W3C


RIF Framework for Logic Dialects

W3C Editor's Draft 22 February07 April 2008

This version:
http://www.w3.org/2005/rules/wg/draft/ED-rif-fld-20080222/http://www.w3.org/2005/rules/wg/draft/ED-rif-fld-20080407/
Latest editor's draft:
http://www.w3.org/2005/rules/wg/draft/rif-fld/
Previous version:
http://www.w3.org/2005/rules/wg/draft/ED-rif-fld-20080219/http://www.w3.org/2005/rules/wg/draft/ED-rif-fld-20080222/ (color-coded diff)
Editors:
Harold Boley, National Research Council Canada
Michael Kifer, State University of New York at Stony Brook

Copyright © 2008 W3C® (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.

Abstract

This document, developed by

the Rule Interchange Format (RIF) Working Group , specifies the general framework for logic-based RIF dialects (RIF-FLD). The framework describes the syntax and semantics of logic-based RIF dialects through a number of generic concepts such as signatures, symbol spaces, semantic structures, and so on. The actual dialects are expected to specialize this framework to produce their concrete syntaxes and semantics.Status of this Document

May Be Superseded

This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

This document is being published as one of a set of 52 documents:

  1. RIF Basic Logic Dialect
  2. RIF Framework for Logic Dialects (this document)

RIF Data Types and Built-Ins RIF Use Cases and Requirements RIF RDF and OWL CompatibilityPlease Comment By 19 February 20082008-04-08

The Rule Interchange Format (RIF) Working Group seeks public feedback on these Working Drafts. Please send your comments to public-rif-comments@w3.org (public archive). If possible, please offer specific changes to the text that would address your concern. You may also wish to check the Wiki Version of this document for internal-review comments and changes being drafted which may address your concerns.

No Endorsement

Publication as a Working Draft does not imply endorsement by the W3C Membership. This is a draft document and may be updated, replaced or obsoleted by other documents at any time. It is inappropriate to cite this document as other than work in progress.

Patents

This document was produced by a group operating under the 5 February 2004 W3C Patent Policy. W3C maintains a public list of any patent disclosures made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent. An individual who has actual knowledge of a patent which the individual believes contains Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.

Contents

1 Overview of RIF-FLD

The RIF Framework for Logic-based Dialects (RIF-FLD) is a formalism for specifying all logic-based dialects of RIF, including 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. RIF-BLD gives precise definitions to these mechanisms, but leaves some concrete details out. Eachomits certain details. Every logic-based RIF dialect that is based on RIF-FLDis expectedrequired to specialize these general mechanisms (even leave out some elements of RIF-FLD) to produce its concrete syntax and model-theoretic semantics.

The framework described in this document is very general and captures most of the popular logic-based 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 document is mostly intended for the designers of future RIF dialects. The reader who is interested in using a particular dialect, such as RIF-BLDAll logic-based RIF dialects are required to be derived from RIF-FLD by specialization, oras explained in implementing suchSections Syntax of a RIF Dialect can go directly to the descriptionas a Specialization of theRIF-FLD and Semantics of a RIF Dialect in question.as a Specialization of RIF-FLD has. In addition to specialization, to lower the following main components: Syntacticbarrier of entry for their intended audiences, some dialects may choose to specify their syntax and semantics in a direct, but equivalent, way, which does not require familiarity with RIF-FLD. For instance, the RIF Basic Logic Dialect is specified both by specialization from RIF-FLD and also directly, without references to the framework. Thus, the reader who is only interested in RIF-BLD can proceed directly to that document.

RIF-FLD has the following main components:

The framework described in this document is very general, and it captures most of the popular logic-based languages found in Databases, Logic Programming, and on the Semantic Web. However, it is expected that the needs of some newly developed dialects may stimulate further evolution of RIF-FLD.Syntactic framework. The syntactic framework defines three main classessix types of RIF terms:

Semantic framework. This framework defines the notion of a semantic structure or interpretation (both terms are used in the literature [Enderton01, Mendelson97], but here we will mostly use the first). Semantic structures are used to interpret RIF formulas and to define logical entailment. As with the syntax, this framework includes a number of mechanisms that RIF logic-based dialects can specialize to suit their needs. These mechanisms include:

Roughly speaking, a set of formulas, G, logically entails another formula, g, if for every semantic structure I in some set S, if I makes G true, then I also makes g true. Almost all known 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 Description Logic, assume that S is the set of all semantic structures. In contrast, logic programming languages, which use default negation, assume that S contains only the so-called "minimal""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 serializing the various parts of the presentation syntax of RIF-FLD.


2 Syntactic Framework

The next subsection explains the overall idea of deriving the syntax of a RIF dialect from the RIF framework. The actual syntax of the RIF framework is given in subsequent subsections.

2.1 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):

  1. The alphabet of RIF-FLD can be restricted.
  2. An assignment of signatures to each constant and variable symbol.

  3. Signatures determine which terms in the dialect are well-formed and which are not.

    The exact way this assignment is definedsignatures are assigned depends on the dialect. TheAn assignment can be explicit or implicit (for instance, derived from the context in which each symbol is used).

  4. The choice of the types of terms supported by the dialect.

  5. The RIF logic framework introduces the following types of terms:

    • constant
    • variable
    • positional
    • with named arguments
    • equality
    • frame
    • class membership
    • subclass
    • external

    A dialect might support all of themthese terms or just a subset.

  6. The choice of symbol spaces supported by the dialect.

  7. Symbol spaces determine the "shapes"syntax of the symbols that are allowed by the syntax ofin the dialect.

  8. The choice of the formulas supported by the dialect.

  9. RIF-FLD allows to buildformulas 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 allowed in the conclusion and theand/or premises of rulesimplications might be restricted, certain quantificationstypes of quantification might be prohibited, classical or default negation (or both) might not be allowed, etc.

2.2 Alphabet

Definition (Alphabet). The alphabet 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 with the symbol "?"."?". The syntax for constant symbols is givenargument 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).

The symbol Group is used to organize RIF-BLD rules into collections and annotate them with metadata.   ☐

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


2.3 TermsSymbol Spaces

Throughout this document, the most basic construct of a logic language is a term . RIF-FLD supports several kinds of terms: constants, variables,xsd: prefix stands for the regular positional terms, plus terms with named argumentsXML Schema namespace URI http://www.w3.org/2001/XMLSchema#, the rdf: prefix stands for http://www.w3.org/1999/02/22-rdf-syntax-ns#, equality, classification terms,and frames .rif: stands for the word " term " willURI of the RIF namespace, http://www.w3.org/2007/rif#. Syntax such as xsd:string should be used to referunderstood as a compact URI [CURIE] -- a macro that expands to any kinda concatenation of terms. Formally, terms are defined as follows: Constantsthe character sequence denoted by the prefix xsd and variables . If t ∈ Const or t ∈ Var then t is a simple term . Positional termsthe string string. If t and t 1 , ..., t n are terms then t(t 1 ... t n )The compact URI notation is not part of the RIF syntax, but rather just a positional term . Positional termsspace-saving device in RIF-FLD generalizethis document.

