase classesinstance, disjunctions in the presentation syntax,rule body can be
eliminated through a standard transformation, such as CONDITION , become XML entities. 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 elementsreplacing
p :- Or(q r) with optional attributes, as shown below. 2.7.1 XML for RIF-BLD Condition Language We now serialize the syntax of Section EBNF for RIF-BLD Condition Language in XML. Classes, roles and their intended meaning - 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 CONDITION formula) - Uniterm (term or atomic formula, positional or with named arguments) - Member (member formula) - Subclass (subclass formula) - Frame (Frame formula) - object (Member/Frame role containinga TERM or an object description) - op (Uniterm role for predicates/functions as operations) - arg (argument role) - upper (Member/Subclass upper class role) - lower (Member/Subclass lower instance/class role) - slot (Uniterm/Frame slot role, prefix version of slot infix ' -> ') - Equal (prefix version of term equation '=') - side (Equal left-hand side and right-hand side role) - Const (slot, individual, function, or predicate symbol, with optional 'type' attribute) - Var (logic variable) For the XML Schema Definition (XSD)pair of the RIF-BLD condition language see Appendix Specificationrules p :- q,
p :- r. Terms with named arguments can be reduced
to positional terms by ordering the XML syntax for symbol spaces utilizesarguments by their names and
incorporating them into the type attribute associated with XML term elements such as Const .predicate name. For instance,
a literal in the xsd:dateTime data typep(bb->1 aa->2) can be represented as
<Const type="xsd:dateTime">2007-11-23T03:55:44-02:30</Const>p_aa_bb(2,1).
3.5 EBNF Grammar for the following example illustrates XML serializationPresentation
Syntax of RIF conditions. Example 3 (A RIF condition and its XML serialization): We use the prefix bks as an abbreviation for http://example.com/books# and curr for http://example.com/currencies# a. RIF condition And ( Exists ?Buyer ( purchase^^rif:local ( ?Buyer ?Seller book^^rif:local ( ?Author bks:LeRif^^rif:iri ) curr:USD^^rif:iri ( 49^^xsd:integer ) ) ?Seller=?Author ) b. XML serialization <And> <formula> <Exists> <declare><Var>Buyer</Var></declare> <formula> <Uniterm> <op><Const type="rif:local">purchase</Const></op> <arg><Var>Buyer</Var></arg> <arg><Var>Seller</Var></arg> <arg> <Uniterm> <op><Const type="rif:local">book</Const></op> <arg><Var>Author</Var></arg> <arg><Const type="rif:iri">bks:LeRif</Const></arg> </Uniterm> </arg> <arg> <Uniterm> <op><Const type="rif:iri">curr:USD</Const></op> <arg><Const type="xsd:integer">49</Const></arg> </Uniterm> </arg> </Uniterm> </formula> </Exists> </formula> <formula> <Equal> <side><Var>Seller</Var></side> <side><Var>Author</Var></side> </Equal> </formula> </And> The following example illustrates XML serialization of RIF conditions that involve terms with named arguments. Example 4 (A RIF condition and its XML serialization): We use the prefix bks to abbreviate http://example.com/books#, the prefix auth for http://example.com/authors#, and curr for http://example.com/currencies#, a. RIF condition: And ( Exists ?Buyer ?P ( ?P # purchase^^rif:local [ buyer^^rif:local -> ?Buyer seller^^rif:local -> ?Seller item^^rif:local -> book^^rif:local ( author^^rif:local -> ?Author title^^rif:local -> bks:LeRif^^rif:iri ) price^^rif:local -> 49^^xsd:integer currency^^rif:local -> curr:USD^^rif:iri ] ) ?Seller=?Author ) b. XML serialization: <And> <formula> <Exists> <declare><Var>Buyer</Var></declare> <declare><Var>P</Var></declare> <formula> <Frame> <object> <Member> <lower><Var>P</Var></lower> <upper><Const type="rif:local">purchase</Const></upper> </Member> </object> <slot><Const type="rif:local">buyer</Const><Var>Buyer</Var></slot> <slot><Const type="rif:local">seller</Const><Var>Seller</Var></slot> <slot> <Const type="rif:local">item</Const> <Uniterm> <op><Const type="rif:local">book</Const></op> <slot><Const type="rif:local">author</Const><Var>Author</Var></slot> <slot><Const type="rif:local">title</Const><Const type="rif:iri">bks:LeRif</Const></slot> </Uniterm> </slot> <slot><Const type="rif:local">price</Const><Const type="xsd:integer">49</Const></slot> <slot><Const type="rif:local">currency</Const><Const type="rif:iri">curr:USD</Const></slot> </Frame> </formula> </Exists> </formula> <formula> <Equal> <side><Var>Seller</Var></side> <side><Var>Author</Var></side> </Equal> </formula> </And> 2.7.2 XML for RIF-BLD Rule Language The following extends the XML syntax in Section XML for RIF-BLD Condition Language , by serializingRIF-BLD
So far, the syntax of Section EBNF forRIF-BLD Rule Languagehas been specified in XML. The Forall element contains the role elements declare and formula ,Mathematical
English. Tool developers, however, may prefer EBNF notation, which
were earlier used withinprovides a more succinct overview of the Exists elementsyntax. Several points
should be kept in Section XML for RIF-BLD Condition Language . The Implies element containsmind regarding this notation.
- The
role elements if and then to designate these two partssyntax of a rule. Classes, roles and their intended meaning - Ruleset (rule collection, containing rule roles) - Forall (quantified formula for 'Forall', containing declare and formula roles) - Implies (implication, containing if and then roles) - if (antecedent role, containing CONDITION) - then (consequent role, containing a Uniterm, Equal, or Frame) Forfirst-order logic is not context-free, so EBNF
does not capture the XML Schema Definition (XSD)syntax of theRIF-BLD Horn rule language see Appendix Specification .precisely. For instance, it
cannot capture the rulesection on well-formedness conditions, i.e., the requirement that each
symbol in Example 5aRIF-BLD can be serializedoccur in XML as shown belowat most one context. As the first element ofa rule set whose second element isresult,
the EBNF grammar defines a business rule for Fred. Example 5 (A RIF rule setstrict superset of RIF-BLD (not
all rules that are derivable using the EBNF grammar are well-formed
rules in XML syntax) <Ruleset> <rule> <Forall> <declare><Var>item</Var></declare> <declare><Var>deliverydate</Var></declare> <declare><Var>scheduledate</Var></declare> <declare><Var>diffduration</Var></declare> <declare><Var>diffdays</Var></declare> <formula> <Implies> <if> <And> <formula> <Uniterm> <op><Const type="rif:local">perishable</Const></op> <arg><Var>item</Var></arg> </Uniterm> </formula> <formula> <Uniterm> <op><Const type="rif:local">delivered</Const></op> <arg><Var>item</Var></arg> <arg><Var>deliverydate</Var></arg> <arg><Const type="rif:iri">ppl:John</Const></arg> </Uniterm> </formula> <formula> <Uniterm> <op><Const type="rif:local">scheduled</Const></op> <arg><Var>item</Var></arg> <arg><Var>scheduledate</Var></arg> </Uniterm> </formula> <formula> <Uniterm> <op><Const type="rif:local">fn:subtract-dateTimes-yielding-dayTimeDuration</Const></op> <arg><Var>deliverydate</Var></arg> <arg><Var>scheduledate</Var></arg> <arg><Var>diffduration</Var></arg> </Uniterm> </formula> <formula> <Uniterm> <op><Const type="rif:local">fn:get-days-from-dayTimeDuration</Const></op> <arg><Var>diffduration</Var></arg> <arg><Var>diffdays</Var></arg> </Uniterm> </formula> <formula> <Uniterm> <op><Const type="rif:iri">op:numeric-greater-than</Const></op> <arg><Var>diffdays</Var></arg> <arg><Const type="xsd:long">10</Const></arg> </Uniterm> </formula> </And> </if> <then> <Uniterm> <op><Const type="xsd:long">reject</Const></op> <arg><Const type="rif:iri">ppl:John</Const></arg> <arg><Var>item</Var></arg> </Uniterm> </then> </Implies> </formula> </Forall> </rule> <rule> <Forall> <declare><Var>item</Var></declare> <formula> <Implies> <if> <Uniterm> <op><Const type="rif:local">unsolicited</Const></op> <arg><Var>item</Var></arg> </Uniterm> </if> <then> <Uniterm> <op><Const type="rif:local">reject</Const></op> <arg><Const type="rif:iri">ppl:Fred</Const></arg> <arg><Var>item</Var></arg> </Uniterm> </then> </Implies> </formula> </Forall> </rule> </Ruleset> 2.8 Translation BetweenRIF-BLD).
