This is an archive of an inactive wiki and cannot be modified.

### Syntax of a RIF Dialect as a Specialization of RIF-FLD

The syntax for a RIF dialect can be obtained from the general syntactic framework of RIF by specializing the following parameters (which are defined in this document):

• The alphabet of RIF-FLD can be restricted.
• An assignment of signatures to each constant symbol.

• Signatures determine which terms in the dialect are well-formed and which are not. The exact way this assignment is defined depends on the dialect. The assignment can be explicit or implicit (for instance, derived from the context in which each symbol is used).
• The choice of the types of terms supported by the dialect.

• The RIF logic framework introduces the following types of terms:
• constant
• variable
• positional
• with named arguments
• equality
• frame
• class membership
• subclass
A dialect might support all of them or a subset.
• The choice of symbol spaces supported by the dialect.

• Symbol spaces determine the "shapes" of the symbols that are allowed by the syntax of the dialect.
• The choice of the formulas supported by the dialect.

• RIF-FLD allows to build formulas of the following kind:
• Atomic
• Conjunction
• Disjunction
• Classical negation
• Default negation
• Rule
• Quantification: universal and existential
A dialect might support all of these formulas or it might impose various restrictions. For instance, the formulas in the conclusion and the premises of rules might be restricted, certain quantifications might be prohibited, classical or default negation (or both) might not be allowed, etc.

### Alphabet

The alphabet of RIF-FLD consists of a countably infinite set of constant symbols Const, a countably infinite set of variable symbols Var (disjoint from Const), a countably infinite set of argument names ArgNames (disjoint from both Const and Var), connective symbols And and Or, quantifiers Exists and Forall, the symbols =, #, ##, :-, ->, Naf, Neg, and auxiliary symbols, such as "(" and ")". The set of connective symbols, quantifiers, =, etc., is disjoint from Const and Var. Variables are written as Unicode strings preceded with the symbol "?". The syntax for constant symbols is given in Section Symbol Spaces.

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

### Terms

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 terms. Formally, terms are defined as follows:

• Constants and variables. If tConst or tVar then t is a simple term.

• Positional terms. If t and t1, ..., tn are terms then t(t1 ... tn) is a positional term. 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.

• Terms with named arguments. A term with named arguments is of the form t(s1->v1 ... sn->vn), where t, v1 , ..., vn are terms (positional, with named arguments, frame, etc.), and s1, ..., sn are (not necessarily distinct) symbols from the set ArgNames. The term t here represents a predicate or a function; s1, ..., sn represent argument names; and v1 , ..., vn 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 both positional and also is considered to have named arguments.

• Equality terms. An equality term has the form t = s, where t and s are terms.

• Classification terms. There are two kinds of classification terms: class membership terms (or just membership terms) and subclass terms.

• t#s is a membership term if t and s are arbitrary terms.

• t##s is a subclass term if t and s are arbitrary terms.

• Frame terms. t[p1->v1 ... pn->vn] is a frame term (or simply a frame) if t, p1, ..., pn, v1, ..., vn, n ≥ 0, are arbitrary terms. As in the case of the terms with named arguments, the order of the properties pi->vi in a frame is immaterial.

Classification and frame terms are used to describe objects in object-based logics like F-logic [KLW95].

The above definition is very general. It makes 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], 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.

The mechanism that allows "carving out" of such subsets is called a signature and works as follows. The RIF-FLD language associates a signature with each symbol (both constant and variable symbols) and uses signatures to define what is called 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.

### Signatures

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. Much of this development 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 the RIF formulas. Instead they are part of a separate language for signatures, which is akin to grammar rules in that it determines which sequences of tokens are in the language and which are not. In some dialects (for example RIF-BLD), signatures are derived from the context and no separate language for signatures is used. Other dialects may choose to specify signatures explicitly. In that case, they will need to define a concrete language for specifying signatures.