The regular notionset of all constant symbols in a termRIF dialect is partitioned into a number of subsets, called symbol spaces, which are used to represent XML Schema data types, data types defined in first-order logic. For instance,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 abovesymbol space to which they belong

Definition allows variables everywhere. Terms with named arguments .(Symbol space). A term with named argumentssymbol space is of the form t(s 1 ->v 1 ... s n ->v n ) , where t , v 1 , ..., v n are terms (positional, witha named arguments, frame, etc.), and s 1 , ..., s n are (not necessarily distinct) symbols fromsubset of the set ArgNamesof all constants, Const. The term t here representssemantic 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, a predicateunique identifier, and, possibly, one or more aliases. More precisely,

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 classification terms: class membership terms (or just membership terms ) and subclass terms . t#s is"symbol space identified by xsd:string").

To refer to a membership term if t and s are arbitrary terms. t##s isconstant in a subclass term if t and s are arbitrary terms. Frame terms . t[p 1 ->v 1 ... p n ->v n ]particular RIF symbol space, we use the following presentation syntax:

     "literal"^^symspace

where literal is a frame term (or simply a frame ) if t , p 1 , ..., p n , v 1 , ..., v n , n ≥ 0, are arbitrary terms. As incalled the caselexical part of the terms with named arguments, the ordersymbol, and symspace is an identifier or an alias of the properties p i ->v i in a framesymbol space. Here literal is immaterial. Classification and frame terms are used to describe objectsa sequence of Unicode characters that must be an element 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.lexical space of the samesymbol can occur in multiple contexts at the same time.space symspace. For instance, if p , a ,"1.2"^^xsd:decimal and b"1"^^xsd:decimal are legal symbols then p(p(a) p(a p c)) is a term. Even variablesbecause 1.2 and general terms1 are allowed to occur inmembers of the positionlexical space of predicates and function symbols, so p(a)(?v(a c) p)the XML Schema data type xsd:decimal. On the other hand, "a+2"^^xsd:decimal is alsonot 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) subsetslegal symbol, since a+2 is not part of RIF-FLD terms, but most languages support much smaller subsets. RIF dialects are expected to carve outthe appropriate subsetslexical space of RIF-FLD terms, andxsd:decimal.

The general formset of all symbol spaces that partition Const is considered to be part of the RIFlogic framework allows a considerable degreelanguage of freedom.RIF-FLD.

The mechanism that allows "carving out"following list of such subsets is called a signature and works as follows. The RIF-FLD language associates a signature with eachsupported symbol (both constant and variable symbols) and uses signaturesspaces will move to define what is called well-formed termsanother document, Data Types and Built-Ins. EachAny existing discrepancies will be fixed at that time.


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

and all the symbol spaces that correspond to select appropriate signatures forthe symbolssubtypes of xsd:string as specified in its alphabet,[XML-SCHEMA2].

and onlyall the termssymbol spaces that are well-formed accordingcorresponds to the selected signatures are allowedsubtypes of xsd:decimal as specified in that particular dialect. 2.4 Signatures[XML-SCHEMA2].

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

This section we introducesymbol space represents XML content. The conceptlexical space of rdf:XMLLiteral is defined in the document [RDF-CONCEPTS].

This symbol space represents text strings with a signaturelanguage tag attached. The lexical space of rif:text is the set of all Unicode strings of the form ...@LANG, whichi.e., strings that end with @LANG where LANG is a key mechanism that allows RIF-FLD to control the contextlanguage identifier as defined in which the various[RFC-3066].

Constant symbols are allowedthat belong to occur. Much ofthis development is inspired bysymbol space are intended to be used in a way similar to RDF resources [ CK95RDF-SCHEMA]. The lexical space consists of all absolute IRIs as specified in [RFC-3987]; it shouldis unrelated to the XML primitive type anyURI. A rif:iri constant must be keptinterpreted as a reference to one and the same object regardless of the context in mindwhich that constant occurs.

Symbols in this symbol space are part of a separate language for signatures, which is akinlocal to grammar rulesthe RIF documents in that it determineswhich sequences of tokens are inthey occur. This means that occurrences of the language and which are not.same rif:local constant in some dialects (for example RIF-BLD ), signaturesdifferent documents are derived fromviewed as unrelated distinct constants, but occurrences of the context and no separate language for signatures is used. Other dialects may choose to specify signatures explicitly.same rif:local constant in that case, they will needthe same document must refer to definethe same object. The lexical space of rif:local is the same as the lexical space of xsd:string.



2.4 Terms

The most basic construct of a concretelogic language for specifying signatures. Let SigNames beis a non-empty, partially-ordered finite or countably infinite setterm. RIF-FLD supports several kinds of symbols, disjoint from Const , Varterms: constants, variables, the regular positional terms, plus terms with named arguments, equality, classification terms, and ArgNames , called signature namesframes. We require that this set includes at leastThe following signature names: atomic --word "term" will be used to represents the syntactic context where atomic formulas are allowedrefer to appear. = -- used for representing contexts where equality terms can appear. # -- a signature name reserved for membership terms. ## -- a signature reserved for subclass terms. -> --any kind of term.

Definition (Term). 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 SigNamesis dialect-specific; it is used in the definitiona statement of one of well-formed terms below. We use the symbol < to representthe 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 ≤ βfollowing forms:

  1. Constants and variables. If either α = βt ∈ Const or α < β. A signature is a statement of the form η{ e 1 , ..., e n , ...} where ηt ∈ SigNames is the name of the signature and { e 1 , ..., e n , ...}Var then t is a countable set of arrow expressionssimple term.
  2. Positional terms. 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, κt and t1, ..., κtn ∈ SigNames , n≥0,are signature namesterms then (κt(t1 ... κtn) ⇒ κis a positional arrow expressionterm.

  3. Positional terms in RIF-FLD generalize the regular notion of a term used in first-order logic. For instance, () ⇒ term and ( term ) ⇒ term are arrow expressions, ifthe above definition allows variables everywhere.

  4. Terms with named arguments. A term with named arguments is a signature name. If κ, κof the form t(s1->v1 ... sn->vn), where t, v1, ..., κvn ∈ SigNames , n≥0,are signature namesterms (positional, with named arguments, frame, etc.), and ps1, ..., psn ∈ ArgNamesare argument names then ( p 1 -> κ 1 ... p(not necessarily distinct) symbols from the set ArgNames.

  5. The term t here represents a predicate or a function; s1, ..., sn -> κrepresent argument names; and v1, ..., vn ) => κ is an arrow expressionrepresent argument values. Terms with named arguments . For instance, ( arg1->term arg2->term ) => termare like regular positional terms except that the arguments are named and their order is an arrow signature expressionimmaterial. Note that a term with no arguments, like f(), is both positional and also is considered to have named arguments.

  6. Equality terms. An equality term has the orderform t = s, where t and s are terms.
  7. Classification terms. There are two kinds of the arguments in arrow expressions with named argumentsclassification terms: class membership terms (or just membership terms) and subclass terms.
    • t#s is immaterial, so any permutation of arguments yields the same expression.a setmembership term if t and s of signaturesare arbitrary terms.
    • t##s is coherent iffa subclass term if t and s contains the special signature atomic{ }are arbitrary terms.

    Classification terms are used to describe class hierarchies.

  8. Frame terms. t[p1->v1 ... pn->vn] is a frame term (or simply a frame) if t, which represents the context of atomic formulas. S contains the signature = { ep1, ..., epn, ...} forv1, ..., vn, n ≥ 0, are arbitrary terms.

    Frame terms are used to describe properties of objects. As in the equality symbol. All arrow expressions e i here havecase of the form (κ κ) ⇒ γ (both arguments in an equation must haveterms with named arguments, the same signature) and at least oneorder 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 containsthe signature # { e 1 , ..., e n ...} where all arrow expressions eproperties pi->vi in a frame is immaterial.

