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This document, developed by the Rule Interchange Format (RIF) Working Group, defines a general RIF Framework for Logic Dialects (RIF-FLD). The framework describes mechanisms for specifying the syntax and semantics of logic RIF dialects through a number of generic concepts such as signatures, symbol spaces, semantic structures, and so on. The actual dialects are required to specialize this framework to produce their syntaxes and semantics.
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The RIF Framework for Logic Dialects (RIF-FLD) is a formalism for specifying all logic dialects of RIF, including the RIF Basic Logic Dialect [RIF-BLD]. It is a logic in which both syntax and semantics are described through a number of mechanisms that are commonly used for various logic languages, but are rarely brought all together. Amalgamation of several different mechanisms is required because the framework must be broad enough to accommodate several different types of logic languages and because various advanced mechanisms are needed to facilitate translation into a common framework. RIF-FLD gives precise definitions to these mechanisms, but allows certain details to vary. The design of RIF envisages that future standard logic dialects will be based on RIF-FLD. Therefore, any logic dialect being developed to become a stardard should either be a specialization of FLD or justify its deviations from (or extensions to) FLD.
The framework described in this document is very general and captures most of the popular logic rule languages found in Databases, Logic Programming, and on the Semantic Web. However, it is anticipated that the needs of future dialects might stimulate further evolution of RIF-FLD. In particular, future extensions might include a logic rendering of actions as found in production and reactive rule languages. This would support semantic Web services languages such as [SWSL-Rules] and [WSML-Rules].
This document is mostly intended for the designers of future RIF dialects. All logic RIF dialects are required to be derived from RIF-FLD by specialization, as explained in Sections Syntax of a RIF Dialect as a Specialization of RIF-FLD and Semantics of a RIF Dialect as a Specialization of RIF-FLD. In addition to specialization, to lower the barrier of entry for their intended audiences, a dialect designer may choose to specify the syntax and semantics in a direct, but equivalent, way, which does not require familiarity with RIF-FLD. For instance, the RIF Basic Logic Dialect [RIF-BLD] is specified by specialization from RIF-FLD and also directly, without relying on the framework. Thus, the reader who is interested in RIF-BLD only can proceed directly to that document.
RIF-FLD has the following main components:
Syntactic framework. The syntactic framework defines six types of RIF terms:
Terms are then used to define several types of RIF-BLD formulas. RIF dialects can choose to permit all or some of the aforesaid categories of terms. The syntactic framework also defines the following specialization mechanisms:
Symbol spaces partition the set of non-logical symbols that correspond to individual constants, predicates, and functions, and each partition is then given its own semantics. A symbol space has an identifier and a lexical space, which defines the "shape" of the symbols in that symbol space. Some symbol spaces in RIF are used to identify Web entities and their lexical space consists of strings that syntactically look like internationalized resource identifiers [RFC-3987], or IRIs (e.g., http://www.w3.org/2007/rif#iri). Other symbol spaces are used to represent the datatypes required by RIF (for example, http://www.w3.org/2001/XMLSchema#integer).
Signatures determine which terms and formulas are well-formed. It is a generalization of the notion of a sort in classical first-order logic [Enderton01]. Each nonlogical symbol (and some logical symbols, like =) has an associated signature. A signature defines, in a precise way, the syntactic contexts in which the symbol is allowed to occur.
For instance, the signature associated with a symbol p might allow p to appear in a term of the form f(p), but disallow it to occur in a term like p(a,b). The signature for f, on the other hand, might allow that symbol to appear in f(p) and f(p,q), but disallow f(p,q,r) and f(f). In this way, it is possible to control which symbols are used for predicates and which for functions, where variables can occur, and so on.
A dialect might impose further restrictions on the form of a particular kind of terms or formulas.
Semantic framework. This framework defines the notion of a semantic structure (also knows as interpretation in the literature [Enderton01, Mendelson97]). Semantic structures are used to interpret formulas and to define logical entailment. As with the syntax, this framework includes a number of mechanisms that RIF logic dialects can specialize to suit their needs. These mechanisms include:
A set of formulas G logically entails another formula g if for every semantic structure I in some set S, if G is true in I then g is also true in I. Almost all logics define entailment this way. The difference lies in which set S they use. For instance, logics that are based on the classical first-order predicate calculus, such as most Description Logics, assume that S is the set of all semantic structures. In contrast, most logic programming languages use default negation. Accordingly, the set S contains only the so-called "minimal" Herbrand models of G and, furthermore, only the minimal models of a special kind. See [Shoham87] for a more detailed exposition of this subject.
XML serialization framework. This framework defines the general principles for mapping the presentation syntax of RIF-FLD to the concrete XML interchange format. This includes:
This document is the latest draft of the RIF-FLD specification. Each RIF dialect that is derived from RIF-FLD will be described in its own document. The first of such dialects, RIF Basic Logic Dialect, is described in [RIF-BLD].
The next subsection explains how to derive the presentation syntax of a RIF dialect from the presentation syntax of the RIF framework. The actual syntax of the RIF framework is given in subsequent subsections.
The presentation syntax for a RIF dialect can be obtained from the general syntactic framework of RIF by specializing the following parameters, which are defined later in this document:
Signatures determine which terms in the dialect are well-formed and which are not.
The exact way signatures are assigned depends on the dialect. An assignment can be explicit or implicit (for instance, derived from the context in which each symbol is used).
The RIF logic framework introduces the following types of terms:
A dialect might support all of these terms or just a subset. For instance, some dialects might not support terms with named arguments or frame terms or certain forms of external terms (e.g., external frames).
Symbol spaces determine the syntax of the constant symbols that are allowed in the dialect.
RIF-FLD allows formulas of the following kind:
A dialect might support all of these formulas or it might impose various restrictions. For instance, the formulas allowed in the conclusion and/or premises of implications might be restricted (e.g., [RIF-BLD] essentially allows Horn rules only), certain types of quantification might be prohibited (e.g., [RIF-BLD] disallows existential quantification in the rule head), classical or default negation (or both) might not be allowed (as in RIF-BLD), etc. A subdialect of RIF-BLD might disallow equality formulas in the conclusions of the rules.
Note that although the presentation syntax of a RIF logic dialect is normative, since semantics is defined in terms of that syntax, the presentation syntax is not intended as a concrete syntax, and conformant systems are not required to implement it.
Definition (Alphabet). The alphabet of the presentation language of RIF-FLD consists of
The set of connective symbols, quantifiers, =, etc., is disjoint from Const and Var. Variables are written as Unicode strings preceded by the symbol "?". The argument names in ArgNames are written as Unicode strings that do not start with a "?". The syntax for constant symbols is given in Section Symbol Spaces.
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 terms. The symbol External indicates that an atomic formula or a function term is defined externally (e.g., a builtin), Dialect is a directive used to indicate the dialect of a RIF document (for those dialects that require this), the symbols Base and Prefix enable abridged representations of IRIs, and the symbol Import is an import directive.
Finally, the symbol Document is used for specifying RIF-FLD documents and the symbol Group is used to organize RIF-FLD formulas into collections. ☐
Throughout this document, we will be using the following abbreviations:
These and other abbreviations will be used as prefixes in the compact URI notation [CURIE], a notation for succinct representation of IRIs. The precise meaning of this notation in RIF is defined in [RIF-DTB].
The set of all constant symbols in a RIF dialect is partitioned into a number of subsets, called symbol spaces, which are used to represent XML Schema datatypes, datatypes defined in other W3C specifications, such as rdf:XMLLiteral, and to distinguish other sets of constants. All constant symbols have a syntax (and sometimes also semantics) imposed by the symbol space to which they belong.
Definition (Symbol space). A symbol space is a named subset of the set of all constants, Const. The semantic aspects of symbol spaces will be described in Section Semantic Framework. Each symbol in Const belongs to exactly one symbol space.
Each symbol space has an associated lexical space and a unique identifier. More precisely,
The identifiers for symbol spaces are not themselves constant symbols in RIF. ☐
To simplify the language, we will often use symbol space identifiers to refer to the actual symbol spaces (for instance, we may use "symbol space xs:string" instead of "symbol space identified by xs:string").
To refer to a constant in a particular RIF symbol space, we use the following presentation syntax:
"literal"^^symspace
where literal is called the lexical part of the symbol, and symspace is an identifier of the symbol space. Here literal is a sequence of Unicode characters that must be an element in the lexical space of the symbol space symspace. For instance, "1.2"^^xs:decimal and "1"^^xs:decimal are syntactically valid constants because 1.2 and 1 are members of the lexical space of the XML Schema datatype xs:decimal. On the other hand, "a+2"^^xs:decimal is not a syntactically valid symbol, since a+2 is not part of the lexical space of xs:decimal.
The set of all symbol spaces that partition Const is considered to be part of the logic language of RIF-FLD.
RIF requires that all dialects include the symbol spaces listed and described in Section Constants and Symbol Spaces of [RIF-DTB] as part of their language. These symbol spaces include constants that belong to several important XML Schema datatypes, certain RDF datatypes, and constant symbols specific to RIF. The latter include the symbol spaces rif:iri and rif:local, which are used to represent internationalized resource identifiers (IRIs) and constant symbols that are not visible outside of the RIF document in which they occur, respectively. Documents that are exchanged through RIF can use additional symbol spaces.
The most basic construct of a logic language is a term. RIF-FLD supports several kinds of terms: constants, variables, the regular positional terms, plus terms with named arguments, equality, classification terms, and frames. The word "term" will be used to refer to any kind of term.
Definition (Term). A term is a statement of one of the following forms:
Positional terms in RIF-FLD generalize the regular notion of a term used in first-order logic. For instance, the above definition allows variables everywhere, as in ?X(?Y ?Z(?V "12"^^xs:integer)), where ?X, ?Y, ?Z, and ?V are variables. Even ?X("abc"^^xs:string ?W)(?Y ?Z(?V "33"^^xs:integer)) is a positional term (as in HiLog [CKW93]).
The term t here represents a predicate or a function; s_{1}, ..., s_{n} represent argument names; and v_{1}, ..., v_{n} represent argument values. Terms with named arguments are like regular positional terms except that the arguments are named and their order is immaterial. Note that a term with no arguments, like f(), is, trivially, both a positional term and a term with named arguments.