- The
RIF-BLD Presentation and XML Syntaxes We now showEBNF syntax is not a concrete syntax: it does not
address the details of how to translate betweenconstants and variables are represented,
and it is not sufficiently precise about the delimiters and escape
symbols. Instead, white space is informally used as a delimiter,
and white space is 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
done on intentionally, since RIF's presentation syntax is intended
as a tool for specifying the semantics and XML syntaxesfor illustration of RIF-BLD. 2.8.1 Translationthe
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 EBNF syntax.
- For all the above reasons, the EBNF syntax is not
normative.
3.5.1 EBNF for RIF-BLD Condition
Language
The translation betweenCondition Language represents formulas that can be used in
the presentation syntax andbody of the XML syntaxRIF-BLD rules. It is intended to be a common part
of a number of RIF dialects, including RIF PRD. The EBNF grammar
for a superset of the RIF-BLD condition language is given by a tableas follows.
Presentation Syntax XML Syntax And ( conjunct 1 . . . conjunct n ) <And> <formula> conjunctFORMULA ::= 'And' '(' FORMULA* ')' |
'Or' '(' FORMULA* ')' |
'Exists' Var+ '(' FORMULA ')' |
ATOMIC |
'External' '(' Atom ')'
ATOMIC ::= 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 ::= Const | Var | Expr | 'External' '(' Expr ')'
Expr ::= UNITERM
Const ::= '"' UNICODESTRING '"^^' SYMSPACE
Name ::= UNICODESTRING
Var ::= '?' UNICODESTRING
The production rule for the non-terminal FORMULA
represents RIF condition formulas (defined earlier). The
connectives And and Or define conjunctions and
disjunctions of conditions, respectively. Exists
introduces existentially quantified variables. Here Var+
stands for the list of variables that are free in FORMULA.
RIF-BLD conditions permit only existential variables, but RIF-FLD
Syntax allows arbitrary quantification, which can be used in
some dialects. A RIF-BLD FORMULA can also be an
ATOMIC term, i.e. an Atom, External
Atom, Equal, Member, Subclass,
or Frame. A TERM can be a constant, variable,
Expr, or External Expr.
The RIF-BLD presentation syntax does not commit to any
particular vocabulary except for using Unicode strings in constant
symbols, as names, and for variables. Constant symbols have the
form: "UNICODESTRING"^^SYMSPACE, where SYMSPACE
is an IRI string that identifies the symbol space of the constant
and UNICODESTRING is a Unicode string from the lexical
space of that symbol space. Names are just denoted by Unicode
character sequences. Variables are denoted by Unicode character
sequences beginning with 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 of
RIF-DTB. Likewise,
an External Expr is a call to an externally
defined function of RIF-DTB.
Example 1 </formula>(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, or
property values. For brevity, we use the compact URI
notation [CURIE],
prefix:suffix, which should be understood as a macro that
expands into a concatenation of the prefix definition and
suffix. Thus, if bks is a prefix that expands
into http://example.com/books# then bks:LeRif
should be understood merely as an abbreviation for
http://example.com/books#LeRif. The compact URI notation
is not part of the RIF-BLD syntax.
Compact URI prefixes:
bks expands into http://example.com/books#
auth expands into http://example.com/authors#
cpt expands into http://example.com/concepts#
Positional terms:
"cpt:book"^^rif:iri("auth:rifwg"^^rif:iri "bks:LeRif"^^rif:iri)
Exists ?X ("cpt:book"^^rif:iri(?X "bks:LeRif"^^rif:iri))
Terms with named arguments:
"cpt:book"^^rif:iri(cpt:author->"auth:rifwg"^^rif:iri
cpt:title->"bks:LeRif"^^rif:iri)
Exists ?X ("cpt:book"^^rif:iri(cpt:author->?X cpt:title->"bks:LeRif"^^rif:iri))
Frames:
"bks:wd1"^^rif:iri["cpt:author"^^rif:iri->"auth:rifwg"^^rif:iri
"cpt:title"^^rif:iri->"bks:LeRif"^^rif:iri]
Exists ?X ("bks:wd2"^^rif:iri["cpt:author"^^rif:iri->?X
"cpt:title"^^rif:iri->"bks:LeRif"^^rif:iri])
Exists ?X ("bks:wd2"^^rif:iri # "cpt:book"^^rif:iri["cpt:author"^^rif:iri->?X
"cpt:title"^^rif:iri->"bks:LeRif"^^rif:iri])
Exists ?I ?X (?I["cpt:author"^^rif:iri->?X "cpt:title"^^rif:iri->"bks:LeRif"^^rif:iri])
Exists ?I ?X (?I # "cpt:book"^^rif:iri["cpt:author"^^rif:iri->?X
"cpt:title"^^rif:iri->"bks:LeRif"^^rif:iri])
Exists ?S ("bks:wd2"^^rif:iri["cpt:author"^^rif:iri->"auth:rifwg"^^rif:iri
?S->"bks:LeRif"^^rif:iri])
Exists ?X ?S ("bks:wd2"^^rif:iri["cpt:author"^^rif:iri->?X
?S->"bks:LeRif"^^rif:iri])
Exists ?I ?X ?S (?I # "cpt:book"^^rif:iri[author->?X ?S->"bks:LeRif"^^rif:iri])
3.5.2 EBNF for RIF-BLD Rule
Language
The presentation syntax for Horn rules extends the syntax in
Section EBNF for
RIF-BLD Condition Language with the following productions.
Group ::= 'Group' IRIMETA? '(' (RULE | Group)* ')'
IRIMETA ::= Frame
RULE ::= 'Forall' Var+ '(' CLAUSE ')' | CLAUSE
CLAUSE ::= Implies | ATOMIC
Implies ::= ATOMIC ':-' FORMULA
For convenient reference, we reproduce the condition language
part of the EBNF below.