  9. Externally defined terms. If t is a term then External(t) is an externally defined term.

  10. Such terms are binary (have two arguments)used for representing builtin functions and at least one has the form (κ γ) ⇒ atomic . Dialects may further specialize this signature. S contains the signature ## { e 1 , ..., e n ...} where all arrow expressions e i havepredicates as well as "procedurally attached" terms or predicates, which might exist in various rule-based systems, but are not specified by RIF.   ☐

The form (κ κ) ⇒ γ (both arguments must haveabove definition is very general. It makes no distinction between constant symbols that represent individuals, predicates, and function symbols. The same signature) andsymbol can occur in multiple contexts at least one of these arrow expressions has the form (κ κ) ⇒ atomic . Dialects may further specialize this signature. S containsthe signature -> { e 1same time. For instance, if p, ..., e n ...}, where all arrow expressions e ia, and b are ternary (have three arguments)symbols then p(p(a) p(a p c)) is a term. Even variables and at least onegeneral terms are allowed to occur in the position of thempredicates and function symbols, so p(a)(?v(a c) p) is of the form (κ 1 κ 2 κ 3 ) ⇒ atomic . Dialects may further specialize this signature. S has at most one signature for any given signature name. Whenever S containsalso a pair of signatures, η Sterm.

Frame, classification, and κ R , such that η<κ then R ⊆ Sother terms can be freely nested, as exemplified by p(?X  q#r[p(1,2)->s](d->e f->g)). Here η S denotes a signature with the name ηSome 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 associated setappropriate subsets of arrow expression S ; similarly κ R is a signature named κ withRIF-FLD terms, and the setgeneral form of expressions R .the requirement that R ⊆ S ensures that symbols that have signature ηRIF logic framework allows a considerable degree of freedom.

Dialects can be used whereveralso restrict the symbols with signature κ are allowed. 2.5 Well-formedcontexts in which the various terms and Formulas Signatures are usedcan occur. The mechanism that allows to control the context in which various symbols are allowed to occur, as explained next. Each variable symbolis associated with exactly one signature from a coherent set of signatures.called a constant symbol can have one or more signatures,signature and different symbols can be associated withworks as follows. The same signature. SinceRIF-FLD language associates a signature names uniquely identifywith each symbol (both constant and variable symbols) and uses signatures in coherent signature sets, we will often referto 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 wedefine well-formed terms. Each RIF dialect is expected to select appropriate signatures for the symbols in its alphabet, and their signatures. Likeonly the constant symbols,terms that are well-formed according to the selected signatures are allowed in that particular dialect.


2.5 Schemas for Externally Defined Terms

can have more than one signature. A constant or variable symbol with signature ηPart of the material in this section will be duplicated in the document Data Types and Built-Ins. This is in order to enable direct specification of RIF dialects, which bypass the references to FLD.

This section introduces the notion of external schemas, which serve as templates for externally defined terms. These schemas determine which externally defined functions or predicates, are acceptable as terms in a well-formed term with signature η. A positionalRIF dialect. Externally defined terms include RIF builtins, which are specified in the document Data Types and Builtins. The notion of an externally defined term t(t 1 ... t n ) , 0≤n,in RIF is well-formed and has a signature σ iff tvery general. It is not necessarily a well-formed term that hasfunction or a signature that containspredicate -- it can be any term, including frames, classification terms, and so on.

Definition (Schema for external term). An arrow expressionexternal schema is a statement of the form (σ(?X1 ... σ... ?Xn ) ⇒ σ; and Each t i; τ) where

The names of the variables in an external schema are immaterial, but their order is. For instance, (?X ?Y;  ?X[foo->?Y]) and has(?V ?W;  ?V[foo->?W]) are considered to be the same schema, but (?X ?Y;  ?X[foo->?Y]) and (?Y ?X;  ?X[foo->?Y]) are viewed as different schemas.

A signature σ iffterm t is a well-formed term that has a signature that containsan arrow expression with named argumentsinstance of an external schema (?X1 ... ?Xn; τ) iff t can be obtained from τ by a simultaneous substitution ?X1/s1 ... ?Xn/sn or the form ( pvariables ?X1 -> σ... ?Xn with terms s1 ... p n -> σsn ) ⇒ σ; and Each t i is a well-formed term whose signature is γ i , such that γ i, ≤ σrespectively. Some of the terms si 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). As a special case, when n=0 we obtain   ☐

Observe that t( ) isa well-formed term with signature σ, if t 's signature contains the arrow expression () ⇒ σ.variable cannot be an equality terminstance of an external schema, since τ in the form t 1 =t 2 is well-formed and hasabove definition cannot be a signature κ iff The signature = has an arrow expression (σ σ) ⇒ κ t i and t 2 are well-formed terms with signatures γ 1 and γ 2 , respectively, suchvariable. It will be seen later that this implies that γ i ≤ σ, i=1,2 .a membershipterm of the form t 1 #t 2External(?X) is not well-formed and hasin RIF.


Definition (Coherent set of external schemas). A signature κ iff The signature # has an arrow expression (σ 1 σ 2 ) ⇒ κset of external schemas is coherent if there can be no term, t i, that is an instance of two distinct schemas.

Note that the coherence condition is easy to verify syntactically and t 2 are well-formed terms with signatures γ 1that it implies that schemas like (?X ?Y;  ?X[foo->?Y]) and γ 2(?Y ?X;  ?X[foo->?Y]), respectively, suchwhich differ only in the order of their variables, cannot be in the same coherent set.    ☐

It important to understand that γ i ≤ σ i , i=1,2 . A subclass termexternal schemas are not part of the form t 1 ##t 2 islogic language in RIF, since they do not appear anywhere in the RIF formulas. 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 forl External(t) are well-formed and has.


2.6 Signatures

In this section we introduce the concept of a signature κ iff, which is a key mechanism that allows RIF-FLD to control the signature ## has an arrow expression (σ σ) ⇒ κ t i and t 2context in which the various symbols are well-formed termsallowed to occur. For instance, a symbol f with signatures γ 1 and γ 2signature {(term term) => term, (term) => term} can occur in terms like f(a b), respectively, such that γ i ≤ σ, i=1,2 .f(f(a b) a), f(f(a)), etc., if a frameand b have signature term of. But f is not allowed to appear in the form t[s 1 ->v 1 ... s n ->v n ]context f(a b a) because there is well-formed and has a signature κ iffno =>-expression in the signature -> has arrow expressions (σ σ 11 σ 12 ) ⇒ κ, ..., (σ σ n1 σ n2 ) ⇒ κ (these n expressions need not be distinct). t , s j , and v j are well-formed terms with signatures γ, γ j1 , and γ j2 , respectively, such that γ ≤ σ and γ ji ≤ σ ji , where j=1,...,n and i=1,2 . Note that, accordingof f to support such a context.

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 oneexample provides intuition behind the use of whosesignatures in RIF-FLD. Much of the development, below, is η, such that η = atomic or η < atomic . Noteinspired by [CK95]. It should be kept in mind that equality, membership, subclass, and frame termssignatures are always atomic formulas, since atomic is always onenot part of their signatures. More general formulasthe logic language in RIF, since they do not appear anywhere in the RIF formulas. Instead they are constructed outpart of atomic formulas withthe helpgrammar: they are used to determine which sequences of logical connectives. A formula is a statement that can have onetokens are in the language and which are not. The actual way by which signatures are assigned to the symbols of the following forms: Atomic : If φlanguage may vary from dialect to dialect. In some dialects (for example RIF-BLD), this assignment is a well-formed atomic formula then itderived from the context in which each symbol occurs and no separate language for signatures is also aused. 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 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 formula. Conjunction :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 φeither α = β or α < β.