For instance, "person"^^xs:string(name->?Y address->?Z), ?X("123"^^xs:integer ?W)(arg->?Y arg2->?Z(?V)), and "Closure"^^rif:local(relation->"http://example.com/Flight"^^rif:iri)(from->?X to->?Y) are terms with named arguments. The second of these terms has a positional term ?X(abc,?W), which occurs in the position of a function, and the third term's function is represented by a named arguments term.
Classification terms are used to describe class hierarchies.
Frame terms are used to describe properties of objects. As in the case of the terms with named arguments, the order of the properties p_{i}->v_{i} in a frame is immaterial.
Such terms are used for representing builtin 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.
This syntax enables very flexible representations for externally defined information sources: not only predicates and functions, but also frames can be used. In this way, external sources can be modeled as frames in an object-oriented way. For instance, External("http://example.com/acme"^^rif:iri["http://example.com/mycompany/president"^^rif:iri(?Year) -> ?Pres]) could be a representation for an external method "http://example.com/mycompany/president"^^rif:iri in an external object "http://example.com/acme"^^rif:iri. ☐
The above definitions are very general. They make no distinction between constant symbols that represent individuals, predicates, and function symbols. The same symbol can occur in multiple contexts at the same time. For instance, if p, a, and b are symbols then p(p(a) p(a p c)) is a term. Even variables and general terms are allowed to occur in the position of predicates and function symbols, so p(a)(?v(a c) p) is also a term.
Frame, classification, and other terms can be freely nested, as exemplified by p(?X q#r[p(1,2)->s](d->e f->g)). Some language environments, like FLORA-2 [FL2], OO jDREW [OOjD], NxBRE [NxBRE], and CycL [CycL] support fairly large (partially overlapping) subsets of RIF-FLD terms, but most languages support much smaller subsets. RIF dialects are expected to carve out the appropriate subsets of RIF-FLD terms, and the general form of the RIF logic framework allows a considerable degree of freedom.
Observe that the argument names of frame terms, p_{1}, ..., p_{n}, are 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 integral part of many inference rules underlying first-order logic. We are not aware of any rule language where terms with named arguments use anything more general than what is defined here.
Dialects can restrict the contexts in which the various terms are allowed by using the mechanism of signatures. The RIF-FLD language associates a signature with each symbol (both constant and variable symbols) and uses signatures to define well-formed terms. Each RIF dialect is expected to select appropriate signatures for the symbols in its alphabet, and only the terms that are well-formed according to the selected signatures are allowed in that particular dialect.
This section introduces the notion of external schemas, which serve as templates for externally defined terms. These schemas determine which externally defined terms are acceptable in a RIF dialect. Externally defined terms include RIF builtins, which are specified in [RIF-DTB], but are more general. They are designed to accommodate the ideas of procedural attachments and querying of external data sources. Because of the need to accommodate many difference possibilities, the RIF logical framework supports a very general notion of an externally defined term. Such a term is not necessarily a function or a predicate -- it can be a frame, a classification term, and so on.
Definition (Schema for external term). An external schema is a statement of the form (?X_{1} ... ?X_{n}; τ) where
The names of the variables in an external schema are immaterial, but their order is important. For instance, (?X ?Y; ?X[foo->?Y]) and (?V ?W; ?V[foo->?W]) are considered to be indistinguishable, but (?X ?Y; ?X[foo->?Y]) and (?Y ?X; ?X[foo->?Y]) are viewed as different schemas.
A term t is an instance of an external schema (?X_{1} ... ?X_{n}; τ) iff t can be obtained from τ by a simultaneous substitution ?X_{1}/s_{1} ... ?X_{n}/s_{n} of the variables ?X_{1} ... ?X_{n} with terms s_{1} ... s_{n}, respectively. Some of the terms s_{i} can be variables themselves. For example, ?Z[foo->f(a ?P)] is an instance of (?X ?Y; ?X[foo->?Y]) by the substitution ?X/?Z ?Y/f(a ?P). ☐
Observe that a variable cannot be an instance of an external schema, since τ in the above definition cannot be a variable. It will be seen later that this implies that a term of the form External(?X) is not well-formed in RIF.
The intuition behind the notion of an external schema, such as (?X ?Y; ?X["foo"^^xs:string->?Y]) or (?V; "pred:isTime"^^rif:iri(?V)), is that ?X["foo"^^xs:string->?Y] or "pred:isTime"^^rif:iri(?V) are invocation patterns for querying external sources, and instances of those schemas correspond to concrete invocations. Thus, External("http://foo.bar.com"^^rif:iri["foo"^^xs:string->"123"^^xs:integer]) and External("pred:isTime"^^rif:iri("22:33:44"^^xs:time) are examples of invocations of external terms -- one querying an external source and another invoking a builtin.
Definition (Coherent set of external schemas). A set of
external schemas is coherent if there is no term,
t, that is an instance of two distinct schemas in the set.
☐
The intuition behind this notion is to ensure that any use of an external term is associated with at most one external schema. This assumption is relied upon in the definition of the semantics of externally defined terms. Note that the coherence condition is easy to verify syntactically and that it implies that schemas like (?X ?Y; ?X[foo->?Y]) and (?Y ?X; ?X[foo->?Y]), which differ only in the order of their variables, cannot be in the same coherent set.
It is important to keep in mind that external schemas are not part of the language in RIF, since they do not appear anywhere in RIF statements. Instead, like signatures, which are defined below, they are best thought of as part of the grammar of the language. In particular, they will be used to determine which external terms, i.e., the terms of the form External(t) are well-formed.
In this section we introduce the concept of a signature, which is a key mechanism that allows RIF-FLD to control the context in which the various symbols are allowed to occur. For instance, a symbol f with signature {(term term) => term, (term) => term} can occur in terms like f(a b), f(f(a b) a), f(f(a)), etc., if a and b have signature term. But f is not allowed to appear in the context f(a b a) because there is no =>-expression in the signature of f to support such a context.
The above example provides intuition behind the use of signatures in RIF-FLD. Much of the development, below, is inspired by [CK95]. It should be kept in mind that signatures are not part of the logic language in RIF, since they do not appear anywhere in RIF-FLD formulas. Instead they are part of the grammar: they are used to determine which sequences of tokens are in the language and which are not. The actual way by which signatures are assigned to the symbols of the language may vary from dialect to dialect. In some dialects (for example [RIF-BLD]), this assignment is derived from the context in which each symbol occurs and no separate language for signatures is used. Other dialects may choose to assign signatures explicitly. In that case, they would require a concrete language for signatures (which would be separate from the language for specifying the logic formulas of the dialect).
Definition (Signature name). Let SigNames be a non-empty, partially-ordered finite or countably infinite set of symbols, called signature names. Since signatures are not part of the logic language, their names do not have to be disjoint from Const, Var, and ArgNames. We require that this set includes at least the following signature names:
Dialects may introduce additional signature names. For instance, RIF Basic Logic Dialect [RIF-BLD] introduces one other signature name, individual. The partial order on SigNames is dialect-specific; it is used in the definition of well-formed terms below.
We use the symbol < to represent the partial order on SigNames. Informally, α < β means that terms with signature α can be used wherever terms with signature β are allowed. We will write α ≤ β if either α = β or α < β.
Definition (Signature). A signature is a statement of the form η{e_{1}, ..., e_{n}, ...} where η ∈ SigNames is the name of the signature and {e_{1}, ..., e_{n}, ...} is a countable set of arrow expressions. Such a set can thus be infinite, finite, or even empty. In RIF-BLD, signatures can have at most one arrow expression. Other dialects (such as HiLog [CKW93], for example) may require polymorphic symbols and thus allow signatures with more than one arrow expression in them.
An arrow expression is defined as follows:
For instance, () ⇒ term and (term) ⇒ term are positional arrow expressions, if term is a signature name.
For instance, (arg1->term arg2->term) => term is an arrow signature expression with named arguments. The order of the arguments in arrow expressions with named arguments is immaterial, so any permutation of arguments yields the same expression. ☐
RIF dialects are always associated with sets of coherent signatures, defined next. The overall idea is that a coherent set of signatures must include all the predefined signatures (such as signatures for equality and classification terms) and the signatures included in a coherent set should not conflict with each other. For instance, two different signatures should not have identical names and if one signature is said to extend another then the arrow expressions of the supersignature should be included among the arrow expressions of the subsignature (a kind of an arrow expression "inheritance").
Definition (Coherent signature set). A set Σ of signatures is coherent iff
All arrow expressions e_{i} here have the form (κ κ) ⇒ γ (the arguments in an equation must be compatible) and at least one of these expressions must have the form (κ κ) ⇒ atomic (i.e., equation terms are also atomic formulas). Dialects may further specialize this signature.
Here all arrow expressions e_{i} are binary (have two arguments) and at least one has the form (κ γ) ⇒ atomic. Dialects may further specialize this signature.
Here all arrow expressions e_{i} have the form (κ κ) ⇒ γ (the arguments must be compatible) and at least one of these arrow expressions has the form (κ κ) ⇒ atomic. Dialects may further specialize this signature.
Here ηA denotes a signature with the name η and the associated set of arrow expressions A; similarly κB is a signature named κ with the set of expressions B. The requirement that B⊆A ensures that symbols that have signature η can be used wherever the symbols with signature κ are allowed. ☐
The requirement that coherent sets of signatures must include the
signatures for =, #, ->, and so on is
just a technicality needed to simplify the definitions. Some of
these signatures may go "unused" in a dialect even though,
technically speaking, they must be present in the signature set
associated with that dialect. If a dialect disallows equality,
classification terms, or frames in its syntax then the
corresponding signatures will remain unused. Such restrictions can
be imposed by specializing RIF-FLD -- see Section Syntax of a RIF Dialect as a
Specialization of RIF-FLD.