FORMULA ::= 'And' '(' FORMULA* ')' |
'Or' '(' FORMULA* ')' |
'Exists' Var+ '(' FORMULA ')' |
ATOMIC |
'External' '(' Atom ')'
ATOMIC ::= 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 ::= Const | Var | Expr | 'External' '(' Expr ')'
Expr ::= UNITERM
Const ::= '"' UNICODESTRING '"^^' SYMSPACE
Name ::= UNICODESTRING
Var ::= '?' UNICODESTRING
A RIF-BLD Group is a nested collection of RIF-BLD rules
annotated with optional metadata, IRIMETA, represented as
Frames. A Group can contain any number of
RULEs along with any number of nested Groups.
Rules are generated by CLAUSE, which can be in the scope
of a Forall quantifier. If a CLAUSE in the
RULE production has a free (non-quantified) variable, it
must occur in the Var+ sequence. 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 treated as a rule with an empty condition part -- in which case
it is usually called a fact. 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 2 (RIF-BLD rules).
This example shows a business rule borrowed from the document
RIF Use Cases and
Requirements:
If an item is perishable and it
is delivered to John more than 10 days after the scheduled delivery
date then the item will be rejected by him.
As before, for better readability we use the compact URI
notation.
Compact URI prefixes:
ppl expands into http://example.com/people#
cpt expands into http://example.com/concepts#
op expands into the yet-to-be-determined IRI for RIF builtin predicates
a. Universal form:
Forall ?item ?deliverydate ?scheduledate ?diffduration ?diffdays (
"cpt:reject"^^rif:iri("ppl:John"^^rif:iri ?item) :-
And("cpt:perishable"^^rif:iri(?item)
"cpt:delivered"^^rif:iri(?item ?deliverydate "ppl:John"^^rif:iri)
"cpt:scheduled"^^rif:iri(?item ?scheduledate)
External("fn:subtract-dateTimes-yielding-dayTimeDuration"^^rif:iri(?deliverydate ?scheduledate ?diffduration))
External("fn:get-days-from-dayTimeDuration"^^rif:iri(?diffduration ?diffdays))
External("op:numeric-greater-than"^^rif:iri(?diffdays "10"^^xsd:integer)))
)
b. Universal-existential form:
Forall ?item (
"cpt:reject"^^rif:iri("ppl:John"^^rif:iri ?item ) :-
Exists ?deliverydate ?scheduledate ?diffduration ?diffdays (
And("cpt:perishable"^^rif:iri(?item)
"cpt:delivered"^^rif:iri(?item ?deliverydate "ppl:John"^^rif:iri)
"cpt:scheduled"^^rif:iri(?item ?scheduledate)
External("fn:subtract-dateTimes-yielding-dayTimeDuration"^^rif:iri(?deliverydate ?scheduledate ?diffduration))
External("fn:get-days-from-dayTimeDuration"^^rif:iri(?diffduration ?diffdays))
External("op:numeric-greater-than"^^rif:iri(?diffdays "10"^^xsd:integer)))
)
)
Example 3 (A RIF-BLD group annotated with metadata).
This example shows a group formula that consists of two RIF-BLD
rules. The first of these rules is copied from Example 2a. The
group is annotated with Dublin Core metadata represented as a
frame.
Compact URI prefixes:
bks expands into http://example.com/books#
auth expands into http://example.com/authors#
cpt expands into http://example.com/concepts#
dc expands into http://dublincore.org/documents/dces/
w3 expands into http://www.w3.org/
Group "http://sample.org"^^rif:iri["dc:publisher"^^rif:iri->"w3:W3C"^^rif:iri
"dc:date"^^rif:iri->"2008-04-04"^^xsd:date]
(
Forall ?item ?deliverydate ?scheduledate ?diffduration ?diffdays (
"cpt:reject"^^rif:iri("ppl:John"^^rif:iri ?item) :-
And("cpt:perishable"^^rif:iri(?item)
"cpt:delivered"^^rif:iri(?item ?deliverydate "ppl:John"^^rif:iri)
"cpt:scheduled"^^rif:iri(?item ?scheduledate)
External("fn:subtract-dateTimes-yielding-dayTimeDuration"^^rif:iri(?deliverydate ?scheduledate ?diffduration))
External("fn:get-days-from-dayTimeDuration"^^rif:iri(?diffduration ?diffdays))
External("op:numeric-greater-than"^^rif:iri(?diffdays "10"^^xsd:integer)))
)
Forall ?item (
"cpt:reject"^^rif:iri("ppl:Fred"^^rif:iri ?item) :- "cpt:unsolicited"^^rif:iri(?item)
)
)
4 Direct Specification of RIF-BLD
Semantics
This normative section specifies the semantics of RIF-BLD
directly, without referring to RIF-FLD.
4.1 Truth Values
The set TV of truth values in RIF-BLD consists of
just two values, t and f.
4.2 Semantic Structures
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.
Definition (Semantic structure). A semantic
structure, I, is a tuple of the form
<TV, DTS, D,
Dind, Dfunc,
IC, IV,
IF, Iframe,
ISF, Isub,
Iisa, I=,
Iexternal,
Itruth>. Here D is a
non-empty set of elements called the domain of
I, and Dind,
Dfunc are nonempty subsets of
D. Dind is used to interpret
the elements of Const, which denote individuals and
Dfunc is used to interpret the elements of
Const that denote 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 the set of primitive data types used in
I (please refer to Section Primitive Data
Types of RIF-FLD for the semantics of data types).
The other components of I are total mappings
defined as follows:
- I C maps Const to
D.
This mapping interprets constant symbols. In addition:
-
- If a constant, c ∈ Const, denotes
an individual then it is required that
IC(c) ∈ Dind.
- If c ∈ Const, denotes a function
symbol (positional or with named arguments) then it is required
that
IC(c) ∈ Dfunc.
- IV maps Var to
Dind.
This mapping interprets variable symbols.
- IF maps D to functions
D*ind → D (here
D*ind is a set of all sequences of any
finite length over the domain Dind)
This mapping interprets positional terms. In addition:
-
- If d ∈ Dfunc then
IF(d) must be a function
D*ind →
Dind.
- This means that when a function symbol is applied to arguments
that are individual object then the result is also an individual
object.
- ISF is a total mapping from
D to the set of total functions of the form
SetOfFiniteSets(ArgNames ×
Dind) → D.
This mapping interprets function symbols with named arguments.
In addition:
-
- If d ∈ Dfunc then
ISF(d) must be a function
SetOfFiniteSets(ArgNames ×
Dind) →
Dind.
- This is analogous to the interpretation of positional terms
with two differences:
-
- Each pair <s,v> ∈ ArgNames ×
Dind 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.
- Iframe is a total mapping from
Dind to total functions of the form
SetOfFiniteBags(Dind ×
Dind) → D.
This mapping interprets frame terms. An argument, d ∈
Dind, to Iframe
represent an object and the finite bag {<a1,v1>,
..., <ak,vk>} represents a bag of attribute-value
pairs for d. We will see shortly how
Iframe 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 ?A->?B] becomes
o[a->b a->b] if variable ?A is
instantiated with the symbol a and ?B with
b.
- Isub gives meaning to the subclass
relationship. It is a total function Dind
× Dind → 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.