Definition (Signature). A signature is a statement of the form η{e1, ..., φen, ...} where η ∈ SigNames is the name of the signature and {e1, ..., en ≥ 0,, ...} 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:

A set S of signatures is coherent iff

  1. S contains the special signature atomic{ }, which represents the context of atomic formulas.
  2. S contains the signature ={e1, ..., en, ...} for the equality symbol.

    All arrow expressions ei 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., some equations should be allowed as atomic formulas). Dialects may further specialize this signature.

  3. S contains the signature #{e1, ..., en...}.

    Here 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.

  4. S contains the signature ##{e1, ..., en...}.

    Here all arrow expressions ei 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.

  5. S contains the signature ->{e1, ..., en...}.
    Here all arrow expressions ei are well-formed formulas then soternary (have three arguments) and at least one of them is And( φof the form (κ1 ... φ nκ2 κ3) ⇒ atomic. AsDialects may further specialize this signature.
  6. S has at most one signature for any given signature name.
  7. Whenever S contains a special case, And() is allowedpair of signatures, ηA and κB, such that η<κ then B⊆A.

  8. Here ηA denotes a signature with the name η and the associated set of arrow expressions A; similarly κB is treated asa tautology, i.e.,signature named κ with the set of expressions B. The requirement that B⊆A formulaensures that is always true. Disjunction : If φ 1 , ..., φ n , n ≥ 0,symbols that have signature η can be used wherever the symbols with signature κ are allowed.   ☐


2.7 Language of a RIF Dialect

The language of a RIF dialect is a set of all well-formed formulas then so is Or( φ 1 ... φ n ) . When n=0, we get Or()as a special case; itdefined below. The language is treated asdetermined by the following parameters:

We have already seen how the alphabet and ψthe 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.


2.8 Well-formed Terms and Formulas

then φ :- ψSince 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).

  1. A well-formed formula. Quantification : If φconstant or variable symbol with signature η is a well-formed formula and ?V 1 , ..., ?V n are variables then Exists ?V 1 ... ?V n ( φ ) and Forall ?Vterm with signature η.
  2. A positional term t(t1 ... ?Vtn ( 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{ }0≤n, whichis also used in RIF-BLD. Consider the term p(p(a) p(a b c)) . If pwell-formed and has the (polymorphic)a signature mysig {( term )⇒ term , ( term term )⇒ term , ( term term term )⇒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, bsuch that γi, c each has the signature term{ } then p(p(a) p(a b c))≤ σi.

    As a special case, when n=0 we obtain that t( ) is a well-formed term with signature term{ } .σ, if instead p had thet's 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 nocontains the arrow expression which allows p to take just one argument). For() ⇒ σ.

  3. A more complex example, let r have the signature mysig3 {(term )⇒ atomicwith named arguments t(p1->t1 ... pn->tn), ( atomic term )⇒ term0≤n, ( term term term )⇒ term }. Then r(r(a) r (a b c))is well-formed. The interesting twist herewell-formed and has a signature σ iff
    • t is a well-formed term that r(a) is an atomic formulahas a signature that occurs ascontains an argument to a function symbol. However, this is allowed by thearrow expression ( atomic term )⇒ term , which is partwith named arguments of r 's signature. If r 's signature were mysig4 {( term )⇒ atomic , ( atomicthe form (p1->σ1 ... pn->σn) ⇒ σ; and
    • Each ti is a well-formed term )⇒ atomicwhose signature is γi, ( term term term )⇒ term } instead, then r(r(a) r(a b c)) would be not onlysuch that γi ≤ σi.

    As a well-formed term, but alsospecial case, when n=0 we obtain that t( ) is a well-formed atomic formula.term with signature σ, if t's signature contains the arrow expression () ⇒ σ.

  4. An even more advanced exampleequality term of signaturesthe form t1=t2 is whenwell-formed and has a signature κ iff
    • The right-hand side ofsignature = has an arrow expression is something other than term or atomic . For instance, let John , Mary , NewYork ,(σ σ) ⇒ κ
    • ti and Boston havet2 are well-formed terms with signatures term{ } ; flightγ1 and parent have signature hγ2 {( term, respectively, such that γi ≤ σ, i=1,2.
  5. A membership term )⇒ atomic }; and closure has signature hhof the form t1 {( h 2 )⇒ p 2 }, where p#t2 is the name ofwell-formed and has a signature κ iff
    • The signature p# has an arrow expression (σ1 σ2 {( term term )⇒ atomic }. Then flight(NewYork Boston) , closure(flight)(NewYork Boston) , parent(John Mary) ,) ⇒ κ
    • t1 and closure(parent)(John Mary) would be well-formed formulas. Such formulast2 are allowed in languages like HiLog [ CKW93 ], which support predicate constructors like closure in the above example. 2.6 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 macrowell-formed terms with signatures γ1 and γ2, respectively, such that expands toγi ≤ σi, i=1,2.
  6. A concatenationsubclass term of the character sequence denoted by the prefix xsdform t1##t2 is well-formed and has a signature κ iff
    • The string stringsignature ## has an arrow expression (σ σ) ⇒ κ
    • t1 and t2 are well-formed terms with signatures γ1 and γ2, respectively, such that γi ≤ σ, i=1,2.
  7. The set of all constant symbols inA RIF dialectframe term of the form t[s1->v1 ... sn->vn] is partitioned intowell-formed and has a number of subsets, called symbol spacessignature κ iff
    • The signature -> has arrow expressions (σ σ11 σ12) ⇒ κ, ..., (σ σn1 σn2) ⇒ κ (these n expressions need not be distinct).
    • t, whichsj, and vj are used to represent XML Schema data types, data types defined in other W3C specifications, such as rdf:XMLLiteralwell-formed terms with signatures γ, γj1, and to distinguish other sets of constants. Constant symbolsγj2, respectively, such that belong to the various symbol spaces have special presentation syntaxγ ≤ σ 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 hasγji ≤ σji, where j=1,...,n and i=1,2.
  8. An associated lexical spaceexternally defined term, External(t), is well-formed and has signature κ iff

Note that, like constant in a particular RIF symbol space, we usesymbols, well-formed terms can have more than one signature. Also note that, according to the following presentation syntax: LITERAL^^SYMSPACE where LITERALabove definition, f() and f are distinct terms.


Definition (Well-formed formula). A well-formed term is also a Unicode string, called the lexical partwell-formed atomic formula iff one of the symbol,its signatures is atomic or it is < atomic. Note that equality, membership, subclass, and SYMSPACEframe terms are atomic formulas, since atomic is an identifierone of the symbol space in the formtheir signatures.

More general formulas are constructed out of an absolute IRI string. LITERAL must be an element inatomic formulas with the lexical spacehelp of logical connectives. A formula is a statement that can have one of the symbol space. For instance, 1.2^^xsd:decimal and 1^^xsd:decimalfollowing forms:

  1. Atomic: If φ is a well-formed atomic formula then it is also a well-formed formula.
  2. Conjunction: If φ1, ..., φn, n ≥ 0, are legal symbols because 1.2 andwell-formed formulas then so is And(φ1 are members of the lexical space of the XML Schema data type xsd:decimal... φn).