An incoherent set of signatures would be one that includes signatures mysig{() ⇒ atomic} and mysig{atomic ⇒ atomic} because it has two different signatures with the same name. Likewise, if this set contains mysig_{1}{() ⇒ atomic} and mysig_{2}{atomic ⇒ atomic} and mysig_{1} < mysig_{1} then it is incoherent because the set of arrow expressions of mysig_{1} does not contain the set of arrow expressions of mysig_{2}.
The presentation language of a RIF dialect is a set of well-formed formulas, as defined in the next section. The language is determined by the following parameters (see Syntax of a RIF Dialect as a Specialization of RIF-FLD):
Each variable symbol is associated with exactly one signature from a coherent set of signatures. A constant symbol can have one or more signatures, and different symbols can be associated with the same signature. (If variables were allowed to have multiple signatures then well-formed terms would not be closed under substitutions. For instance, a term like f(?X,?X) could be well-formed, but f(a,a) could be ill-formed.)
We have already seen how the alphabet and the symbol spaces are used to define RIF terms. The next section shows how signatures and external schemas are used to further specialize this notion to define well-formed RIF-FLD terms.
Since signature names uniquely identify signatures in coherent signature sets, we will often refer to signatures simply by their names. For instance, if one of f's signatures is atomic{ }, we may simply say that symbol f has signature atomic.
Definition (Well-formed term).
As a special case, when n=0 we obtain that t( ) is a well-formed term with signature σ, if t's signature contains the arrow expression () ⇒ σ.
As a special case, when n=0 we obtain that t( ) is a well-formed term with signature σ, if t's signature contains the arrow expression () ⇒ σ.
Note that, according to the definition of coherent sets of schemas, a term can be an instance of at most one external schema. ☐
Note that, like constant symbols, well-formed terms can have more than one signature. Also note that, according to the above definition, f() and f are distinct terms.
Definition (Well-formed formula). A well-formed term is also
a well-formed atomic formula iff one of its
signatures is atomic or is ≤ atomic. Note
that equality, membership, subclass, and frame terms are atomic
formulas, since atomic is one of their signatures.
More general formulas are constructed out of atomic formulas with the help of logical connectives. A well-formed formula is a statement that can have one of the forms (1) -- (9) below. The Group and Document formulas, defined in (8) and (9), are aggregate formulas while the formulas in (1) -- (7) are non-aggregate. This distinction manifests itself in that Group and Document cannot be part of non-aggregate formulas, and Document cannot be part of a group.
As a special case, And() is allowed and is treated as a tautology, i.e., a formula that is always true.
As a special case, Or() is treated as a contradiction, i.e., a formula that is always false.
Group formulas are intended to represent sets of formulas. Note that some of the φ_{i}'s can be group formulas themselves, which means that groups can be nested.
The Base directive does not affect the semantics. It defines a syntactic shortcut for expanding relative IRIs into full IRIs, as described in in Section Constants and Symbol Spaces of [RIF-DTB].
Like the Base directive, the Prefix directives do not affect the semantics of RIF documents. Instead, they define shorthands to allow more concise representation of IRI constants. This mechanism is explained in [RIF-DTB], Section Constants and Symbol Spaces.
RIF-FLD defines the semantics for the directive Import(t) only. The directive Import(t p) is reserved for RIF dialects, which might use it to import non-RIF logical entities, such as RDF data and OWL ontologies [RIF-RDF+OWL]. The profile might specify what kind of entity is being imported and under what semantics (for instance, the various RDF entailment regimes can be specified using different profiles).
A document formula can contain at most one Dialect and at most one Base directive. The Dialect directive, if present, must be first, followed by an optional Base directive, followed by any number of Prefix directives, followed by any number of Import directives.
In the definition of a formula, the component formulas φ, φ_{i}, ψ_{i}, and Γ are said to be subformulas of the respective formulas (conjunction, disjunction, nagation, implication, group, etc.) that are built using these components. ☐
Observe that the restrictions in (1) -- (8) above imply that groups and documents cannot be nested inside non-aggregate formulas and documents cannot be nested inside groups.
Example 1 (Signatures, well-formed terms and formulas).
We illustrate the above definitions with the following examples. In addition to atomic, let there be another signature, term{ }, which is intended here to represent the context of the arguments to positional terms or atomic formulas.
Consider the term p(p(a) p(a b c)). If p has the (polymorphic) signature mysig{(term)⇒term, (term term)⇒term, (term term term)⇒term} and a, b, c each has the signature term{ } then p(p(a) p(a b c)) is a well-formed term with signature term{ }. If instead p had the signature mysig2{(term term)⇒term, (term term term)⇒term} then p(p(a) p(a b c)) would not be a well-formed term since then p(a) would not be well-formed (in this case, p would have no arrow expression which allows p to take just one argument).
For a more complex example, let r have the signature mysig3{(term)⇒atomic, (atomic term)⇒term, (term term term)⇒term}. Then r(r(a) r(a b c)) is well-formed. The interesting twist here is that r(a) is an atomic formula that occurs as an argument to a function symbol. However, this is allowed by the arrow expression (atomic term)⇒ term, which is part of r's signature. If r's signature were mysig4{(term)⇒atomic, (atomic term)⇒atomic, (term term term)⇒term} instead, then r(r(a) r(a b c)) would be not only a well-formed term, but also a well-formed atomic formula.
An even more interesting example arises when the right-hand side of an arrow expression is something other than term or atomic. For instance, let John, Mary, NewYork, and Boston have signatures term{ }; flight and parent have signature h_{2}{(term term)⇒atomic}; and closure has signature hh_{1}{(h_{2})⇒p_{2}}, where p_{2} is the name of the signature p_{2}{(term term)⇒atomic}. Then flight(NewYork Boston), closure(flight)(NewYork Boston), parent(John Mary), and closure(parent)(John Mary) would be well-formed formulas. Such formulas are allowed in languages like HiLog [CKW93], which support predicate constructors like closure in the above example. ☐
RIF-FLD 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 RIF formula, which is not a document-formula. Both items inside the annotation are optional. The id part represents the identifier of the term (or formula) to which the annotation is attached and φ is the rest of the annotation. RIF-FLD does not impose any restrictions on φ apart from what is stated above. This means that φ may include variables, function symbols, rif:local constants, and so on.
Document formulas with and without annotations will be referred to as RIF-FLD documents.
A convention is used to avoid a syntactic ambiguity in the above definition. For instance, in (* id φ *) t[w -> v] the annotation can be attributed to the term t or to the entire frame t[w -> v]. Similarly, for an annotated HiLog-like term of the form (* id φ *) f(a)(b,c), the annotation can be attributed to the entire term f(a)(b,c) or to just f(a). The convention adopted in RIF-FLD is that any annotation is syntactically associated with the largest RIF-FLD term or formula that appears to the right of that annotation. Therefore, in our examples the annotation (* id φ *) is considered to be attached to the entire frame t[w -> v] and to the entire term f(a)(b,c).
Example 2 (A RIF-FLD document with nested groups and annotations).
We illustrate formulas, including documents and groups, with the following complete example (with apologies to Shakespeare for the imperfect rendering of the intended meaning in logic). For better readability, we use the shortcut notation defined in [RIF-DTB]. The example also illustrates attachment of annotations.
Document( Prefix(dc http://http://purl.org/dc/terms/) Prefix(ex http://example.org/ontology#) Prefix(hamlet http://www.shakespeare-literature.com/Hamlet/) (* hamlet:assertions hamlet:assertions[dc:title->"Hamlet" dc:creator->"Shakespeare"] *) Group( Exists ?X (And(?X # ex:RottenThing ex:partof(?X <http://www.denmark.dk>))) Forall ?X (Or(hamlet:tobe(?X) Naf hamlet:tobe(?X))) Forall ?X (And(Exists ?B (And(ex:has(?X ?B) ?B # ex:business)) Exists ?D (And(ex:has(?X ?D) ?D # ex:desire))) :- ?X # ex:man) (* hamlet:facts *) Group( hamlet:Yorick # ex:poor hamlet:Hamlet # ex:prince ) ) )
Observe that the above set of formulas has a nested subset with its own annotation, hamlet:facts, which contains only a global IRI. ☐
Until now, to specify the syntax of RIF-FLD we relied on "mathematical English," a special form of English for communicating mathematical definitions, examples, etc. We will now specify the syntax using the familiar EBNF notation. The following points about the EBNF notation should be kept in mind:
In view of the above, the EBNF grammar can be viewed as just an intermediary between the mathematical English and the XML. However, it also gives a succinct overview of the syntax of RIF-FLD and as such can be useful for dialect designers and users alike.
Document ::= IRIMETA? 'Document' '(' Dialect? Base? Prefix* Import* Group? ')' Dialect ::= 'Dialect' '(' Name ')' Base ::= 'Base' '(' IRI ')' Prefix ::= 'Prefix' '(' Name IRI ')' Import ::= IRIMETA? 'Import' '(' IRICONST PROFILE? ')' Group ::= IRIMETA? 'Group' '(' (FORMULA | Group)* ')' Implies ::= IRIMETA? FORMULA ':-' FORMULA FORMULA ::= IRIMETA? 'And' '(' FORMULA* ')' | IRIMETA? 'Or' '(' FORMULA* ')' | Implies | IRIMETA? 'Exists' Var* '(' FORMULA ')' | IRIMETA? 'Forall' Var* '(' FORMULA ')' | IRIMETA? 'Neg' FORMULA | IRIMETA? 'Naf' FORMULA | FORM FORM ::= IRIMETA? (Var | ATOMIC | 'External' '(' ATOMIC ')') ATOMIC ::= Const | Atom | Equal | Member | Subclass | Frame Atom ::= UNITERM UNITERM ::= TERM '(' (TERM* | (Name '->' TERM)*) ')' Equal ::= TERM '=' TERM Member ::= TERM '#' TERM Subclass ::= TERM '##' TERM Frame ::= TERM '[' (TERM '->' TERM)* ']' TERM ::= IRIMETA? (Var | EXPRIC | 'External' '(' EXPRIC ')') EXPRIC ::= Const | Expr | Equal | Member | Subclass | Frame Expr ::= UNITERM Const ::= '"' UNICODESTRING '"^^' SYMSPACE | CONSTSHORT PROFILE ::= TERM Name ::= UNICODESTRING Var ::= '?' UNICODESTRING SYMSPACE ::= ANGLEBRACKIRI | CURIE IRIMETA ::= '(*' IRICONST? (Frame | 'And' '(' Frame* ')')? '*)'
The RIF-FLD 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 the 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 used in the examples in this document. Names are Unicode character sequences. Variables are composed of UNICODESTRING symbols prefixed with a ?-sign.