- Iisa gives meaning to class
membership. It is a total function Dind ×
Dind → 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.
- I= is a total function
Dind × Dind →
D.
It gives meaning to the equality operator.
- Itruth is a total mapping
D → TV.
<formula> conjunctIt is used to define truth valuation for formulas.
- Iexternal is a mapping from the
coherent set of schemas for externally defined functions to total
functions D* → D. For each external
schema σ = (?X1
... ?Xn
</formula> </And>; τ) in the coherent set of such schemas
associated with the language,
Iexternal(σ) is a function of the
form Dn → D.
For every external schema, σ, associated with the
language, Iexternal(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 builtin predicate or function,
Iexternal(σ) is specified in the
document Data Types and
Builtins so that:
- If σ is a schema of a builtin function then
Iexternal(σ) must be the function
defined in the aforesaid document.
- If σ is a schema of a builtin predicate then
Itruth ο
(Iexternal(σ)) (the composition
of Itruth and
Iexternal(σ), a truth-valued
function) must be as specified in Data Types and Builtins.
For convenience, we also define the following mapping
I from terms to D:
- I(k) =
IC(k), if k is a symbol
in Const
- I(
disjunct 1 . . . disjunct n?v) <Or> <formula> disjunct 1 </formula> . . . <formula> disjunct n </formula> </Or> Exists variable 1 . . . variable n=
IV( body ) <Exists> <declare> variable 1 </declare> . . . <declare>?v), if ?v is a
variable n </declare> <formula> body </formula> </Exists> predfuncin Var
- I(
argumentf(t1 . . . argument... tn) <Uniterm> <op> predfunc </op> <arg> argument) =
IF(I(f))(I(t1 </arg> . . . <arg> argument),...,I(tn </arg> </Uniterm> predfunc))
- I(
keyf(s1 -> filler->v1 . . . key...
sn -> filler->vn) <Uniterm> <op> predfunc </op> <slot> key 1 filler 1 </slot> . . . <slot> key n filler n </slot> </Uniterm> inst [ key 1 -> filler 1 . . . key n -> filler n ] <Frame> <object> inst </object> <slot> key 1 filler 1 </slot> . . . <slot> key n filler n </slot> </Frame> inst # class [ key 1 -> filler 1 . . . key n -> filler n ] <Frame> <object> <Member> <lower> inst </lower> <upper> class </upper> </Member> </object> <slot> key 1 filler 1 </slot> . . . <slot> key n filler n </slot> </Frame> sub ## super [ key 1 -> filler 1 . . . key n -> filler n ] <Frame> <object> <Subclass> <lower> sub </lower> <upper> super </upper> </Subclass> </object> <slot> key 1 filler 1 </slot> . . . <slot> key n filler n </slot> </Frame> inst # class <Member> <lower> inst </lower> <upper> class </upper> </Member> sub ## super <Subclass> <lower> sub </lower> <upper> super </upper> </Subclass> left) =
right <Equal> <side> left </side> <side> right </side> </Equal> name ^^ space <Const type=" space "> name </Const> ? name <Var> name </Var> 2.8.2 Translation of RIF-BLD Rule Language The translation between the presentation syntax and the XML syntax of the RIF-BLD Rule Language is given by a table that extends the translation table of Section Translation of RIF-BLD Condition Language as follows. Presentation Syntax XML Syntax RulesetISF(I(f))({<s1,I( clausev1 . . . clause)>,...,<sn,I(vn)>})
-
Here we use {...} to denote a set of argument/value pairs.
- I(o[a1->v1 ...
ak->vk])
<Ruleset> <rule> clause=
Iframe(I(o))({<I(a1 </rule> . . . <rule> clause n </rule> </Ruleset> Forall variable),I(v1 . . . variable)>,
...,
<I(an),I( rule ) <Forall> <declare> variable 1 </declare> . . . <declare> variablevn </declare> <formula> rule </formula> </Forall> conclusion :- condition <Implies> <if> condition </if> <then> conclusion </then> </Implies> 2.9 Subdialects of RIF-BLD This is)>})
-
Here {...} denotes a proposal to specify RIF-CORE etc. by just removing syntactic constructs from RIF-BLD (hence, through The effect of the syntax , restricting the semantics). The point is that it makes more sense for most engines to support only some subdialects of BLD, and that subdialects and fragments of BLD are reused in the definition of other RIF dialects. *** The syntactic structurebag of RIF-BLD suggests several useful subdialects: RIF-CORE . This subdialectattribute/value pairs.
- I(c1##c2) =
Isub(I(c1),
I(c2))
- I(o#c) =
Iisa(I(o),
I(c))
- I(x=y) =
I=(I(x),
I(y))
- I(External(t)) =
Iexternsl(σ)(I(s1),
..., I(sn)), if t is
obtained from RIF-BLD by removing support for: equality formulas in the rule conclusions (while still allowing them in conditions) terms with named arguments ??? membership, subclass, and frame terms ??? RIF-CORE+equality . This subdialect extends RIF-CORE by adding support for equality formulas inan
instance of the rule conclusions. RIF-CORE+named argumentsexternal schema σ = (?X1
... ?Xn; τ) by substitution
?X1/s1
... ?Xn/s1.
This subdialect extends RIF-CORENote that, by adding syntactic support for terms with named arguments.definition, External(t) is well formed
only if frames are not included in RIF-CORE / RIF-CORECOND then extensions of RIF-CORE / RIF-CORECOND with frames are added here. 3 RIF-BLD Semantics 3.1 The Semantics of RIF-BLD as a Specialization of RIF-FLD This section defines the precise relationship between the semantics of RIF-BLD and the semantic framework of RIF-FLD. The remaining sections describe the semantics of RIF-BLD without referring to the general framework -- except for Primitive Data Types whose definitiont is not duplicated here. The semanticsan instance of the RIF Basic Logic Dialect is definedan external schema.
Furthermore, by specialization from the semantics ofthe [:FLD/Semantics:Semantic Framework for Logic Dialects] of RIF. Section [:FLD/Semantics#sec-rif-dialect-semantics:Semantics of a RIF Dialect as a Specializationdefinition of RIF-FLD] in that document lists the parameterscoherent sets of the semantic framework, which we need to specialize for RIF-BLD. Recall that the semanticsexternal schemas,
t can be an instance of a dialectat most one such schema, so
I(External(t)) is derived from these notions by specializing the following parameters.well-defined.
The effect of data types. The syntax . RIF-BLD does not support negation. This is the only obvious simplification with respect to RIF-FLD as far asdata types in
DTS impose the semanticsfollowing restrictions. If dt
is concerned. Truth values . The set TV of truth values in RIF-BLD consistsa symbol space identifier of just two values, t and f such that f < t t . Clearly, < t isa total order here.data types . RIF-BLD supports alltype, let
LSdt denote the data types listed inlexical space of
dt, VSdt denote its value space,
and Ldt: LSdt →
VSdt the lexical-to-value-space mapping
(for the definitions of these concepts, see Section Primitive Data
Types of RIF-FLD: xsd:long xsd:integer xsd:decimal xsd:string xsd:time xsd:dateTime rdf:XMLLiteral rif:text Logical entailment . Recall that logical entailment in RIF-FLD is defined with respect to an unspecified set of intended semantic structuresRIF-FLD). Then the following must hold:
- VSdt ⊆ Dind;
and
- For each constant "lit"^^dt ∈
LSdt,
IC("lit"^^dt) =
Ldt(lit).