  3. On the other hand, a+2^^xsd:decimalAs a special case, And() is notallowed and is treated as a legal symbol, since a+2tautology, i.e., a formula that is not part of the lexical space of xsd:decimalalways true.

  4. Disjunction: If φ1, ..., φn, n ≥ 0, are well-formed formulas then so is Or(φ1 ... φn).

  5. The set of all symbol spacesWhen n=0, we get Or() as a special case; it is treated as a contradiction, i.e., a formula that partition Constis considered to be part of the logic language used by RIFalways false.

  6. Classical negation: If φ is a well-formed formula then Neg φ is a well-formed formula.
  7. Default negation: If φ is a well-formed formula then Naf φ is a well-formed formula.
  8. Rule sets. RIF supportsimplication: If φ and ψ are well-formed formulas then φ :- ψ is a well-formed formula.
  9. Quantification: If φ is a well-formed formula and ?V1, ..., ?Vn are variables then the following symbol spaces. Rule sets thatformulas are exchanged through RIF can use additional symbol spaces as explained below. xsd:string ( http://www.w3.org/2001/XMLSchema#string )well-formed:
  10. Group: If φ is a frame term and ρ1, ..., ρn are RIF-FLD formulas or group formulas (they can be mixed) then Group φ (ρ1 ... ρn) and all the symbol spaces that correspondsGroup (ρ1 ... ρn) are group formulas.

    Group formulas are intended to the subtypesrepresent sets of xsd:decimal asformulas annotated with metadata. This metadata is 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 spacesusing an optional frame term φ. Note that some 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 ).ρi's can be group formulas themselves, which means that groups can be nested. This symbol space represents XML content. The lexical spaceallows one to attach metadata to various subsets of rdf:XMLLiteral is definedformulas, which may be inside larger sets of formulas, which in turn may be annotated.   ☐


Example 1 (Signatures, well-formed terms and formulas).

We illustrate the document [ RDF-CONCEPTS ]. rif:text (for text strings with language tags attached). This symbol space represents text stringsabove definitions with a language tag attached. The lexical space of rif:text is the set of all Unicode strings ofthe form ...@LANG , i.e., strings that end with @LANG where LANG is a language identifier as definedfollowing examples. In [ RFC-3066 ]. rif:iri (for internationalized resource identifiers or IRI s). Constant symbols that belong to this symbol space are intendedaddition to atomic, let there be used in a way similaranother signature, term{ }, which is intended here to RDF resources [ RDF-SCHEMA ].represent the lexical space consistscontext of all absolute IRIs as specified in [ RFC-3987 ]; it is unrelatedthe arguments to positional terms or atomic formulas.

Consider the XML primitive type anyURIterm 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 rif:iri constant, b, c each has the signature term{ } then p(p(a) p(a b c)) is supposed toa 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 interpreted asa referencewell-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 andargument).

For a more complex example, let r have the same object regardless ofsignature mysig3{(term)⇒atomic, (atomic term)⇒term, (term term term)⇒term}. Then r(r(a) r(a b c)) is well-formed. The context in whichinteresting twist here is that constant occurs. rif:local (for constant symbolsr(a) is an atomic formula that are not visible outside ofoccurs as an argument to a particular set of RIF formulas). Symbols in this symbol space are used locally in their respective rule sets.function symbol. However, this means that occurrences ofis allowed by the same rif:local -constant in different rule sets are viewed as unrelated distinct constants, but occurrencesarrow 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 same constant in the same rule set must refer to the same object. The lexical spaceright-hand side of rif:localan arrow expression is the same as the lexical space of xsd:stringsomething other than term or atomic. Notes on RIF-compliant supportFor symbol spaces. A RIF-compliant inference engine must support the following symbol spaces: xsd:string , xsd:decimalinstance, let John, xsd:timeMary, xsd:dateNewYork, xsd:dateTimeand Boston have signatures term{ }; flight and parent have signature h2{(term term)⇒atomic}; and closure has signature hh1{(h2)⇒p2}, rdf:XMLLiteralwhere p2 is the name of the signature p2{(term term)⇒atomic}. Then flight(NewYork Boston), rif:textclosure(flight)(NewYork Boston), rif:iriparent(John Mary), rif:local . Such an engine can support additional symbol spaces. A RIF-producing system includes a RIF compliant inference engineand a transformation from the language of that engine into valid RIF XML format.closure(parent)(John Mary) would be well-formed formulas. Such an engine must support all the symbol spaces thatformulas are mentionedallowed in languages like HiLog [CKW93], which support predicate constructors like closure in the documents producedabove example.   ☐

Example 2 (A nested RIF-FLD group annotated with metadata).

We illustrate formulas, groups, and metadata by the aforesaid transformation. In particular,following complete example. For better readability, we use the compact URI notation which assumes that prefixes are macro-expanded into IRIs. As explained earlier, this transformation must not produce invalid constant symbols, i.e., symbols whose lexical partis just a space-saving device and not an elementpart of the RIF syntax.

Compact URI prefixes:
  
  dc     expands into http://dublincore.org/documents/dces/
  ex     expands into http://example.org/ontology# 
  hamlet expands into http://www.shakespeare-literature.com/Hamlet/ 

  Group "hamlet:assertions"^^rif:iri["dc:title"^^rif:iri->"Hamlet"^^xsd:string,
                                     "dc:creator"^^rif:iri->"Shakespeare"^^xsd:string]
  (
      Exists ?X (And(?X # "ex:RottenThing"^^rif:iri
                     "ex:part-of"^^rif:iri(?X "http://www.denmark.dk"^^rif:iri)))
      Forall ?X (Or("hamlet:to-be"^^rif:iri(?X)  Naf "hamlet:to-be"^^rif:iri(?X)))
      Forall ?X (And(Exists ?B (And("ex:has"^^rif:iri(?X ?B) ?B#"ex:business"^^rif:iri))
                     Exists ?D (And("ex:has"^^rif:iri(?X ?D) ?D#"ex:desire"^^rif:iri)))
                   :- ?X#"ex:man"^^rif:iri)
      Group "hamlet:facts"^^rif:iri[ ]
      (
         "hamlet:Yorick"^^rif:iri#"ex:poor"^^rif:iri
         "hamlet:Hamlet"^^rif:iri#"ex:prince"^^rif:iri
      )
  )

Observe that the above set of formulas has a nested subset with its own metadata, "hamlet:facts"^^rif:iri[ ], which contains only a global IRI.   ☐


2.9 EBNF Grammar for the Presentation Syntax of RIF-FLD

Up to now we used Mathematical English to specify the lexical spacesyntax of RIF-FLD. We will now use the symbol's symbol space. A RIF-consuming system includes a RIF-compliant inference engine andfamiliar EBNF notation in order to provide a transformation from RIF XMLsuccinct overview of the syntax. The following points about the EBNF notation have to be kept in mind:


  Group          ::= 'Group' IRIMETA? '(' (FORMULA | Group)* ')'
  IRIMETA        ::= Frame
  FORMULA        ::= 'And' '(' FORMULA* ')' |
                     'Or' '(' FORMULA* ')' |
                     Implies |
                     'Exists' Var+ '(' FORMULA ')' |
                     'Forall' Var+ '(' FORMULA ')' |
                     'Neg' FORMULA |
                     'Naf' FORMULA |
                     ATOMIC |
                     'External' '(' ATOMIC ')'
  Implies        ::= FORMULA ':-' FORMULA
  ATOMIC         ::= 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           ::= Const | Var | Expr | 'External' '(' Expr ')' |
                     Equal | Member | Subclass | Frame
  Expr           ::= UNITERM
  Const          ::= '"' UNICODESTRING '"^^' SYMSPACE
  Name           ::= UNICODESTRING
  Var            ::= '?' UNICODESTRING



3 Semantic Framework

3.1 Semantics of a RIF Dialect as a Specialization of RIF-FLD

The RIF-FLD semantic framework defines the notions of semantic structures and of models of RIF formulas. The semantics of a dialect is derived from these notions by specializing the following parameters.