RIF-FLD 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 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].
Recall that the presentation syntax of RIF-FLD allows the use of macros, which are specified via the Prefix and Base directives, and various shortcuts for integers, strings, and rif:local symbols. The semantics, below, is described using the full syntax, i.e., we assume that all shortcuts and macros have already been expanded, as defined in [RIF-DTB], Section Constants and Symbol Spaces.
The RIF-FLD semantic framework defines the notions of semantic structures and of models for RIF-FLD formulas. The semantics of a dialect is derived from these notions by specializing the following parameters.
The syntax of a dialect may limit the kinds of terms that are allowed. For instance, if a dialect's syntax excludes frames or terms with named arguments then the parts of the semantic structures whose purpose is to interpret those types of terms (I_{frame} and I_{NF} in this case) become redundant.
The RIF-FLD semantic framework allows formulas to have truth values from an arbitrary partially ordered set of truth values, TV. A concrete dialect must select a concrete partially or totally ordered set of truth values.
A datatype is a symbol space whose symbols have a fixed interpretation in any semantic structure. RIF-FLD defines a set of core datatypes that each dialect is required to include as part of its syntax and semantics. However, RIF-FLD does not limit dialects to just the core types: they can introduce additional datatypes, and each dialect must define the exact set of datatypes that it includes.
Logical entailment in RIF-FLD is defined with respect to an unspecified set of intended models. A RIF dialect must define which models are considered to be intended. For instance, one dialect might specify that all models are intended (which leads to classical first-order entailment), another may consider only the minimal models as intended, while a third one might only use well-founded or stable models [GRS91, GL88].
These notions are defined in the remainder of this specification.
Definition (Set of truth values). Each RIF dialect must define the set of truth values, denoted by TV. This set must have a partial order, called the truth order, denoted <_{t}. In some dialects, <_{t} can be a total order. We write a ≤_{t} b if either a <_{t} b or a and b are the same element of TV. In addition,
RIF dialects can have additional truth values. For instance, the semantics of some versions of NAF, such as well-founded negation, requires three truth values: t, f, and u (undefined), where f <_{t} u <_{t} t. Handling of contradictions and uncertainty usually requires at least four truth values: t, u, f, and i (inconsistent). In this case, the truth order is partial: f <_{t} u <_{t} t and f <_{t} i <_{t} t.
Definition (Primitive datatype). A primitive datatype (or just a datatype, for short) is a symbol space that has
Semantic structures are always defined with respect to a particular set of datatypes, denoted by DTS. In a concrete dialect, DTS always includes the datatypes supported by that dialect. All RIF dialects must support the primitive datatypes that are listed in Section Datatypes of [RIF-DTB]. Their value spaces and the lexical-to-value-space mappings fot these datatypes are described in the same section.
Although the lexical and the value spaces might sometimes look
similar, one should not confuse them. Lexical spaces define the
syntax of the constant symbols in the RIF language. Value spaces
define the meaning of the constants. The lexical and the
value spaces are often not even isomorphic. For example,
1.2^^xs:decimal and 1.20^^xs:decimal are two
legal -- and distinct -- constants in RIF because 1.2 and
1.20 belong to the lexical space of xs:decimal.
However, these two constants are interpreted by the same
element of the value space of the xs:decimal type.
Therefore, 1.2^^xs:decimal = 1.20^^xs:decimal is
a RIF tautology. Likewise, RIF semantics for datatypes implies
certain inequalities. For instance, abc^^xs:string ≠
abcd^^xs:string is a tautology, since the
lexical-to-value-space mapping of the xs:string type maps
these two constants into distinct elements in the value space of
xs:string.
The central step in specifying a model-theoretic semantics for a logic-based language is defining the notion of a semantic structure. Semantic structures are used to assign truth values to RIF-FLD formulas.
Definition (Semantic structure). A semantic structure, I, is a tuple of the form <TV, DTS, D, I_{C}, I_{V}, I_{F}, I_{frame}, I_{NF}, I_{sub}, I_{isa}, I_{=}, I_{external}, I_{truth}>. Here D is a non-empty set of elements called the domain of I. We will continue to use Const to refer to the set of all constant symbols and Var to refer to 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.
The other components of I are total mappings defined as follows:
This mapping interprets constant symbols.
This mapping interprets variable symbols.
This mapping interprets positional terms.
This is analogous to the interpretation of positional terms with two differences:
To see why such repetition can occur, note that argument names may repeat: p(a->b a->c). This can be understood as treating a as a set-valued argument. Identical argument/value pairs can then arise as a result of a substitution. For instance, p(a->?A a->?B) becomes p(a->b a->b) if the variables ?A and ?B are both instantiated with the symbol b.
This mapping interprets frame terms. An argument, d ∈ D, to I_{frame} represents an object and a finite bag {<a1,v1>, ..., <ak,vk>} represents a bag (multiset) of attribute-value pairs for d. We will see shortly how I_{frame} is used to determine the truth valuation of frame terms.
Bags are employed here because the order of the attribute/value pairs in a frame is immaterial and the pairs may repeat. For instance, 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.
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.
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.
It gives meaning to the equality operator.
It is used to define truth valuation for formulas.
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 builtin predicate or function, I_{external}(σ) is specified in [RIF-DTB] so that:
For convenience, we also define the following mapping I :
Here we use {...} to denote a bag of argument/value pairs.
Here {...} denotes a bag of attribute/value pairs.
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 signatures. For every signature, sg, supported by a dialect, there is a subset D_{sg} ⊆ D, called the domain of the signature. Terms that have a given signature, sg, must be mapped by I to D_{sg}, and if a term has more than one signature it must be mapped into the intersection of the corresponding signature domains. To ensure this, the following is required:
The effect of datatypes. The datatype identifiers in DTS impose the following restrictions. If 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. Then the following must hold:
That is, I_{C} must map the constants of a datatype dt in accordance with L_{dt}. ☐
RIF-FLD does not impose special requirements on I_{C} for constants in the symbol spaces that do not correspond to the identifiers of the primitive datatypes in DTS. Dialects may have such requirements, however. An example of such a restriction could be a requirement that no constant in a particular symbol space (such as rif:local) can be mapped to VS_{dt} of a datatype dt.
RIF-FLD annotations are stripped before the mappings that constitue RIF-FLD semantic structures are applied. Likewise, they are stripped before applying the truth valuation, TVal_{I}, defined in the next section. Thus, identifiers and metadata have no effect on the formal semantics.
Note that although annotations associated with RIF-FLD formulas are ignored by the semantics, they can be extracted by XML tools. Since annotations are represented by frame terms, they can be reasoned with by the rules.
This section defines how a semantic structure, I, determines the truth value TVal_{I}(φ) of a RIF-FLD formula, φ, where φ is any formula other than a document formula. Truth valuation of document formulas is defined in the next section.
To this end, we define a mapping, TVal_{I}, from the set of all non-document formulas to TV. Note that the definition implies that TVal_{I}(φ) is defined only if the set DTS of the datatypes of I includes all the datatypes mentioned in φ.
Definition (Truth valuation). Truth valuation for well-formed formulas in RIF-FLD is determined using the following function, denoted TVal_{I}:
To ensure that equality has precisely the expected properties, it is required that
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, glb_{t}(TVal_{I}(c1 ## c2), TVal_{I}(c2 ## c3)) ≤_{t} TVal_{I}(c1 ## c3).
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:
Since the bag of attribute/value pairs represents the conjunction of all the pairs, the following is required:
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 empty conjunction is treated as a tautology, so TVal_{I}(And()) = t.
The empty disjunction is treated as a contradiction, so TVal_{I}(Or()) = f.
The symbol ~ here is the idempotent operator of negation on TV introduced in Section Truth Values. Note that both classical and default negation are interpreted the same way in any concrete semantic structure. The difference between the two kinds of negation comes into play when logical entailment is defined.
Here lub_{t} (respectively, glb_{t}) is taken over all interpretations I* of the form <TV, DTS, D, I_{C}, I*_{V}, I_{F}, I_{frame}, I_{NF}, I_{sub}, I_{isa}, I_{=}, I_{external}, I_{truth}>, which are 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 ?v_{1},... ,?v_{n}.
If Γ is a group formula of the form Group(φ_{1} ... φ_{n}) then
This means that a group of formulas is treated as a conjunction. ☐
Note that rule implications and equality formulas are always two-valued, even if TV has more than two values.
Document formulas are interpreted using semantic multi-structures.
Definition (Semantic multi-structures). A semantic multi-structure is a set {I^{Δ1}, ..., I^{Δn}}, n>0, where I^{Δ1}, ..., I^{Δn} are semantic structures adorned with document formulas. These structures must be identical in all respects except that the mappings I_{C}^{Δ1}, ..., I_{C}^{Δn} 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 adornment. ☐
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'. In this case, 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 another formula, which is directly imported into Δ. ☐
The above definition considers only one-argument import directive, since two-argument directives are expected to be defined on a case-by-case basis by other specifications that need to be integrated with RIF.
The notion of semantic multi-structures will now be used to define a semantics for RIF documents.
Definition (Truth valuation of document formulas). Let Δ be a document formula and let Δ_{1}, ..., Δ_{k} be all the RIF-FLD document formulas that are imported (directly or indirectly, according to the previous definition) into Δ. Let Γ, Γ_{1}, ..., 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^{Δk}, ...} be a semantic multi-structure that contains semantic structures adorned with at least the documents Δ, Δ_{1}, ..., 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 not covered by RIF-FLD. Their semantics is supposed to be defined by other documents, such as [RIF-RDF+OWL]. ☐
The above definitions make the intent behind the rif:local constants clear: occurrences of these
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.