That dialects of RIFis, IC must make this notion concrete.map the constants of a
data type dt in accordance with
Ldt.
RIF-BLD does not impose restrictions on
IC for RIF-BLD, this set is definedconstants in one ofthe two following equivalent ways: as a setlexical spaces
that do not correspond to primitive datatypes in DTS.
☐
4.3 Interpretation of
all models; or as the unique minimal model. These two definitions are equivalentFormulas
Definition
(Truth valuation). Truth valuation for
entailment of RIF-BLD conditions by RIF-BLD rulesets, since all ruleswell-formed formulas in RIF-BLD are Horn -- itis a classical result of Van Emden and Kowalski [ vEK76 ]. 3.2determined using the following
function, denoted TValI:
- Positional atomic formulas:
TValI(r(t1 ...
tn)) =
Itruth(I(r(t1
... tn)))
- Atomic formulas with named arguments:
TValI(p(s1->v1 ...
sk->vk)) =
Itruth
Values(I(p(s1->v1
... sk->vk))).
- Equality:
TValI(x = y) =
Itruth(I(x = y)).
- Subclass:
TValI(sc ## cl) =
Itruth(I(sc ## cl)).
To ensure that the operator ## is a tuple oftransitive, i.e.,
c1 ## c2 and c2 ## c3 imply
c1 ## c3, the form < TVfollowing is required:
For all c1, DTSc2,
c3 ∈ D, if
TValI C ,(c1 ## c2) =
TValI V ,(c2 ## c3) = t
then TValI F ,(c1 ## c3) =
t.
- Membership:
TValI
frame ,(o # cl) =
I SF ,truth(I sub(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
TValI(o # cl) =
TValI(cl ## scl) = t
then
TValI isa ,(o # scl) =
t.
- Frame:
TValI(o[a1->v1 ...
ak->vk]) =
,Itruth >. Here D is a non-empty set of elements called the domain of(I , and there is(o[a1->v1
... a proper subset, D ind ⊂ D , which is used to interpret individuals. We use Const to refer tok->vk])).
Since the set of all constant symbols and Var to referdifferent attribute/value pairs are supposed to be
understood as conjunctions, the set of all variable symbols. TV denotes the set of truth values that the semantic structure uses and DTSfollowing is the set of primitive data types used inrequired:
TValI (please refer to Section Primitive Data Types of RIF-FLD for the semantics of data types). The other components of(o[a1->v1 ...
ak->vk]) = t if and only if
TValI are total mappings(o[a1->v1])
= ... =
TValI(o[ak->vk])
= t.
- Externally defined
as follows:atomic formula:
TValI C maps Const to elements(External(t)) =
Itruth(Iexternal(σ)(I(s1),
..., I(sn))), if t is an
atomic formula that is an instance of Dthe external schema σ =
(?X1 ... ?Xn; τ) by substitution
?X1/s1
... ?Xn/s1.
This mapping interprets constant symbols.Note that, by definition, External(t) is well-formed
only if a constant, c ∈ Const , occurs int is an instance of an external schema.
Furthermore, by the positiondefinition of coherent sets of external schemas,
t can be an individual then itinstance of at most one such schema, so
I(External(t)) is required thatwell-defined.
- Conjunction:
TValI
C(And(c ) ∈ D ind .1 ...
cn)) = t if and only if
TValI V maps Var to elements of D ind(c1) = ... =
TValI(cn) = t. This mapping interprets variable symbols.Otherwise,
TValI(And(c1 ...
cn)) = f maps D to functions D* → D (here D*.
-
The empty conjunction is treated as a set of all sequences of any finite length over the domain Dtautology, so
TValI(And()) This mapping interprets positional terms.= t.
- Disjunction:
TValI
SF interprets terms with named arguments. It is a total mapping from Const to the set of total functions of the form SetOfFiniteBags( ArgNames × DOr(c1 ...
cn) → D) = f if and only if
TValI(c1) = ... =
TValI(cn) = f. Otherwise,
TValI(Or(c1 ...
cn)) = t.
-
This is analogous to the interpretation of positional terms with two differences: Each pair <s,v> ∈ ArgNames × D represents a 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 bag of argument/value pairs rather than a finite ordered sequence of simple elements. Bags (multisets) are used here because the order ofThe argument/value pairs in a term with named argumentsempty disjunction is immaterialtreated as a contradiction, so
TValI(Or()) = f.
- Quantification:
- TValI(Exists ?v1
... ?vn (φ)) = t if and
the pairs may repeat.only if for
instance, p(a->b a->b)some I*, described below,
TValI*(φ) = t.
- TValI
frame(Forall ?v1
... ?vn (φ)) = t if and only if for
every I*, described below,
TValI*(φ) = t.
Here I* is a total mapping from D to total functionssemantic structure of the form
SetOfFiniteBags ( D × D ) →<TV, DTS, D . This mapping interprets frame terms. An argument,,
D ∈ind, Dfunc,
toI frame represent an object and the finite bag {<a1,v1>, ..., <ak,vk>} represents a bag of attribute-value pairs for d . We will see shortly howC, I*V,
IF, Iframe,
ISF, Isub,
Iisa, I=,
Iexternsl,
Itruth>, which is used to determineexactly like
I, except that the truth valuation of frame terms. Bags aremapping
I*V, is used here because the orderinstead of
the attribute/value pairs in a frameIV. I*V is
immaterial and pairs may repeat. For instance, o[a->b a->b]defined to coincide with IV on all
variables except, possibly, on
?v1,...,?vn.
- Rule implication:
- TValI(conclusion :-
condition) = t, if either
TValI(conclusion)=t or
TValI(condition)=f.
- TValI
sub gives meaning to the subclass relationship. It(conclusion :-
condition) = f otherwise.
- Groups of rules:
If Γ is a total function D × D → D .group formula of the operator ## is required to be transitive, i.e., c1 ## c2form Group φ
(ρ1 ... ρn) or Group (ρ1
... ρn) then
- TValI(Γ) = t if and
c2 ## c3 must imply c1 ## c3only if
TValI(ρ1) = t, ...,
TValI(ρn) = t.
- TValI(Γ) = f
otherwise.
This is ensured bymeans that a restriction in Section Interpretationgroup of Formulas . I isa gives meaning to class membership. Itrules is treated as a total function D × D → D .conjunction.
The relationships # and ## are required to havemetadata is ignored for purposes of the usual property that all membersRIF-BLD semantics.
A model of a subclass are also membersgroup of the superclass, i.e., o # cl and cl ## scl must imply o # sclrules, Γ, is a
semantic structure I such that
TValI(Γ) = t. In this case, we
write I |= Γ. ☐
Note that although metadata associated with RIF-BLD formulas is
ensuredignored by the semantics, it can be extracted by XML tools. Since
metadata is represented by frame terms, it can be reasoned with by
RIF-BLD rules.
4.4 Logical Entailment
We now define what it means for a restriction in Section Interpretationset of Formulas . I = gives meaningRIF-BLD rules to the equality. Itentail
a RIF-BLD condition. We say that a RIF-BLD condition formula
φ is existentially closed, if and only if every
variable, ?V, in φ occurs in a total function D × D → Dsubformula of the
form Exists ...?V...(ψ).