  1. The effect of the syntax.

  2. The syntax of a dialect may limit the kinds of terms that are supported. For instance, if the dialect does not support frames or terms with named arguments then the parts of the semantic structures whose purpose is to interpret the unsupported types of terms become redundant.

  3. Truth values.

  4. 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.

  5. Data types.

  6. A data type is a symbol space that haswhose symbols have a fixed interpretation in any semantic structure. RIF-FLD defines a set of core data types that each dialect is expected to support, but its semantics does not limit support to just the core types. RIF dialects can introduce additional data types, and each dialect is expected to define the exact set of data types that it supports.

  7. Logical entailment.

  8. 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 so-calledwell-founded or stable models. All of the abovemodels [GRS91, GL88].

These notions are defined in the remainder of this document.

3.2 Truth Values

Definition (Set of truth values). Each RIF dialect is expected to define the set of truth values, denoted by TV. This set must have a partial order, called the truth order, denoted <t. As a special case,In some dialects, <t can be a total order in some dialects.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 the 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.

3.3 Primitive Data Types

Definition (Primitive data type). A primitive data type (or just a data type, for short) is a symbol space that has

Semantic structures are always defined with respect to a particular set of data types, denoted by DTS. In a concrete dialect, DTS always includes the data types supported by that dialect. All RIF dialects are expected to support the following primitive data types:

Their value spaces and the lexical-to-value-space mappings are defined as follows:

The value space and the lexical-to-value-space mapping for rif:text defined here are compatible with RDF's semantics for strings with named tags [RDF-SEMANTICS].


The above list of supported data types will move to the document Data Types and Built-Ins. Any existing discrepancies will be fixed at that time.


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 that belong to the various primitive data types. In contrast,language. Value spaces define the meaning of thosethe constants. The lexical and the value spaces are often not even isomorphic. For instance,example, 1.2^^xsd:decimal and 1.20^^xsd:decimal are two legal -- and distinct -- constants in RIF because 1.2 and 1.20 belong to the lexical space of xsd:decimal. However, these two constants are interpreted by the same element of the value space of the xsd:decimal type. Therefore, 1.2^^xsd:decimal = 1.20^^xsd:decimal is a RIF tautology. Likewise, RIF semantics for data types implies certain inequalities. For instance, abc^^xsd:string ≠ abcd^^xsd:string is a tautology, since the lexical-to-value-space mapping of the xsd:string type maps these two constants into distinct elements in the value space of xsd:string.

3.4 Semantic Structures

The central step in specifying a model-theoretic semantics for a logic-based language is defining the notion of a semantic structure, also known as an interpretation. 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, IC, IV, IF, Iframe, ISF, Isub, Iisa, I=, Iexternal, Itruth >.>. 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 the set of primitive data types used in I.

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

  1. IC maps Const to elements of D.

  2. This mapping interprets constant symbols.

  3. IV maps Var to elements of D.

  4. This mapping interprets variable symbols.

  5. IF maps D to functions D* → D (here D* is a set of all sequences of any finite length over the domain D)

  6. This mapping interprets positional terms.

  7. ISF interprets terms with named arguments. It is a total mapping from ConstD to the set of total functions of the form SetOfFiniteBags(ArgNames × D) → D.

    This is analogous to the interpretation of positional terms with two differences:

  8. Iframe is a total mapping from D to total functions of the form SetOfFiniteBags(D × D) → D.
  9. Isub gives meaning to the subclass relationship. It is a total function D × D → D.

  10. 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.

  11. Iisa gives meaning to class membership. It is a total function D × D → D.

  12. The relationships # and ## are requiredrelationships # 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.

  13. I= is a total function D × D → D.

    It gives meaning to the equality operator.

  14. Itruth is a total mapping D → TV.

    It is used to define truth valuation for formulas.

  15. Iexternal is a mapping from the coherent set of schemas for externally defined functions to total functions D* → D. For each external schema σ = (?X1 ... ?Xn; τ) in the coherent set of such schemas associated with the language, Iexternal(σ) is a function of the form Dn → D.

    For every external schema, σ, associated with the language, Iexternal(σ) is assumed to havebe specified externally in some document (hence the usual property that all members ofname external schema). In particular, if σ is a subclass are also membersschema of a RIF builtin predicate or function, Iexternal(σ) is specified in the superclass, i.e., o # cldocument Data Types and cl ## scl must imply o # scl . ThisBuiltins so that:

It is used to define truth valuation of formulas.For convenience, we also define the following mapping I :

  1. I (k)(k) = IC (k),(k), if k is a symbol in Const
  2. I (?v)(?v) = IV (?v), if ?v(?v), if ?v is a variable in Var
  3. I (f(t(f(t1 ... tn )))) = IF(I (f))((f))(I (t(t1),...,I (t(tn))
  4. I (f( s(f(s1 ->v->v1 ... sn ->v->vn )))) = ISF(I (f))({ <s(f))({<s1,I (v(v1 ) > ,..., <s)>,...,<sn,I (v(vn ) > }))>})

  5. Here we use {...} to denote a bag of argument/value pairs.

  6. I(o[a1 ->v->v1 ... ak ->v->vk]) = Iframe(I(o ))({ <))({<I(a1),I (v(v1 ) > ,)>, ..., <I(an),I (v(vn ) > }))>})

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

  8. I(c1##c2) = Isub(I(c1), I(c2))
  9. I(o#c) = Iisa(I(o), I(c))
  10. I( x=yx=y) = I=(I(x), I(y))
  11. I(External(t)) = Iexternal(σ)(I(s1), ..., I(sn)), if t is an instance of the external schema σ = (?X1 ... ?Xn; τ) by substitution ?X1/s1 ... ?Xn/s1.

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

The effect of signatures. For every signature, sg, supported by the dialect, there is a subset Dsg ⊆ D, called the domain of the signature. Terms that have a given signature, sg, are supposed tomust be mapped by I to Dsg, and if a term has more than one signature it is supposed tomust be mapped into the intersection of the corresponding signature domains. To ensure this, the following is required:

  1. If sg < sg' then Dsg⊆Dsg'.
  2. If k is a constant that has signature sg then IC(k) ∈ Dsg.
  3. If ?v is a variable that has signature sg then IV(?v) ∈ Dsg.
  4. If sg has an arrow expression of the form (s1 ... sn)⇒s then, for every d∈Dsg, IF(d) must map Ds1× ... ×Dsn to Ds.
  5. If sg has an arrow expression of the form ( p1->s1p1->s1 ... pn->snpn->sn)⇒s then, for every d∈Dsg, ISF(d) must map the set {<p1{<p1,Ds1 > ,>, ..., <pn<pn,Dsn > }>} to Ds.
  6. If the signature ->-> has an arrow expressions ( sg , s(sg,s1 , r,r1 )⇒ k)⇒k, ..., ( sg , s(sg,sn , r,rn )⇒ k)⇒k, then, for every d∈Dsg, Iframe(d) must map { <{<Ds1,Dr1 > ,>, ..., <Dsn,Drn > }>} to Dk.
  7. If the signature # has an arrow expression ( s r )⇒ k(s r)⇒k then Iisa must map Ds×Dr to Dk.
  8. If the signature ## has an arrow expression ( s s )⇒ k(s s)⇒k then Isub must map Ds×Ds to Dk.
  9. If the signature = has an arrow expression ( s s )⇒ k(s s)⇒k then I= must map Ds×Ds to Dk.