In the remainder of this specification, every formula is assumed to
be part of some document. It a formula is not physically part of
any document, it will be said to belong to a special query
document. This allows us to define
TVal_{I}(φ), where I is a
multi-structure, for arbitrary formulas, not just for document
formulas: If φ is a formula that is not a document-formula
and I is a semantic multi-structure that contains a
component I^{Δ} that corresponds to
the document of the formula φ, then
TVal_{I}(φ) is defined as
TVal_{IΔ}(φ).
Otherwise, TVal_{I}(φ) is undefined.
Definition (Models). A multi-structure I is a
model of a formula, φ, written as
I|=φ, iff
TVal_{I}(φ) is defined and equals t.
☐
The semantics of a set of formulas, Γ, is the set of its intended semantic multi-structures. RIF-FLD does not specify what these intended multi-structures are, leaving this to RIF dialects. Different logic theories may have different criteria for what is considered an intended semantic multi-structure.
For the classical first-order logic, every model is an intended semantic multi-structure. For [RIF-BLD], which is based on Horn rules, intended multi-structures are defined only for sets of rules: an intended semantic multi-structure of a RIF-BLD set of formulas, Γ, is the unique minimal Herbrand model of Γ. For the dialects in which rule bodies may contain literals negated with the negation-as-failure connective Naf, only some of the minimal Herbrand models of a set of rules are intended. Each logic dialect of RIF must define the set of intended semantic multi-structures precisely. The two most common such theories are the well-founded models [GRS91] and stable models [GL88].
The following example illustrates the notion of intended semantic structures. Suppose Γ consists of a single rule formula p :- Naf q. If Naf were interpreted as classical negation, then this rule would be simply equivalent to Or(p q), and so it would have two kinds of models: those where p is true and those where q is true. In contrast to first-order logic, most rule-based systems do not consider p and q symmetrically. Instead, they view the rule p :- Naf q as a statement that p must be true if it is not possible to establish the truth of q. Since it is, indeed, impossible to establish the truth of q, such theories would derive p even though it does not logically follow from Or(p q). The logic underlying rule-based systems also assumes that only the minimal Herbrand models are intended (minimality here is with respect to the set of true facts). Furthermore, although our example has two minimal Herbrand models -- one where p is true and q is false, and the other where p is false, but q is true, only the first model is considered to be intended.
The above concept of intended models and the corresponding notion of logical entailment with respect to the intended models, defined below, is due to [Shoham87].
We will now define what it means for a set of RIF-FLD formulas to entail another RIF-FLD formula. This notion is typically used for defining queries to knowledge bases and for other tasks, such as testing subsumption of concepts (e.g., in OWL). We assume that each set of formulas has an associated set of intended semantic structures.
Definition
(Logical entailment). Let Γ be a RIF-FLD formula and
φ another RIF-FLD formula. We say that Γ
entails φ, written as
Γ |= φ, if and only if, for every intended
semantic multi-structure I of Γ for which
both TVal_{I}(Γ) and
TVal_{I}(φ) are defined, it is the case
that TVal_{I}(Γ) ≤_{t}
TVal_{I}(φ). ☐
This general notion of entailment covers both first-order logic and the non-monotonic logics that underlie many rule-based languages [Shoham87].
Note that one consequence of the multi-document semantics is that
local constants specified in one document cannot be queried from
another document. In particular, they cannot appear in entailed
formulas. For instance, if one document, Δ', has the fact
"http://example.com/ppp"^^rif:iri("abc"^^rif:local) while
another document formula, Δ, imports Δ' and has
the rule "http://example.com/qqq"^^rif:iri(?X) :-
"http://example.com/ppp"^^rif:iri(?X) , then Δ |=
"http://example.com/qqq"^^rif:iri("abc"^^rif:local) does
not hold. This is because "abc"^^rif:local in
Δ' and "abc"^^rif:local in the formula on the
right-hand side of |= are treated as different constants
by semantic multi-structures.
The RIF-FLD XML serialization framework defines
As explained in the overview section, the design of RIF envisions that the presentation syntaxes of future logic RIF dialects will be specializations of the presentation syntax of RIF-FLD. This means that every well-formed formula in the presentation syntax of a standard logic RIF dialect must also be well-formed in RIF-FLD. The goal of the XML serialization framework is to provide a similar yardstick for the RIF XML syntax. This amounts to the requirement that any conformant XML document for a logic RIF dialect must also be a conformant XML document for RIF-FLD (conformance is defined below). In terms of the presentation-to-XML syntax mappings, this means that each mapping for a logic RIF dialect must be a restriction of the corresponding mapping for RIF-FLD. For instance, the mapping from the presentation syntax of RIF-BLD to XML in [RIF-BLD] is a restriction of the presentation-syntax-to-XML mapping for RIF-FLD. In this way, RIF-FLD provides a framework for extensibility and mutual compatibility between XML syntaxes of RIF dialects.
Recall that the syntax of RIF-FLD is not context-free and thus cannot be fully captured by EBNF or 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 XML document in a logic dialect). A valid RIF-FLD document in the XML syntax is an XML document that is valid with respect to the XML schema in Appendix XML Schema for FLD.
If a dialect, D, specializes RIF-FLD then its XML schema is a specialization of the XML schema of RIF-FLD. A valid XML document in D is then one that is valid with respect to the XML schema of D. ☐
Definition (Conformant XML document in a logic dialect). A conformant FLD document in the XML syntax is a valid FLD document in the XML syntax that is the image of a well-formed RIF-FLD document in the presentation syntax (see Definition Well-formed formula in Section Well-formed Terms and Formulas) under the presentation-to-XML syntax mapping χ_{fld} defined in Section Mapping from the RIF-FLD Presentation Syntax to the XML Syntax.
If a dialect, D, specializes RIF-FLD then an XML document is conformant with respect to D if and only if it is a valid document in D and it is an image of a well-formed document in the presentation syntax of D.
Note that if D requires the directive Dialect(D) as part of its syntax then this implies that any D-conformant document must have this directive. ☐
A round-tripping of a conformant document in a dialect, D, is a semantics-preserving mapping to a document in any language L followed by a semantics-preserving mapping from the L-document back to a conformant D-document. While semantically equivalent, the original and the round-tripped D-documents need not be identical. Metadata should survive round-tripping.
RIF-FLD uses [XML1.0] for its XML syntax. The XML serialization for RIF-FLD 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 EBNF of the presentation syntax, such as FORMULA, become XML Schema groups in Appendix XML Schema for FLD. 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.
The RIF serialization framework for the syntax of Section EBNF Grammar for the Presentation
Syntax of RIF-FLD uses the following XML tags. While there is a
RIF-FLD element tag for the Import directive and an
attribute for the Dialect directive, there are none for
the Base and Prefix directives: they are handled
as discussed in Section Mapping of the RIF-FLD Rule Language.
- Document (document, with optional 'dialect' attribute, containing optional directive and payload roles) - directive (directive role, containing Import) - payload (payload role, containing Group) - Import (importation, containing location and optional profile) - location (location role, containing IRICONST) - profile (profile role, containing PROFILE) - Group (nested collection of sentences) - sentence (sentence role, containing FORMULA or Group) - Forall (quantified formula for 'Forall', containing declare and formula roles) - Exists (quantified formula for 'Exists', containing declare and formula roles) - declare (declare role, containing a Var) - formula (formula role, containing a FORMULA) - Implies (implication, containing if and then roles) - if (antecedent role, containing FORMULA) - then (consequent role, containing FORMULA) - And (conjunction) - Or (disjunction) - Neg (strong negation, containing a formula role) - Naf (negation as failure, containing a formula role) - 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.
The XML Schema Definition of RIF-FLD is given in Appendix XML Schema for FLD.
The XML syntax for symbol spaces uses the type attribute associated with the XML element Const. For instance, a literal in the xs:dateTime datatype is represented as <Const type="&xs;dateTime">2007-11-23T03:55:44-02:30</Const>. RIF-FLD also uses the ordered attribute to indicate that the children of args and slot elements are ordered.
Example 3 (Serialization of a nested RIF-FLD group with annotations).
This example shows an XML serialization for the formulas in Example 2. For convenience of reference, the original formulas are included at the top. For better readability, we again use the shortcut syntax defined in [RIF-DTB].