I Truth isDefinition
(Logical entailment). Let Γ be a total mapping D → TV .RIF-BLD group formula
and φ an existentially closed RIF-BLD condition formula.
We say that Γ entails φ, written as
Γ |= φ, if and only if for every model of
Γ it is used to define truth valuation of formulas. We also definethe following mapping I :case that TValI (k)(φ)
= t.
Equivalently, we can say that Γ |= φ holds
iff whenever I C (k), if k is a symbol in Const I (?v) = |= Γ it
follows that also I V (?v), if ?v |= φ.
☐
5 XML Serialization Syntax for
RIF-BLD
The XML serialization for RIF-BLD is alternating or
fully striped [ANF01]. A variable in Var I (f(t 1 ... t n )) = I F ( I (f))( I (t 1 ),..., I (t n )) I (f( s 1 ->v 1 ... s n ->v n )) = I SF ( I (f))({ <s 1 , I (v 1 ) > ,..., <s nfully striped serialization views XML documents as
objects and divides all XML tags into class descriptors, called
type tags, I (v n ) > }) Hereand property descriptors, called role
tags. We use {...} to denotecapitalized names for type tags and lowercase
names for role tags.
5.1 XML for RIF-BLD Condition
Language
XML serialization of the presentation syntax of Section EBNF for RIF-BLD Condition
Language uses the following tags.
- 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 bag of argument/value pairs. I ( o[a 1 ->v 1 ... a k ->v k ] ) = Icontent role)
- content (content role, containing an Atom, for predicates, or Expr, for functions)
- Member (member formula)
- Subclass (subclass formula)
- Frame ( I ( o ))({ < I ((Frame formula)
- object (Member/Frame role, containing a 1 ), I (v 1 ) > , ..., < I (TERM or an object description)
- op (Atom/Expr role for predicates/functions as operations)
- arg (positional argument role)
- upper (Member/Subclass upper class role)
- lower (Member/Subclass lower instance/class role)
- slot (Atom/Expr/Frame slot role, containing a n ), I (v n ) > }) Here {...} denotesProp)
- Prop (Property, prefix version of slot infix '->')
- key (Prop key role, containing a bagConst)
- val (Prop val role, containing a TERM)
- Equal (prefix version 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))term equation '=')
- Expr (expression formula, positional or with named arguments)
- side (Equal left-hand side and right-hand side role)
- Const (individual, function, or predicate symbol, with optional 'type' attribute)
- Name (name of named argument)
- Var (logic variable)
For the effectXML Schema Definition (XSD) of data types.the data types in DTS imposeRIF-BLD condition
language see Appendix XML Schema
for BLD.
The following restrictions. If dt is aXML syntax for symbol space identifier ofspaces utilizes the type
attribute associated with XML term elements such as Const.
For instance, a literal in the xsd:dateTime data type, let LS dt denotetype can
be represented as
<Const type="xsd:dateTime">2007-11-23T03:55:44-02:30</Const>.
Example 4 (A RIF condition and its XML serialization).
This example illustrates XML serialization for RIF conditions.
As before, the lexical space of dt , VS dt denotecompact URI notation is used for better
readability.
Compact URI prefixes:
bks expands into http://example.com/books#
cpt expands into http://example.com/concepts#
curr expands into http://example.com/currencies#
RIF condition
And (Exists ?Buyer ("cpt:purchase"^^rif:iri(?Buyer
?Seller
"cpt:book"^^rif:iri(?Author "bks:LeRif"^^rif:iri)
"curr:USD"^^rif:iri("49"^^xsd:integer)))
?Seller=?Author )
XML serialization
<And>
<formula>
<Exists>
<declare><Var>Buyer</Var></declare>
<formula>
<Atom>
<op><Const type="rif:iri">cpt:purchase</Const></op>
<arg><Var>Buyer</Var></arg>
<arg><Var>Seller</Var></arg>
<arg>
<Expr>
<op><Const type="rif:iri">cpt:book</Const></op>
<arg><Var>Author</Var></arg>
<arg><Const type="rif:iri">bks:LeRif</Const></arg>
</Expr>
</arg>
<arg>
<Expr>
<op><Const type="rif:iri">curr:USD</Const></op>
<arg><Const type="xsd:integer">49</Const></arg>
</Expr>
</arg>
</Atom>
</formula>
</Exists>
</formula>
<formula>
<Equal>
<side><Var>Seller</Var></side>
<side><Var>Author</Var></side>
</Equal>
</formula>
</And>
Example 5 (A RIF condition and its value space,XML serialization).
This example illustrates XML serialization of RIF conditions
that involve terms with named arguments.
Compact URI prefixes:
bks expands into http://example.com/books#
auth expands into http://example.com/authors#
cpt expands into http://example.com/concepts#
curr expands into http://example.com/currencies#
RIF condition:
And L dt : LS dt → VS dt the lexical-to-value-space mapping (for(Exists ?Buyer ?P (
?P#"cpt:purchase"^^rif:iri["cpt:buyer"^^rif:iri->?Buyer
"cpt:seller"^^rif:iri->?Seller
"cpt:item"^^rif:iri->"cpt:book"^^rif:iri(cpt:author->?Author
cpt:title->"bks:LeRif"^^rif:iri)
"cpt:price"^^rif:iri->"49"^^xsd:integer
"cpt:currency"^^rif:iri->"curr:USD"^^rif:iri])
?Seller=?Author)
XML serialization:
<And>
<formula>
<Exists>
<declare><Var>Buyer</Var></declare>
<declare><Var>P</Var></declare>
<formula>
<Frame>
<object>
<Member>
<lower><Var>P</Var></lower>
<upper><Const type="rif:iri">cpt:purchase</Const></upper>
</Member>
</object>
<slot>
<Prop>
<key><Const type="rif:iri">cpt:buyer</Const></key>
<val><Var>Buyer</Var></val>
</Prop>
</slot>
<slot>
<Prop>
<key><Const type="rif:iri">cpt:seller</Const></key>
<val><Var>Seller</Var></val>
</Prop>
</slot>
<slot>
<Prop>
<key><Const type="rif:iri">cpt:item</Const></key>
<val>
<Expr>
<op><Const type="rif:iri">cpt:book</Const></op>
<slot>
<Prop>
<key><Name>cpt:author</Name></key>
<val><Var>Author</Var></val>
</Prop>
</slot>
<slot>
<Prop>
<key><Name>cpt:title</Name></key>
<val><Const type="rif:iri">bks:LeRif</Const></val>
</Prop>
</slot>
</Expr>
</val>
</Prop>
</slot>
<slot>
<Prop>
<key><Const type="rif:iri">cpt:price</Const></key>
<val><Const type="xsd:integer">49</Const></val>
</Prop>
</slot>
<slot>
<Prop>
<key><Const type="rif:iri">cpt:currency</Const></key>
<val><Const type="rif:iri">curr:USD</Const></val>
</Prop>
</slot>
</Frame>
</formula>
</Exists>
</formula>
<formula>
<Equal>
<side><Var>Seller</Var></side>
<side><Var>Author</Var></side>
</Equal>
</formula>
</And>
5.2 XML for RIF-BLD Rule
Language
We now extend the definitions of these concepts, seeRIF-BLD serialization from Section Primitive Data Types of RIF-FLD). ThenXML for RIF-BLD Condition
Language by including rules as described in Section EBNF for RIF-BLD Rule
Language. The extended serialization uses the following
must hold: VS dt ⊆ D ; andadditional tags.