The effect of data types. The data types in DTS impose the following restrictions. If dt is a symbol space identifier of a data type, let LSdt denote the lexical space of dt, VSdt denote its value space, and Ldt: LSdt → VSdt the lexical-to-value-space mapping. Then the following must hold:

That is, IC must map the constants of a data type dt in accordance with Ldt.   ☐

RIF-FLD does not impose special requirements toon IC for constants in the lexicalsymbol spaces that do not correspond to 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 VSdt of a data type dt.

3.5 Interpretation of Formulas

Definition (Truth valuation). Truth valuation for well-formed formulas in RIF-BLDRIF-FLD is determined using the following function, denoted TValI:

  1. Constants: TValI(k) = Itruth(I(k)), if k ∈ Const.
  2. Variables: TValI(?v) = Itruth(I(?v)), if ?v ∈ Var.
  3. Positional atomic formulas: TValI(r(t1 ... tn )))) = Itruth(I(r(t1 ... tn )))))).
  4. Atomic formulas with named arguments: TValI(p(s1 ->v->v1 ... sk ->v->vk )))) = Itruth(I(p(s1 -> v-> v1 ... sk ->v->vk ))).))).
  5. Equality: TValI(x = y) = Itruth(I(x = y)).

    To ensure that equality has precisely the expected properties, it is required that

    Itruth(I(x = y)) = t if and only if I(x) = I(y) and that Itruth(I(x = y)) = f otherwise.
  6. Subclass: TValI(sc ## cl) = Itruth(I(sc ## cl)).

  7. 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,   glbt(TValI(c1 ## c2), TValI(c2 ## c3 ))  ))  ≤t  TValI(c1 ## c3).

  8. Membership: TValI(o # cl) = Itruth(I(o # cl)).

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

    For all o, cl, scl ∈ D,   glbt(TValI(o # cl), TValI(cl ## scl ))  ))  ≤t  TValI(o # scl).
  9. Frame: TValI(o[a1 ->v->v1  ... a... ak ->v->vk]) = Itruth(I(o[a1 ->v->v1  ... a... ak ->v->vk])).

    Since the different attribute/value pairs are supposed to be understood as conjunctions, the following is required:

    TValI(o[a1 ->v->v1  ... a... ak ->v->vk]) = glbt(TValI(o[a1 ->v->v1]), ..., TValI(o[ak ->v->vk] )))).
  10. Externally defined atomic formula: TValI(External(t)) = Itruth(Iexternal(σ)(I(s1), ..., I(sn))), if t is an atomic formula that is an instance of the external schema σ = (?X1 ... ?Xn; τ) by substitution ?X1/s1 ... ?Xn/s1.

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

  11. Conjunction: TValI( And( cAnd(c1 ... cn)) = glbt(TValI (c(c1), ..., TValI (c(cn)).

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

  12. Disjunction: TValI(Or(c1 ... cn)) = lubt(TValI(c1), ..., TValI(cn)).

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

  13. Negation: TValI(Neg φ)φ) = ~TValI (φ)(φ) and TValI(Naf φ)φ) = ~TValI (φ) where(φ).

    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.

  14. Quantification:

    Here lubt (respectively, glbt) is taken over all interpretations I* of the form <TV, DTS, D, IC, I*V, IF, Iframe, ISF, Isub, Iisa, I=, Iexternal, Itruth >,>, which are exactly like I, except that the mapping I*V, is used instead of IV.   I*V is defined to coincide with I V on all variables except, possibly,defined to coincide with IV on all variables except, possibly, on ?v1,... ,?vn.

  15. Rule implication:
  16. Groups of formulas:

    If Γ is a group formula of the form Group φ (ρ1 ... ρn) or Group (ρ1 ... ρn) then

    TValI(Γ) = glbt(TValI(ρ1), ..., TValI(ρn)).

    This means that a group of formulas is treated as a conjunction. The metadata is ignored for semantic purposes.

Note that rule implications and equality formulas are always two-valued, even if TV has more than two values.

A model of a group of formulas Γ is a semantic structure I such that TValI(Γ) = t.     ☐

Note that although metadata associated with RIF formulas is ignored by the semantics, it can be extracted by XML tools. Since metadata is represented by frame terms, it can be reasoned with by RIF dialects, such as RIF-BLD.


3.6 Intended Semantic Structures

The semantics of a set of formulas, Γ, is the set of its intended semantic structures. RIF-FLD does not specify what these intended structures are, leaving this to RIF dialects. Different logic theories may have different criteria for what is considered an intended semantic structure.

For the classical first-order logic, every semantic structure is intended. For RIF-BLD, which is based on ?v 1 ,..., ?v n . Rules : TVal I ( head  :- body ) = t , if TVal I ( head ) ≥ t TVal I ( body );   TVal I ( head  :- body ) = f   otherwise. Note that rules and equality formulasHorn rules, intended semantic structures are two-valued even if TV has more than two values.defined only for sets of rules: an intended semantic structure of a RIF-BLD set 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 Rof formulasrules are intended. Each dialect of RIF is asupposed to define the notion of intended semantic structure I such that TVal I (φ) = t   for every   φ∈ R . 3.6structures precisely. The two most common theories of intended semantic structures are the so called 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, not, 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 semantics ofrule p :- Naf q as a set of formulas , R ,statement that p must be true if it is not possible to establish the settruth of its intended semantic structuresq. RIF-FLDSince it is, indeed, impossible to establish the truth of q, such theories would derive p even though it does not specify what theselogically follow from Or(p q). The logic underlying rule-based systems also assumes that only the minimal Herbrand models are intended structures are, leaving this(minimality here is with respect to RIF dialects. There are different theoriesthe set of howtrue 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 setsmodels and the corresponding notion of semantic structures are supposedlogical entailment with respect to look like. Forthe classical first-order logic, every semantic structure is intended. For RIF-BLD, which is based on Horn rules,intended semantic structures aremodels, defined onlybelow, is due to [Shoham87].

3.7 Logical Entailment

We will now define what it means for rulesets:a set of RIF formulas to entail a RIF formula. We assume that each set of formulas has an associated set of intended semantic structure ofstructures.


Definition (Logical entailment). Let Γ be a RIF-BLD ruleset R is the unique minimal Herbrand model of R . For the dialects in which rule bodies may contain literals negated with the negation-as-failure connective nafRIF group formula and φ a RIF formula. We say that Γ entails φ, written as Γ |= φ, if and only some of the minimal Herbrand models of a rule set are intended. Each dialectif for every intended semantic structure I of RIFΓ it is supposed to definethe case that TValI(Γ) ≤ TValI(φ).   ☐

This general notion of intended semantic structures precisely. The two most common theories of intended semantic structures are the so called well-founded models [ GRS91 ]entailment covers both first-order logic and stable modelsnon-monotonic logics that underlie many rule-based languages [ GL88Shoham87].