Presentation syntax: Document( Dialect(FOL) Prefix(dc http://http://purl.org/dc/terms/) Prefix(ex http://example.org/ontology#) Prefix(hamlet http://www.shakespeare-literature.com/Hamlet/) (* hamlet:assertions hamlet:assertions[dc:title->"Hamlet" dc:creator->"Shakespeare"] *) Group( Exists ?X (And(?X # ex:RottenThing ex:partof(?X <http://www.denmark.dk>))) Forall ?X (Or(hamlet:tobe(?X) Naf hamlet:tobe(?X))) Forall ?X (And(Exists ?B (And(ex:has(?X ?B) ?B # ex:business)) Exists ?D (And(ex:has(?X ?D) ?D # ex:desire))) :- ?X # ex:man) (* hamlet:facts *) Group( hamlet:Yorick # ex:poor hamlet:Hamlet # ex:prince ) ) ) XML serialization: <!DOCTYPE Document [ <!ENTITY dc "http://purl.org/dc/terms/"> <!ENTITY ex "http://example.org/ontology#"> <!ENTITY hamlet "http://www.shakespeare-literature.com/Hamlet/"> <!ENTITY rif "http://www.w3.org/2007/rif#"> <!ENTITY xs "http://www.w3.org/2001/XMLSchema#"> ]> <Document dialect="FOL"> <payload> <Group> <meta> <Frame> <object> <Const type="&rif;iri">hamlet:assertions</Const> </object> <slot ordered="yes"> <Const type="&rif;iri">&dc;title</Const> <Const type="&xs;string">Hamlet</Const> </slot> <slot ordered="yes"> <Const type="&rif;iri">&dc;creator</Const> <Const type="&xs;string">Shakespeare</Const> </slot> </Frame> </meta> <sentence> <Exists> <declare><Var>X</Var></declare> <formula> <And> <formula> <Member> <instance><Var>X</Var></instance> <class><Const type="&rif;iri">ex:RottenThing</Const></class> </Member> </formula> <formula> <Atom> <op><Const type="&rif;iri">ex:partof</Const></op> <args ordered="yes"> <Var>X</Var> <Const type="&rif;iri">http://www.denmark.dk</Const> </args> </Atom> </formula> </And> </formula> </Exists> </sentence> <sentence> <Forall> <declare><Var>X</Var></declare> <formula> <Or> <formula> <Atom> <op><Const type="&rif;iri">hamlet:tobe</Const></op> <args ordered="yes"><Var>X</Var></args> </Atom> </formula> <formula> <Naf> <formula> <Atom> <op><Const type="&rif;iri">hamlet:tobe</Const></op> <args ordered="yes"><Var>X</Var></args> </Atom> </formula> </Naf> </formula> </Or> </formula> </Forall> </sentence> <sentence> <Forall> <declare><Var>X</Var></declare> <formula> <Implies> <if> <Member> <instance><Var>X</Var></instance> <class><Const type="&rif;iri">ex:man</Const></class> </Member> </if> <then> <And> <formula> <Exists> <declare><Var>B</Var></declare> <formula> <And> <formula> <Atom> <op><Const type="&rif;iri">ex:has</Const></op> <args> <Var>X</Var> <Var>B</Var> </args> </Atom> </formula> <formula> <Member> <instance><Var>B</Var></instance> <class><Const type="&rif;iri">ex:business</Const></class> </Member> </formula> </And> </formula> </Exists> </formula> <formula> <Exists> <declare><Var>D</Var></declare> <formula> <And> <formula> <Atom> <op><Const type="&rif;iri">ex:has</Const></op> <args> <Var>X</Var> <Var>D</Var> </args> </Atom> </formula> <formula> <Member> <instance><Var>D</Var></instance> <class><Const type="&rif;iri">ex:desire</Const></class> </Member> </formula> </And> </formula> </Exists> </formula> </And> </then> </Implies> </formula> </Forall> </sentence> <sentence> <Group> <meta> <Frame> <object> <Const type="&rif;iri">hamlet:facts</Const> </object> </Frame> </meta> <sentence> <Member> <instance><Const type="&rif;iri">hamlet:Yorick</Const></instance> <class><Const type="&rif;iri">ex:poor</Const></class> </Member> </sentence> <sentence> <Member> <instance><Const type="&rif;iri">hamlet:Hamlet</Const></instance> <class><Const type="&rif;iri">ex:prince</Const></class> </Member> </sentence> </Group> </sentence> </Group> </payload> </Document>
This section defines a normative mapping, χ_{fld}, from the presentation syntax of Section EBNF Grammar for the Presentation Syntax of RIF-FLD to the XML syntax of RIF-FLD. The mapping is given via tables where each row specifies the mapping of a particular syntactic pattern in the presentation syntax. These patterns appear in the first column of the tables and the bold-italic symbols represent metavariables. The second column represents the corresponding XML patterns, which may contain applications of the mapping χ_{fld} to these metavariables. When an expression χ_{fld}(metavar) occurs in an XML pattern in the right column of a translation table, it should be understood as a recursive application of χ_{fld} to the presentation syntax represented by the metavariable. The XML syntax result of such an application is substituted for the expression χ_{fld}(metavar). A sequence of terms containing metavariables with subscripts is indicated by an ellipsis. A metavariable or a well-formed XML subelement is marked as optional by appending a bold-italic question mark, ?, to its right.
The χ_{fld} mapping from the presentation syntax to the XML syntax of the non-annotated RIF-FLD Language is given by the table below. Each row indicates a translation χ_{fld}(Presentation) = XML. Since the presentation syntax of RIF-FLD is context sensitive, the mapping must differentiate between the terms that occur in the position of the individuals and the terms that occur as atomic formulas. To this end, in the translation table, the positional and named argument terms that occur in the context of atomic formulas are denoted by the expressions of the form pred(...) and the terms that occur as individuals are denoted by expressions of the form func(...). In the table, each metavariable for an (unnamed) positional argument_{i} is assumed to be instantiated to values unequal to the instantiations of named arguments unicodestring_{j} -> filler_{j}. Regarding the last but first row, we assume that shortcuts for constants [RIF-DTB] have already been expanded to their full form ("..."^^symspace).
Note that while the Import and the Dialect directives are handled by the presentation-to-XML syntax mapping, the Prefix and Base directives are not. Instead, these directives should be handled by macro-expanding the associated shortcuts (compact URIs). Namely, a prefix name declared in a Prefix directive is expanded into the associated IRI, while relative IRIs are completed using the IRI declared in the Base directive. The mapping χ_{fld} applies only to such macro-expanded documents. RIF-FLD also allows other treatments of Prefix and Base provided that they produce equivalent XML documents. One such treatment is employed in the examples in this document, especially Example 3. It replaces prefix names with definitions of XML entities as follows. Each Prefix declaration becomes an ENTITY declaration [XML1.0] within a DOCTYPE DTD attached to the RIF-FLD Document. The Base directive is mapped to the xml:base attribute [XML-Base] in the XML Document tag. Compact URIs of the form prefix:suffix are then mapped to &prefix;suffix.
Presentation Syntax | XML Syntax |
---|---|
Document( Dialect(name)? Import(loc_{1} prfl_{1}?) . . . Import(loc_{n} prfl_{n}?) group ) |
<Document dialect="name"?> <directive> <Import> <location>χ_{fld}(loc_{1})</location> <profile>χ_{fld}(prfl_{1})</profile>? </Import> </directive> . . . <directive> <Import> <location>χ_{fld}(loc_{n})</location> <profile>χ_{fld}(prfl_{n})</profile>? </Import> </directive> <payload>χ_{fld}(group)</payload> </Document> |
Group( clause_{1} . . . clause_{n} ) |
<Group> <sentence>χ_{fld}(clause_{1})</sentence> . . . <sentence>χ_{fld}(clause_{n})</sentence> </Group> |
Forall variable_{1} . . . variable_{n} ( body ) |
<Forall> <declare>χ_{fld}(variable_{1})</declare> . . . <declare>χ_{fld}(variable_{n})</declare> <formula>χ_{fld}(body)</formula> </Forall> |
conclusion :- condition |
<Implies> <if>χ_{fld}(condition)</if> <then>χ_{fld}(conclusion)</then> </Implies> |
And ( conjunct_{1} . . . conjunct_{n} ) |
<And> <formula>χ_{fld}(conjunct_{1})</formula> . . . <formula>χ_{fld}(conjunct_{n})</formula> </And> |
Or ( disjunct_{1} . . . disjunct_{n} ) |
<Or> <formula>χ_{fld}(disjunct_{1})</formula> . . . <formula>χ_{fld}(disjunct_{n})</formula> </Or> |
Neg form |
<Neg> <formula>χ_{fld}(form)</formula> </Neg> |
Naf form |
<Naf> <formula>χ_{fld}(form)</formula> </Naf> |
Exists variable_{1} . . . variable_{n} ( body ) |
<Exists> <declare>χ_{fld}(variable_{1})</declare> . . . <declare>χ_{fld}(variable_{n})</declare> <formula>χ_{fld}(body)</formula> </Exists> |
External ( atomframexpr ) |
<External> <content>χ_{fld}(atomframexpr)</content> </External> |
pred ( argument_{1} . . . argument_{n} ) |
<Atom> <op>χ_{fld}(pred)</op> <args ordered="yes"> χ_{fld}(argument_{1}) . . . χ_{fld}(argument_{n}) </args> </Atom> |
func ( argument_{1} . . . argument_{n} ) |
<Expr> <op>χ_{fld}(func)</op> <args ordered="yes"> χ_{fld}(argument_{1}) . . . χ_{fld}(argument_{n}) </args> </Expr> |
pred ( unicodestring_{1} -> filler_{1} . . . unicodestring_{n} -> filler_{n} ) |
<Atom> <op>χ_{fld}(pred)</op> <slot ordered="yes"> <Name>unicodestring_{1}</Name> χ_{fld}(filler_{1}) </slot> . . . <slot ordered="yes"> <Name>unicodestring_{n}</Name> χ_{fld}(filler_{n}) </slot> </Atom> |
func ( unicodestring_{1} -> filler_{1} . . . unicodestring_{n} -> filler_{n} ) |
<Expr> <op>χ_{fld}(func)</op> <slot ordered="yes"> <Name>unicodestring_{1}</Name> χ_{fld}(filler_{1}) </slot> . . . <slot ordered="yes"> <Name>unicodestring_{n}</Name> χ_{fld}(filler_{n}) </slot> </Expr> |
inst [ key_{1} -> filler_{1} . . . key_{n} -> filler_{n} ] |
<Frame> <object>χ_{fld}(inst)</object> <slot ordered="yes"> χ_{fld}(key_{1}) χ_{fld}(filler_{1}) </slot> . . . <slot ordered="yes"> χ_{fld}(key_{n}) χ_{fld}(filler_{n}) </slot> </Frame> |
inst # class |
<Member> <instance>χ_{fld}(inst)</instance> <class>χ_{fld}(class)</class> </Member> |
sub ## super |
<Subclass> <sub>χ_{fld}(sub)</sub> <super>χ_{fld}(super)</super> </Subclass> |
left = right |
<Equal> <left>χ_{fld}(left)</left> <right>χ_{fld}(right)</right> </Equal> |
"unicodestring"^^space |
<Const type="space">unicodestring</Const> |
?unicodestring |
<Var>unicodestring</Var> |
The χ_{fld} mapping from RIF-FLD annotations in the presentation syntax to the XML syntax is specified by the table below. It extends the translation table of Section Mapping of the Non-annotated RIF-FLD Language. The metavariable Typetag in the presentation and XML syntaxes stands for any of the class names And, Or, External, Document, or Group, Quantifier for Exists or Forall, and Negation for Neg or Naf. The dollar sign, $, stands for any of the binary infix operator names #, ##, =, or :-, while Binop stands for their respective class names Member, Subclass, Equal, or Implies. The metavariable attr? is used with Typetag to capture the optional dialect attribute (with its value) of Document. Again, each metavariable for an (unnamed) positional argument_{i} is assumed to be instantiated to values unequal to the instantiations of named arguments unicodestring_{j} -> filler_{j}.