- Group (nested collection of rules annotated with metadata)
- meta (meta role, containing metadata, which is represented as a Frame)
- rule (rule role, containing RULE)
- Forall (quantified formula for each constant lit^^dt ∈ LS dt , I C ( lit^^dt ) = L dt ( lit ). That is, I C must map'Forall', containing declare and formula roles)
- Implies (implication, containing if and then roles)
- if (antecedent role, containing FORMULA)
- then (consequent role, containing ATOMIC)
The constantsXML Schema Definition of a data type dtRIF-BLD is given in accordance with L dtAppendix
XML Schema for BLD.
Example 6 (Serializing a RIF-BLD does not impose restrictions on I Cgroup annotated with
metadata).
This example shows a serialization for constants inthe lexical spaces that do not correspond to primitive datatypes in DTS . 3.4 Interpretation of Formulas Truth valuationgroup from Example 3.
For well-formed formulas in RIF-BLD is determined usingconvenience, the following function, denoted TVal I : Positional atomic formulas : TVal I ( r(t 1 ... t n )) = I Truth ( I ( r(t 1 ... t n ))) Atomic formulas with named arguments : TVal I ( p(s 1 ->v 1 ... s k ->v k )) = I Truth ( I ( p(s 1 -> v 1 ... s k ->v k ))). Equality : TVal I ( x = y ) = I Truth ( I ( x = y )). To ensure that equality has preciselygroup is reproduced at the expected properties, ittop and then is
required that I Truth ( Ifollowed by its serialization.
Compact URI prefixes:
bks expands into http://example.com/books#
auth expands into http://example.com/authors#
cpt expands into http://example.com/concepts#
dc expands into http://dublincore.org/documents/dces/
w3 expands into http://www.w3.org/
Presentation syntax:
Group "http://sample.org"^^rif:iri["dc:publisher"^^rif:iri->"w3:W3C"^^rif:iri
"dc:date"^^rif:iri->"2008-04-04"^^xsd:date]
(
x = y )) = t if and only if IForall ?item ?deliverydate ?scheduledate ?diffduration ?diffdays (
x"cpt:reject"^^rif:iri("ppl:John"^^rif:iri ?item) :-
And("cpt:perishable"^^rif:iri(?item)
"cpt:delivered"^^rif:iri(?item ?deliverydate "ppl:John"^^rif:iri)
"cpt:scheduled"^^rif:iri(?item ?scheduledate)
External("fn:subtract-dateTimes-yielding-dayTimeDuration"^^rif:iri(?deliverydate ?scheduledate ?diffduration))
External("fn:get-days-from-dayTimeDuration"^^rif:iri(?diffduration ?diffdays))
External("op:numeric-greater-than"^^rif:iri(?diffdays "10"^^xsd:integer)))
)
= IForall ?item (
y"cpt:reject"^^rif:iri("ppl:Fred"^^rif:iri ?item) :- "cpt:unsolicited"^^rif:iri(?item)
)
and that I Truth ( I ( x = y )) = f otherwise. Subclass : TVal I ( sc ## cl)
= I Truth ( I ( sc ## cl )).XML syntax:
<Group>
<meta>
<Frame>
<object>
<Const type="rif:iri">http://sample.org</Const>
</object>
<slot>
<Prop>
<key><Const type="rif:iri">dc:publisher</Const></key>
<val><Const type="rif:iri">w3:W3C</Const></val>
</Prop>
</slot>
<slot>
<Prop>
<key><Const type="rif:iri">dc:date</Const></key>
<val><Const type="xsd:date">2008-04-04</Const></val>
</Prop>
</slot>
</Frame>
</meta>
<rule>
<Forall>
<declare><Var>item</Var></declare>
<declare><Var>deliverydate</Var></declare>
<declare><Var>scheduledate</Var></declare>
<declare><Var>diffduration</Var></declare>
<declare><Var>diffdays</Var></declare>
<formula>
<Implies>
<if>
<And>
<formula>
<Atom>
<op><Const type="rif:iri">cpt:perishable</Const></op>
<arg><Var>item</Var></arg>
</Atom>
</formula>
<formula>
<Atom>
<op><Const type="rif:iri">cpt:delivered</Const></op>
<arg><Var>item</Var></arg>
<arg><Var>deliverydate</Var></arg>
<arg><Const type="rif:iri">ppl:John</Const></arg>
</Atom>
</formula>
<formula>
<Atom>
<op><Const type="rif:iri">cpt:scheduled</Const></op>
<arg><Var>item</Var></arg>
<arg><Var>scheduledate</Var></arg>
</Atom>
</formula>
<formula>
<External>
<content>
<Atom>
<op><Const type="rif:iri">fn:subtract-dateTimes-yielding-dayTimeDuration</Const></op>
<arg><Var>deliverydate</Var></arg>
<arg><Var>scheduledate</Var></arg>
<arg><Var>diffduration</Var></arg>
</Atom>
</content>
</External>
</formula>
<formula>
<External>
<content>
<Atom>
<op><Const type="rif:iri">fn:get-days-from-dayTimeDuration</Const></op>
<arg><Var>diffduration</Var></arg>
<arg><Var>diffdays</Var></arg>
</Atom>
</content>
</External>
</formula>
<formula>
<External>
<content>
<Atom>
<op><Const type="rif:iri">op:numeric-greater-than</Const></op>
<arg><Var>diffdays</Var></arg>
<arg><Const type="xsd:long">10</Const></arg>
</Atom>
</content>
</External>
</formula>
</And>
</if>
<then>
<Atom>
<op><Const type="xsd:long">reject</Const></op>
<arg><Const type="rif:iri">ppl:John</Const></arg>
<arg><Var>item</Var></arg>
</Atom>
</then>
</Implies>
</formula>
</Forall>
</rule>
<rule>
<Forall>
<declare><Var>item</Var></declare>
<formula>
<Implies>
<if>
<Atom>
<op><Const type="rif:iri">cpt:unsolicited</Const></op>
<arg><Var>item</Var></arg>
</Atom>
</if>
<then>
<Atom>
<op><Const type="rif:iri">cpt:reject</Const></op>
<arg><Const type="rif:iri">ppl:Fred</Const></arg>
<arg><Var>item</Var></arg>
</Atom>
</then>
</Implies>
</formula>
</Forall>
</rule>
</Group>
5.3 Translation Between the RIF-BLD
Presentation and XML Syntaxes
We now show how to ensure thattranslate between the operator ##presentation and XML
syntaxes of RIF-BLD.