Let SigNames be a non-empty, partially-ordered finite or countably infinite set of symbols, disjoint from Const, Var, and ArgNames, called signature names. We require that this set includes at least the following signature names:

• atomic -- used to represents the syntactic context where atomic formulas are allowed to appear.

• = -- used for representing contexts where equality terms can appear.

• # -- a signature name reserved for membership terms.

• ## -- a signature reserved for subclass terms.

• -> -- a signature reserved for frame terms.

Dialects are expected to introduce additional signature names. For instance, RIF-BLD introduces one other signature name, term. 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.

A signature is a statement of the form η{e1, ..., en, ...} where η ∈ SigNames is the name of the signature and {e1, ..., en, ...} 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:

• If κ, κ1, ..., κnSigNames, n≥0, are signature names then (κ1 ... κn) ⇒ κ is a positional arrow expression. For instance, () ⇒ term and (term) ⇒ term are arrow expressions, if term is a signature name.

• If κ, κ1, ..., κnSigNames, n≥0, are signature names and p1, ..., pnArgNames are argument names then (p1->κ1 ... pn->κn) => κ is an arrow expression with named arguments. 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.

A set S of signatures is coherent iff

• S contains the special signature atomic{ }, which represents the context of atomic formulas.

• S contains the signature ={e1, ..., en, ...} for the equality symbol. All arrow expressions ei here have the form (κ κ) ⇒ γ (both arguments in an equation must have the same signature) and at least one of these expressions must have the form (κ κ) ⇒ atomic (i.e., some equations should be allowed as atomic formulas). Dialects may further specialize this signature.

• S contains the signature #{e1, ..., en...} where all arrow expressions ei are binary (have two arguments) and at least one has the form (κ γ) ⇒ atomic. Dialects may further specialize this signature.

• S contains the signature ##{e1, ..., en...} where all arrow expressions ei have the form (κ κ) ⇒ γ (both arguments must have the same signature) and at least one of these arrow expressions has the form (κ κ) ⇒ atomic. Dialects may further specialize this signature.

• S contains the signature ->{e1, ..., en...}, where all arrow expressions ei are ternary (have three arguments) and at least one of them is of the form (κ1 κ2 κ3 ) ⇒ atomic. Dialects may further specialize this signature.

• S has at most one signature for any given signature name.

• Whenever S contains a pair of signatures, ηS and κR, such that η<κ then RS. Here ηS denotes a signature with the name η and the associated set of arrow expression S; similarly κR is a signature named κ with the set of expressions R. The requirement that RS ensures that symbols that have signature η can be used wherever the symbols with signature κ are allowed.

### Well-formed Terms and Formulas

Signatures are used to control the context in which various symbols are allowed to occur, as explained next.

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. 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.

Next we define well-formed terms and their signatures. Like the constant symbols, well-formed terms can have more than one signature.

• A constant or variable symbol with signature η is a well-formed term with signature η.

• A positional term t(t1 ... tn), 0≤n, is well-formed and has a signature σ iff

• t is a well-formed term that has a signature that contains an arrow expression of the form (σ1 ... σn) ⇒ σ; and

• Each ti is a well-formed term whose signature is γi, such that γi, ≤ σi.

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.

• A term with named arguments t(p1->t1 ... pn->tn), 0≤n, is well-formed and has a signature σ iff

• t is a well-formed term that has a signature that contains an arrow expression with named arguments of the form (p1->σ1 ... pn->σn) ⇒ σ; and

• Each ti is a well-formed term whose signature is γi, such that γi, ≤ σi.

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.

• An equality term of the form t1=t2 is well-formed and has a signature κ iff

• The signature = has an arrow expression (σ σ) ⇒ κ

• ti and t2 are well-formed terms with signatures γ1 and γ2, respectively, such that γi ≤ σ, i=1,2.