Presentation Syntax | XML Syntax |
---|---|
(* iriconst? frameconj? *) Typetag ( e_{1} . . . e_{n} ) |
<Typetag attr?> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? e_{1}' . . . e_{n}' </Typetag> where attr, e_{1}', . . ., e_{n}' are defined by the equation χ_{fld}(Typetag(e_{1} . . . e_{n})) = <Typetag attr?>e_{1}' . . . e_{n}'</Typetag> |
(* iriconst? frameconj? *) Quantifier variable_{1} . . . variable_{n} ( body ) |
<Quantifier> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? <declare>χ_{fld}(variable_{1})</declare> . . . <declare>χ_{fld}(variable_{n})</declare> <formula>χ_{fld}(body)</formula> </Quantifier> |
(* iriconst? frameconj? *) Negation e |
<Negation> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? χ_{fld}(e) </Negation> |
(* iriconst? frameconj? *) pred ( argument_{1} . . . argument_{n} ) |
<Atom> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? <op>χ_{fld}(pred)</op> <args ordered="yes"> χ_{fld}(argument_{1}) . . . χ_{fld}(argument_{n}) </args> </Atom> |
(* iriconst? frameconj? *) func ( argument_{1} . . . argument_{n} ) |
<Expr> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? <op>χ_{fld}(func)</op> <args ordered="yes"> χ_{fld}(argument_{1}) . . . χ_{fld}(argument_{n}) </args> </Expr> |
(* iriconst? frameconj? *) pred ( unicodestring_{1} -> filler_{1} . . . unicodestring_{n} -> filler_{n} ) |
<Atom> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? <op>χ_{fld}(pred)</op> <slot ordered="yes"> <Name>unicodestring_{1}</Name> χ_{fld}(filler_{1}) </slot> . . . <slot ordered="yes"> <Name>unicodestring_{n}</Name> χ_{fld}(filler_{n}) </slot> </Atom> |
(* iriconst? frameconj? *) func ( unicodestring_{1} -> filler_{1} . . . unicodestring_{n} -> filler_{n} ) |
<Expr> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? <op>χ_{fld}(func)</op> <slot ordered="yes"> <Name>unicodestring_{1}</Name> χ_{fld}(filler_{1}) </slot> . . . <slot ordered="yes"> <Name>unicodestring_{n}</Name> χ_{fld}(filler_{n}) </slot> </Expr> |
(* iriconst? frameconj? *) inst [ key_{1} -> filler_{1} . . . key_{n} -> filler_{n} ] |
<Frame> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? <object>χ_{fld}(inst)</object> <slot ordered="yes"> χ_{fld}(key_{1}) χ_{fld}(filler_{1}) </slot> . . . <slot ordered="yes"> χ_{fld}(key_{n}) χ_{fld}(filler_{n}) </slot> </Frame> |
(* iriconst? frameconj? *) e_{1} $ e_{2} |
<Binop> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? e_{1}' e_{2}' </Binop> where Binop, e_{1}', e_{2}' are defined by the equation χ_{fld}(e_{1} $ e_{2}) = <Binop>e_{1}' e_{2}'</Binop> |
(* iriconst? frameconj? *) unicodestring^^symspace |
<Const type="symspace"> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? unicodestring </Const> |
(* iriconst? frameconj? *) ?unicodestring |
<Var> <id>χ_{fld}(iriconst)</id>? <meta>χ_{fld}(frameconj)</meta>? unicodestring </Var> |
RIF does not require or expect conformant systems to implement the presentation syntax of a RIF dialect. Instead, conformance is described in terms of semantics-preserving transformations.
Let Τ be a set of datatypes that includes the datatypes specified
in [RIF-DTB], and suppose Ε is
a set of external predicates and functions that includes the
built-ins listed in [RIF-DTB].
Let D be a RIF dialect (e.g., [RIF-BLD]). We say that a formula φ is a
D_{Τ,Ε} formula iff
A RIF processor is a conformant D_{Τ,Ε} consumer iff it implements a semantics-preserving mapping, μ, from the set of all D_{Τ,Ε} formulas to the language L of the processor.
Formally, this means that for any pair φ, ψ of D_{Τ,Ε} formulas for which φ |=_{D} ψ is defined, φ |=_{D} ψ iff μ(φ) |=_{L} μ(ψ). Here |=_{D} denotes the logical entailment in the RIF dialect D and |=_{L} is the logical entailment in the language L of the RIF processor.
A RIF processor is a conformant D_{Τ,Ε} producer iff it implements a semantics-preserving mapping, ν, from a subset of the language L of the processor to the set of D_{Τ,Ε} formulas.
Formally, this means that for any pair φ, ψ of formulas in the aforesaid subset of L for which φ |=_{L} ψ is defined, φ |=_{L} ψ iff ν(φ) |=_{D} ν(ψ).
A conformant document in a logic RIF dialect D
is one which conforms to all the syntactic constraints of D,
including the ones that cannot be checked by an XML Schema
validator (see Definition Conformant XML document in a logic dialect).
The namespace of RIF is http://www.w3.org/2007/rif#.
XML schemas for the RIF-FLD language are defined below and are also available here with additional examples. For modularity, we define a Baseline schema and a Skyline schema. Baseline is the schema module that provides the foundation up to FORMULAs without Implies. Skyline provides the full schema by augmenting Baseline with the Implies FORMULA as well as with Group and Document.
<?xml version="1.0" encoding="UTF-8"?> <xs:schema xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns="http://www.w3.org/2007/rif#" targetNamespace="http://www.w3.org/2007/rif#" elementFormDefault="qualified" version="Id: FLDBaseline.xsd, v. 0.96, 2008-07-15, hboley/dhirtle"> <xs:annotation> <xs:documentation> This is the Baseline module of FLD. It is the foundation of the full schema defined through the Skyline module. The Baseline XML schema is based on the following EBNF (compared to the full EBNF of RIF-FLD, Group and Document are omitted, and 'Implies' is missing from the production for FORMULA). FORMULA ::= IRIMETA? 'And' '(' FORMULA* ')' | IRIMETA? 'Or' '(' FORMULA* ')' | IRIMETA? 'Exists' Var* '(' FORMULA ')' | IRIMETA? 'Forall' Var* '(' FORMULA ')' | IRIMETA? 'Neg' FORMULA | IRIMETA? 'Naf' FORMULA | FORM FORM ::= IRIMETA? (Var | ATOMIC | 'External' '(' ATOMIC ')') ATOMIC ::= Const | Atom | Equal | Member | Subclass | Frame Atom ::= UNITERM UNITERM ::= TERM '(' (TERM* | (Name '->' TERM)*) ')' Equal ::= TERM '=' TERM Member ::= TERM '#' TERM Subclass ::= TERM '##' TERM Frame ::= TERM '[' (TERM '->' TERM)* ']' TERM ::= IRIMETA? (Var | EXPRIC | 'External' '(' EXPRIC ')') EXPRIC ::= Const | Expr | Equal | Member | Subclass | Frame Expr ::= UNITERM Const ::= '"' UNICODESTRING '"^^' SYMSPACE | CONSTSHORT Name ::= UNICODESTRING Var ::= '?' UNICODESTRING SYMSPACE ::= ANGLEBRACKIRI | CURIE IRIMETA ::= '(*' IRICONST? (Frame | 'And' '(' Frame* ')')? '*)' </xs:documentation> </xs:annotation> <xs:group name="FORMULA"> <!-- 'Implies' omitted from Baseline schema, allowing its modular use FORMULA ::= IRIMETA? 'And' '(' FORMULA* ')' | IRIMETA? 'Or' '(' FORMULA* ')' | IRIMETA? 'Exists' Var* '(' FORMULA ')' | IRIMETA? 'Forall' Var* '(' FORMULA ')' | IRIMETA? 'Neg' FORMULA | IRIMETA? 'Naf' FORMULA | FORM --> <xs:choice> <xs:element ref="And"/> <xs:element ref="Or"/> <xs:element ref="Exists"/> <xs:element ref="Forall"/> <xs:element ref="Neg"/> <xs:element ref="Naf"/> <xs:group ref="FORM"/> </xs:choice> </xs:group> <xs:complexType name="External-FORMULA.type"> <!-- sensitive to FORMULA (Atom | Frame) context--> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element name="content" type="content-FORMULA.type"/> </xs:sequence> </xs:complexType> <xs:complexType name="content-FORMULA.type"> <!-- sensitive to FORMULA (Atom | Frame) context--> <xs:sequence> <xs:choice> <xs:element ref="Atom"/> <xs:element ref="Frame"/> </xs:choice> </xs:sequence> </xs:complexType> <xs:element name="And"> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="formula" minOccurs="0" maxOccurs="unbounded"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Or"> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="formula" minOccurs="0" maxOccurs="unbounded"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Exists"> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="declare" minOccurs="0" maxOccurs="unbounded"/> <xs:element ref="formula"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Forall"> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="declare" minOccurs="0" maxOccurs="unbounded"/> <xs:element ref="formula"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Neg"> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="formula" minOccurs="1" maxOccurs="1"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Naf"> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="formula" minOccurs="1" maxOccurs="1"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="formula"> <xs:complexType> <xs:sequence> <xs:group ref="FORMULA"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="declare"> <xs:complexType> <xs:sequence> <xs:element ref="Var"/> </xs:sequence> </xs:complexType> </xs:element> <xs:group name="FORM"> <!-- FORM ::= IRIMETA? (Var | ATOMIC | 'External' '(' ATOMIC ')') --> <xs:choice> <xs:element ref="Var"/> <xs:group ref="ATOMIC"/> <xs:element name="External" type="External-FORM.type"/> </xs:choice> </xs:group> <xs:complexType name="External-FORM.type"> <!-- sensitive to FORM (ATOMIC) context--> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element name="content" type="content-FORM.type"/> </xs:sequence> </xs:complexType> <xs:complexType name="content-FORM.type"> <!-- sensitive to FORM (ATOMIC) context--> <xs:sequence> <xs:group ref="ATOMIC"/> </xs:sequence> </xs:complexType> <xs:group name="ATOMIC"> <!-- ATOMIC ::= Const | Atom | Equal | Member | Subclass | Frame --> <xs:choice> <xs:element ref="Const"/> <xs:element ref="Atom"/> <xs:element ref="Equal"/> <xs:element ref="Member"/> <xs:element ref="Subclass"/> <xs:element ref="Frame"/> </xs:choice> </xs:group> <xs:element name="Atom"> <!