5.3.1 Translation of RIF-BLD Condition
Language
The translation between the presentation syntax and the XML
syntax of the RIF-BLD Condition Language is transitive, i.e., c1 ## c2 and c2 ## c3 imply c1 ## c3 ,specified by the followingtable
below. Since the presentation syntax of RIF-BLD is required: For all c1 , c2 , c3 ∈ D , min t ( TVal I ( c1 ## c2 ), TVal I ( c2 ## c3 )) ≤ t TVal I ( c1 ## c3 ). Membership : TVal I ( o # cl ) = I Truth ( I ( o # cl )).context
sensitive, the translation must differentiate between the terms
that occur in the position of the individuals from terms that occur
as atomic formulas. To ensurethis end, in the translation table, the
positional and named argument terms that all membersoccur in the context of
a subclassatomic formulas are also membersdenoted by the expressions of the superclass, i.e., o # clform
pred(...) and cl ## scl implies o # scl ,the following is required: For all o , cl , scl ∈ D , min t ( TVal I ( o # cl ), TVal I ( cl ## scl )) ≤ t TVal I ( o # scl ). Frame : TVal I ( o[a 1 ->v 1 ... a k ->v k ]terms that occur as individuals are
denoted by expressions of the form func(...).
The prime symbol (for instance,
variable') = I Truth ( Iindicates that the
translation function defined by the table must be applied
recursively (i.e., to variable in our example).
| Presentation Syntax |
XML Syntax |
And (
o[aconjunct1
->v. . .
conjunctn
)
|
<And>
<formula>conjunct1 ... a k ->v k ] )). Since the different attribute/value pairs are supposed to be understood as conjunctions, the following is required: TVal I'</formula>
. . .
<formula>conjunctn'</formula>
</And>
|
Or (
o[adisjunct1
. . .
disjunctn
)
|
<Or>
<formula>disjunct1 ->v'</formula>
. . .
<formula>disjunctn'</formula>
</Or>
|
Exists
variable1
... a k ->v k ] ) = min t. . .
variablen (
TVal Ibody
)
|
<Exists>
<declare>variable1'</declare>
. . .
<declare>variablen'</declare>
<formula>body'</formula>
</Exists>
|
pred (
o[aargument1
->v. . .
argumentn
)
|
<Atom>
<op>pred'</op>
<arg>argument1 ] ), ..., TVal I'</arg>
. . .
<arg> argumentn'</arg>
</Atom>
|
External (
o[a k ->v k ] )) Conjunction : TVal Iatomexpr
)
|
<External>
<content>atomexpr'</content>
</External>
|
func (
And( cargument1
... c. . .
argumentn
)
|
) = min t ( TVal I (c<Expr>
<op>func'</op>
<arg>argument1 ), ..., TVal I (c'</arg>
. . .
<arg> argumentn )). Disjunction : TVal I'</arg>
</Expr>
|
pred (
Or( cunicode1 ... c-> filler1
. . .
unicoden -> fillern
)
|
) = max t ( TVal I (c<Atom>
<op>pred'</op>
<slot>
<Prop>
<key><Name>unicode1 ), ..., TVal I (c n )). Quantification : TVal I ( Exists ?v</Name></key>
<val>filler1 ... ?v'</val>
</Prop>
</slot>
. . .
<slot>
<Prop>
<key><Name>unicoden (φ)) = max t ( TVal I* (φ)) and TVal I</Name></key>
<val>fillern'</val>
</Prop>
</slot>
</Atom>
|
func (
Forall ?vunicode1 ... ?v-> filler1
. . .
unicoden -> fillern
(φ)) = min t ( TVal I* (φ)). Here max t (respectively, min t)
|
is taken over all interpretations I * of the form < TV , DTS , D , I C , I * V , I F , I frame , I SF , I sub , I isa , I Truth >, which are exactly like I , except that the mapping I * V , is used instead of I V<Expr>
<op>func'</op>
<slot>
<Prop>
<key><Name>unicode1</Name></key>
<val>filler1'</val>
</Prop>
</slot>
. . .
<slot>
<Prop>
<key><Name>unicoden</Name></key>
<val>fillern'</val>
</Prop>
</slot>
</Expr>
|
inst [
key1 -> filler1
. I * V is defined to coincide with I V on all variables except, possibly, on ?v. .
keyn -> fillern
]
|
<Frame>
<object>inst'</object>
<slot>
<Prop>
<key>key1 ,..., ?v'</key>
<val>filler1'</val>
</Prop>
</slot>
. . .
<slot>
<Prop>
<key>keyn'</key>
<val>fillern'</val>
</Prop>
</slot>
</Frame>
|
inst # class [
key1 -> filler1
. Rules : TVal I ( conclusion :- condition ) = t , if TVal I ( conclusion ) ≥ t TVal I ( condition ); TVal I ( conclusion :- condition ) = f otherwise. A model of a set Ψ of formulas is a semantic structure I such that TVal I (φ). .
keyn -> fillern
]
|
<Frame>
<object>
<Member>
<lower>inst'</lower>
<upper>class'</upper>
</Member>
</object>
<slot>
<Prop>
<key>key1'</key>
<val>filler1'</val>
</Prop>
</slot>
. . .
<slot>
<Prop>
<key>keyn'</key>
<val>fillern'</val>
</Prop>
</slot>
</Frame>
|
sub ## super [
key1 -> filler1
. . .
keyn -> fillern
]
|
<Frame>
<object>
<Subclass>
<lower>sub'</lower>
<upper>super'</upper>
</Subclass>
</object>
<slot>
<Prop>
<key>key1'</key>
<val>filler1'</val>
</Prop>
</slot>
. . .
<slot>
<Prop>
<key>keyn'</key>
<val>fillern'</val>
</Prop>
</slot>
</Frame>
|
inst # class
|
<Member>
<lower>inst'</lower>
<upper>class'</upper>
</Member>
|
sub ## super
|
<Subclass>
<lower>sub'</lower>
<upper>super'</upper>
</Subclass>
|
left = t for every φ∈Ψ. In this case, we write I |= Ψ. 3.5 Logical Entailment We now define what it means for a set of RIF-BLD rules to entail a RIF-BLD condition. Let R be a setright
|
<Equal>
<side>left'</side>
<side>right'</side>
</Equal>
|
unicode^^space
|
<Const type="space">unicode</Const>
|
?unicode
|
<Var>unicode</Var>
|
5.3.2 Translation of RIF-BLD rules and φ an existentially closed RIF-BLD condition formula. We say that R entails φ, written as R |= φ, if and only if for every semantic structure I of R and every ψ ∈ R , itRule
Language
The translation between the presentation syntax and the XML
syntax of the RIF-BLD Rule Language is given by the case that TVal I (ψ) ≤ TVal I (φ). Equivalently, we can say that R |= φ holds iff whenever I |= R it follows that also I |= φ. 4table below,
which extends the translation table of Section Translation of
RIF-BLD Condition Language.
| Presentation Syntax |
XML Syntax |
Group (
clause1
. . .
clausen
)
|
<Group>
<rule>clause1'</rule>
. . .
<rule>clausen'</rule>
</Group>
|
Group metaframe (
clause1
. . .
clausen
)
|
<Group>
<meta>metaframe'</meta>
<rule>clause1'</rule>
. . .
<rule>clausen'</rule>
</Group>
|
Forall
variable1
. . .
variablen (
rule
)
|
<Forall>
<declare>variable1'</declare>
. . .
<declare>variablen'</declare>
<formula>rule'</formula>
</Forall>
|
conclusion :- condition
|
<Implies>
<if>condition'</if>
<then>conclusion'</then>
</Implies>
|