• A membership term of the form t1#t2 is well-formed and has a signature κ iff

• The signature # has an arrow expression (σ1 σ2) ⇒ κ

• ti and t2 are well-formed terms with signatures γ1 and γ2, respectively, such that γi ≤ σi, i=1,2.

• A subclass term of the form t1##t2 is well-formed and has a signature κ iff

• The signature ## has an arrow expression (σ σ) ⇒ κ

• ti and t2 are well-formed terms with signatures γ1 and γ2, respectively, such that γi ≤ σ, i=1,2.

• A frame term of the form t[s1->v1 ... sn->vn] is well-formed and has a signature κ iff

• The signature -> has arrow expressions (σ σ11 σ12) ⇒ κ, ..., (σ σn1 σn2) ⇒ κ (these n expressions need not be distinct).

• t, sj, and vj are well-formed terms with signatures γ, γj1, and γj2, respectively, such that γ ≤ σ and γji ≤ σji, where j=1,...,n and i=1,2.

Note that, according to the above definition, f() and f are distinct terms. We define atomic formulas as follows:

• A term is a well-formed atomic formula iff it is a well-formed term one of whose signatures is η, such that η = atomic or η < atomic.

Note that equality, membership, subclass, and frame terms are always atomic formulas, since atomic is always one of their signatures.

More general formulas are constructed out of atomic formulas with the help of logical connectives. A formula is a statement that can have one of the following forms:

• Atomic: If φ is a well-formed atomic formula then it is also a well-formed formula.

• Conjunction: If φ1, ..., φn, n ≥ 0, are well-formed formulas then so is And(φ1 ... φn). As a special case, And() is allowed and is treated as a tautology, i.e., a formula that is always true.

• Disjunction: If φ1, ..., φn, n ≥ 0, are well-formed formulas then so is Or(φ1 ... φn). When n=0, we get Or() as a special case; it is treated as a formula that is always false.

• Classical negation: If φ is a well-formed formula then Neg φ is a well-formed formula.

• Default negation: If φ is a well-formed formula then Naf φ is a well-formed formula.

• Rule: If φ and ψ are well-formed formulas then φ :- ψ is a well-formed formula.

• Quantification: If φ is a well-formed formula and ?V1, ..., ?Vn are variables then Exists ?V1 ... ?Vn(φ) and Forall ?V1 ... ?Vn(φ) are well-formed formulas.

Example 1 (The use of signatures)

We illustrate the above definitions with the following examples. In addition to atomic, let there be another signature, term{ }, which is also used in RIF-BLD.

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 advanced example of signatures is 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 h2{(term term)⇒atomic}; and closure has signature hh1{(h2)⇒p2}, where p2 is the name of the signature p2{(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.

### Symbol Spaces

Throughout this document, the xsd: prefix stands for the XML Schema namespace URI http://www.w3.org/2001/XMLSchema#, the rdf: prefix stands for http://www.w3.org/1999/02/22-rdf-syntax-ns#, and rif: stands for the URI of the RIF namespace, http://www.w3.org/2007/rif#. Syntax such as xsd:string should be understood as a compact URI [CURIE] -- a macro that expands to a concatenation of the character sequence denoted by the prefix xsd and the string string.

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 data types, data types defined in other W3C specifications, such as rdf:XMLLiteral, and to distinguish other sets of constants. Constant symbols that belong to the various symbol spaces have special presentation syntax and semantics.

Formally, 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 an identifier.

• The lexical space of a symbol space is a non-empty set of Unicode character strings.

• The identifier of a symbol space is an absolute IRI.

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 xsd:string" instead of "symbol space identified by xsd:string").

To refer to a constant in a particular RIF symbol space, we use the following presentation syntax:

```     LITERAL^^SYMSPACE
```

where LITERAL is a Unicode string, called the lexical part of the symbol, and SYMSPACE is an identifier of the symbol space in the form of an absolute IRI string. LITERAL must be an element in the lexical space of the symbol space. For instance, 1.2^^xsd:decimal and 1^^xsd:decimal are legal symbols because 1.2 and 1 are members of the lexical space of the XML Schema data type xsd:decimal. On the other hand, a+2^^xsd:decimal is not a legal symbol, since a+2 is not part of the lexical space of xsd:decimal.