-- Atom ::= UNITERM --> <xs:complexType> <xs:sequence> <xs:group ref="UNITERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:group name="UNITERM"> <!-- UNITERM ::= TERM '(' (TERM* | (Name '->' TERM)*) ')' --> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="op"/> <xs:choice> <xs:element ref="args" minOccurs="0" maxOccurs="1"/> <xs:element name="slot" type="slot-UNITERM.type" minOccurs="0" maxOccurs="unbounded"/> </xs:choice> </xs:sequence> </xs:group> <xs:element name="op"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="args"> <xs:complexType> <xs:sequence> <xs:group ref="TERM" minOccurs="0" maxOccurs="unbounded"/> </xs:sequence> <xs:attribute name="ordered" type="xs:string" fixed="yes"/> </xs:complexType> </xs:element> <xs:complexType name="slot-UNITERM.type"> <!-- sensitive to UNITERM (Name) context--> <xs:sequence> <xs:element ref="Name"/> <xs:group ref="TERM"/> </xs:sequence> <xs:attribute name="ordered" type="xs:string" fixed="yes"/> </xs:complexType> <xs:element name="Equal"> <!-- Equal ::= TERM '=' TERM --> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="left"/> <xs:element ref="right"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="left"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="right"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Member"> <!-- Member ::= TERM '#' TERM --> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="instance"/> <xs:element ref="class"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Subclass"> <!-- Subclass ::= TERM '##' TERM --> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="sub"/> <xs:element ref="super"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="instance"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="class"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="sub"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="super"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Frame"> <!-- Frame ::= TERM '[' (TERM '->' TERM)* ']' --> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="object"/> <xs:element name="slot" type="slot-Frame.type" minOccurs="0" maxOccurs="unbounded"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="object"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:complexType name="slot-Frame.type"> <!-- sensitive to Frame (TERM) context--> <xs:sequence> <xs:group ref="TERM"/> <xs:group ref="TERM"/> </xs:sequence> <xs:attribute name="ordered" type="xs:string" fixed="yes"/> </xs:complexType> <xs:group name="TERM"> <!-- TERM ::= IRIMETA? (Var | EXPRIC | 'External' '(' EXPRIC ')') --> <xs:choice> <xs:element ref="Var"/> <xs:group ref="EXPRIC"/> <xs:element name="External" type="External-TERM.type"/> </xs:choice> </xs:group> <xs:complexType name="External-TERM.type"> <!-- sensitive to TERM (EXPRIC) context--> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element name="content" type="content-TERM.type"/> </xs:sequence> </xs:complexType> <xs:complexType name="content-TERM.type"> <!-- sensitive to TERM (EXPRIC) context--> <xs:sequence> <xs:group ref="EXPRIC"/> </xs:sequence> </xs:complexType> <xs:group name="EXPRIC"> <!-- EXPRIC ::= Const | Expr | Equal | Member | Subclass | Frame --> <xs:choice> <xs:element ref="Const"/> <xs:element ref="Expr"/> <xs:element ref="Equal"/> <xs:element ref="Member"/> <xs:element ref="Subclass"/> <xs:element ref="Frame"/> </xs:choice> </xs:group> <xs:element name="Expr"> <!-- Expr ::= UNITERM --> <xs:complexType> <xs:sequence> <xs:group ref="UNITERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Const"> <!-- Const ::= '"' UNICODESTRING '"^^' SYMSPACE | CONSTSHORT --> <xs:complexType mixed="true"> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> </xs:sequence> <xs:attribute name="type" type="xs:anyURI" use="required"/> </xs:complexType> </xs:element> <xs:element name="Name" type="xs:string"> <!-- Name ::= UNICODESTRING --> </xs:element> <xs:element name="Var"> <!-- Var ::= '?' UNICODESTRING --> <xs:complexType mixed="true"> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> </xs:sequence> </xs:complexType> </xs:element> <xs:group name="IRIMETA"> <!-- IRIMETA ::= '(*' IRICONST? (Frame | 'And' '(' Frame* ')')? '*)' --> <xs:sequence> <xs:element ref="id" minOccurs="0" maxOccurs="1"/> <xs:element ref="meta" minOccurs="0" maxOccurs="1"/> </xs:sequence> </xs:group> <xs:element name="id"> <xs:complexType> <xs:sequence> <xs:element name="Const" type="IRICONST.type"/> <!-- type="&rif;iri" --> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="meta"> <xs:complexType> <xs:choice> <xs:element ref="Frame"/> <xs:element name="And" type="And-meta.type"/> </xs:choice> </xs:complexType> </xs:element> <xs:complexType name="And-meta.type"> <!-- sensitive to meta (Frame) context--> <xs:sequence> <xs:element name="formula" type="formula-meta.type" minOccurs="0" maxOccurs="unbounded"/> </xs:sequence> </xs:complexType> <xs:complexType name="formula-meta.type"> <!-- sensitive to meta (Frame) context--> <xs:sequence> <xs:element ref="Frame"/> </xs:sequence> </xs:complexType> <xs:complexType name="IRICONST.type" mixed="true"> <!-- sensitive to location/id context--> <xs:sequence/> <xs:attribute name="type" type="xs:anyURI" use="required" fixed="http://www.w3.org/2007/rif#iri"/> </xs:complexType> </xs:schema>
<?xml version="1.0" encoding="UTF-8"?> <xs:schema xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns="http://www.w3.org/2007/rif#" targetNamespace="http://www.w3.org/2007/rif#" elementFormDefault="qualified" version="Id: FLDSkyline.xsd, v. 0.96, 2008-07-16, hboley/dhirtle"> <xs:annotation> <xs:documentation> This is the Skyline schema module of FLD. It is split off from the Baseline schema for modularity. The Skyline XML schema is based on the following EBNF (which adds Group and Document, and brings 'Implies' into FORMULA): Document ::= IRIMETA? 'Document' '(' Dialect? Base? Prefix* Import* Group? ')' Dialect ::= 'Dialect' '(' Name ')' Base ::= 'Base' '(' IRI ')' Prefix ::= 'Prefix' '(' Name IRI ')' Import ::= IRIMETA? 'Import' '(' IRICONST PROFILE? ')' Group ::= IRIMETA? 'Group' '(' (FORMULA | Group)* ')' Implies ::= IRIMETA? FORMULA ':-' FORMULA FORMULA ::= IRIMETA? 'And' '(' FORMULA* ')' | IRIMETA? 'Or' '(' FORMULA* ')' | Implies | IRIMETA? 'Exists' Var* '(' FORMULA ')' | IRIMETA? 'Forall' Var* '(' FORMULA ')' | IRIMETA? 'Neg' FORMULA | IRIMETA? 'Naf' FORMULA | FORM PROFILE ::= TERM Note that this is an extension of the syntax for the Baseline schema (FLDBaseline.xsd). </xs:documentation> </xs:annotation> <!-- The Skyline schema includes the Baseline schema from the same directory --> <xs:include schemaLocation="FLDBaseline.xsd"/> <!-- The Skyline schema extends, with Implies, the FORMULA group of the Baseline schema --> <xs:redefine schemaLocation="FLDBaseline.xsd"> <!-- FORMULA ::= IRIMETA? 'And' '(' FORMULA* ')' | IRIMETA? 'Or' '(' FORMULA* ')' | Implies | IRIMETA? 'Exists' Var* '(' FORMULA ')' | IRIMETA? 'Forall' Var* '(' FORMULA ')' | IRIMETA? 'Neg' FORMULA | IRIMETA? 'Naf' FORMULA | FORM --> <xs:group name="FORMULA"> <xs:choice> <xs:group ref="FORMULA"/> <xs:element ref="Implies"/> </xs:choice> </xs:group> </xs:redefine> <xs:element name="Document"> <!-- Document ::= IRIMETA? 'Document' '(' Dialect? Base? Prefix* Import* Group? ')' --> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="directive" minOccurs="0" maxOccurs="unbounded"/> <xs:element ref="payload" minOccurs="0" maxOccurs="1"/> </xs:sequence> <xs:attribute name="dialect" type="xs:string"/> </xs:complexType> </xs:element> <xs:element name="directive"> <!-- Base and Prefix represented directly in XML --> <xs:complexType> <xs:sequence> <xs:element ref="Import"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="payload"> <xs:complexType> <xs:sequence> <xs:element ref="Group"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Import"> <!-- Import ::= IRIMETA? 'Import' '(' IRICONST PROFILE? ')' --> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="location"/> <xs:element ref="profile" minOccurs="0" maxOccurs="1"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="location"> <xs:complexType> <xs:sequence> <xs:element name="Const" type="IRICONST.type"/> <!-- type="&rif;iri" --> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="profile"> <xs:complexType> <xs:sequence> <xs:group ref="TERM"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="Group"> <!-- Group ::= IRIMETA? 'Group' '(' (FORMULA | Group)* ')' --> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="sentence" minOccurs="0" maxOccurs="unbounded"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="sentence"> <xs:complexType> <xs:choice> <xs:group ref="FORMULA"/> <xs:element ref="Group"/> </xs:choice> </xs:complexType> </xs:element> <xs:element name="Implies"> <!-- Implies ::= IRIMETA? FORMULA ':-' FORMULA --> <xs:complexType> <xs:sequence> <xs:group ref="IRIMETA" minOccurs="0" maxOccurs="1"/> <xs:element ref="if"/> <xs:element ref="then"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="if"> <xs:complexType> <xs:sequence> <xs:group ref="FORMULA"/> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="then"> <xs:complexType> <xs:sequence> <xs:group ref="FORMULA"/> </xs:sequence> </xs:complexType> </xs:element> </xs:schema>