The set of all symbol spaces that partition Const is considered to be part of the logic language used by RIF rule sets.

RIF supports the following symbol spaces. Rule sets that are exchanged through RIF can use additional symbol spaces as explained below.

• xsd:string (http://www.w3.org/2001/XMLSchema#string)

and all the symbol spaces that corresponds to the subtypes of xsd:string as specified in [XML-SCHEMA2].

• xsd:decimal (http://www.w3.org/2001/XMLSchema#decimal)

and all the symbol spaces that corresponds to the subtypes of xsd:decimal as specified in [XML-SCHEMA2].

• xsd:time (http://www.w3.org/2001/XMLSchema#time).

• xsd:date http://www.w3.org/2001/XMLSchema#dateTime).

• xsd:dateTime http://www.w3.org/2001/XMLSchema#dateTime).

The lexical spaces of the above symbol spaces are defined in the document [XML-SCHEMA2].

• rdf:XMLLiteral (http://www.w3.org/1999/02/22-rdf-syntax-ns#XMLLiteral).

This symbol space represents XML content. The lexical space of rdf:XMLLiteral is defined in the document [RDF-CONCEPTS].

• rif:text (for text strings with language tags attached).

This symbol space represents text strings with a language tag attached. The lexical space of rif:text is the set of all Unicode strings of the form ...@LANG, i.e., strings that end with @LANG where LANG is a language identifier as defined in [RFC-3066].

• rif:iri (for internationalized resource identifiers or IRIs).

Constant symbols that belong to this symbol space are intended to be used in a way similar to RDF resources [RDF-SCHEMA]. The lexical space consists of all absolute IRIs as specified in [RFC-3987]; it is unrelated to the XML primitive type anyURI. A rif:iri constant is supposed to be interpreted as a reference to one and the same object regardless of the context in which that constant occurs.

• rif:local (for constant symbols that are not visible outside of a particular set of RIF formulas).

Symbols in this symbol space are used locally in their respective rule sets. This means that occurrences of the same rif:local-constant in different rule sets are viewed as unrelated distinct constants, but occurrences of the same constant in the same rule set must refer to the same object. The lexical space of rif:local is the same as the lexical space of xsd:string.

Notes on RIF-compliant support for symbol spaces.

• A RIF-compliant inference engine must support the following symbol spaces: xsd:string, xsd:decimal, xsd:time, xsd:date, xsd:dateTime, rdf:XMLLiteral, rif:text, rif:iri, rif:local. Such an engine can support additional symbol spaces.

• A RIF-producing system includes a RIF compliant inference engine and a transformation from the language of that engine into valid RIF XML format. Such an engine must support all the symbol spaces that are mentioned in the documents produced by the aforesaid transformation. In particular, this transformation must not produce invalid constant symbols, i.e., symbols whose lexical part is not an element of the lexical space of the symbol's symbol space.

• A RIF-consuming system includes a RIF-compliant inference engine and a transformation from RIF XML to the language of the engine. A consumer engine is not required to support all symbol spaces that are subspaces of the symbol spaces supported by the producer engine. For instance, a RIF-producer system might support xsd:short, a subspace of xsd:decimal, but RIF consumers do not need to support xsd:short. The consumer is allowed to replace the constants in an unsupported symbol space with the corresponding constant symbols in a supported superspace. For example, "123"^^xsd:short can be replaced with "123"^^xsd:decimal and "abc123"^^xsd:IDREF with "abc123"^^xsd:string. Such substitutions are permitted because they do not affect the inferences that can be made from RIF documents (see Section RIF Semantic Framework).