4 XML Serialization Framework

The following example illustrates the notion of intended semantic structures. Suppose R consists ofRIF XML serialization framework defines a single rule p :- naf q . If naf were interpreted as classical negation, not , then this rule would be simply equivalentnormative mapping from the RIF-FLD presentation syntax to p \/ q , and so it would have two kinds of models: those where p is trueXML, and those where q is true.also a normative XML Schema for that XML syntax. As explained in contrast to first-order logic, most rule-based systems do not consider p and q symmetrically. Instead, they viewthe rule p :- naf q as a statementoverview section, RIF requires that pthe presentation syntax of any logic-based RIF dialect must be true if it is not possible to establish the trutha specialization of q . Since it is, indeed, impossible to establishthe truthpresentation syntax of q , such theories would derive p even though it does not logically follow from p \/ q .RIF-FLD, i.e., every well-formed formula in the logic underlying rule-based systemspresentation syntax of a RIF dialect must be well-formed also assumes that onlyin RIF-FLD. The minimal Herbrand models are intended (minimality heregoal of the XML serialization framework is with respectto provide a similar yardstick for the RIF XML syntax. This amounts to the setrequirement that any valid XML document for a logic-based RIF dialect must also be a valid XML document for RIF-FLD. In this way, RIF-FLD provides a framework for extensibility and mutual compatibility between XML syntaxes of true facts). Furthermore, although our example has two minimal Herbrand models -- one where pRIF dialects.

This section is true and qincomplete in the present draft. The next draft will include full treatment of the XML serialization framework.


The XML serialization for RIF-BLD is false,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. We use capitalized names for type tags and lowercase names for role tags. The other where p is false, but q is true, onlyRIF serialization framework uses the first modelfollowing XML tags.

- Group     (nested collection of formulas annotated with metadata)
- meta      (meta role, containing metadata, which is  considered to be intended. The above conceptrepresented as a Frame)
- 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)
- arg       (argument role)
- upper     (Member/Subclass upper class role)
- lower     (Member/Subclass lower instance/class role)
- slot      (Atom/Expr/Frame slot role, containing a Prop)
- Prop      (Property, prefix version of slot infix '->')
- key       (Prop key role, containing a Const)
- val       (Prop val role, containing a TERM)
- Equal     (prefix version of  intended modelsterm equation '=')
- Expr      (expression formula, positional or with named arguments)
- side      (Equal left-hand side and  the corresponding notionright-hand side role)
- Const     (individual, function, or predicate symbol, with optional 'type' attribute)
- Name      (name of  logical entailmentnamed argument)
- Var       (logic variable)



Example 3 (Serialization of a nested RIF-FLD group annotated with respect tometadata).

This example shows an XML serialization for the intended models, defined below, is due to [ Shoham87 ]. 3.7 Logical Entailment We will now define what it meansformulas in Example 2. For a setconvenience of RIFreference, the original formulas to entail a RIF formula.are included at the top. For better readability, we assume that each ruleset has an associated set of intended semantic structures. Let R be a set of RIF formulas and φ a closed RIF formula.again use the compact URI syntax.

Compact URI prefixes:
 
  dc     expands into http://dublincore.org/documents/dces/
  ex     expands into http://example.org/ontology#
  hamlet expands into http://www.shakespeare-literature.com/Hamlet/ 

Presentation syntax:

  Group "hamlet:assertions"^^rif:iri["dc:title"^^rif:iri->"Hamlet"^^xsd:string,
                                     "dc:creator"^^rif:iri->"Shakespeare"^^xsd:string]
  (
      Exists ?X (And(?X # "ex:RottenThing"^^rif:iri
                     "ex:part-of"^^rif:iri(?X "http://www.denmark.dk"^^rif:iri)))
      Forall ?X (Or("hamlet:to-be"^^rif:iri(?X)  Naf "hamlet:to-be"^^rif:iri(?X)))
      Forall ?X (And(Exists ?B (And("ex:has"^^rif:iri(?X ?B) ?B#"ex:business"^^rif:iri))
                     Exists ?D (And("ex:has"^^rif:iri(?X ?D) ?D#"ex:desire"^^rif:iri)))
                   :- ?X#"ex:man"^^rif:iri)
      Group "hamlet:facts"^^rif:iri[ ]
      (
         "hamlet:Yorick"^^rif:iri#"ex:poor"^^rif:iri
         "hamlet:Hamlet"^^rif:iri#"ex:prince"^^rif:iri
      )
  )


XML serialization:

   <Group>
    <meta>
      <Frame>
        <object>
          <Const type="rif:iri">hamlet:assertions</Const>
        </object>
        <slot>
          <Prop>
            <key><Const type="rif:iri">dc:title</Const></key>
            <val><Const type="xsd:string">Hamlet</Const></val>
          </Prop>
        </slot>
        <slot>
          <Prop>
            <key><Const type="rif:iri">dc:creator</Const></key>
            <val><Const type="xsd:string">Shakespeare</Const></val>
          </Prop>
        </slot>
      </Frame>
    </meta>
    <formula>
     <Exists>
       <declare><Var>X</Var></declare>
       <formula>
         <And>
           <formula>
             <Member>
               <lower><Var>X</Var></lower>
               <upper><Const type="rif:iri">ex:RottenThing</Const></upper>
             </Member>
           </formula>
           <formula>
             <Atom>
               <op><Const type="rif:iri">ex:part-of</Const></op>
               <arg><Var>X</Var></arg>
               <arg><Const type="rif:iri">http://www.denmark.dk</Const></arg>
             </Atom>
           </formula>
         </And>
       </formula>
     </Exists>
    </formula>
    <formula>
     <Forall>
       <declare><Var>X</Var></declare>
       <formula>
         <Or>
           <formula>
             <Atom>
               <op><Const type="rif:iri">hamlet:to-be</Const></op>
               <arg><Var>X</Var></arg>
             </Atom>
           </formula>
           <formula>
             <Naf>
               <formula>
                 <Atom>
                   <op><Const type="rif:iri">hamlet:to-be</Const></op>
                   <arg><Var>X</Var></arg>
                 </Atom>
               </formula>
             </Naf>
           </formula>
         </Or>
       </formula>
     </Forall>
    </formula>
    <formula>
     <Forall>
       <declare><Var>X</Var></declare>
       <formula>
         <Implies>
           <if>
             <Member>
               <lower><Var>X</Var></lower>
               <upper><Const type="rif:iri">ex:man</Const></upper>
             </Member>
           </if>
           <then>
             <And>
               <formula>
                 <Exists>
                   <declare><Var>B</Var></declare>
                   <And>
                     <formula>
                       <Atom>
                         <op><Const type="rif:iri">ex:has</Const></op>
                         <arg><Var>X</Var></arg>
                         <arg><Var>B</Var></arg>
                       </Atom>
                     </formula>
                     <formula>
                       <Member>
                         <lower><Var>B</Var></lower>
                         <upper><Const type="rif:iri">ex:business</Const></upper>
                       </Member>
                     </formula>
                   </And>
                 </Exists>
               </formula>
               <formula>
                 <Exists>
                   <declare><Var>D</Var></declare>
                   <And>
                     <formula>
                       <Atom>
                         <op><Const type="rif:iri">ex:has</Const></op>
                         <arg><Var>X</Var></arg>
                         <arg><Var>D</Var></arg>
                       </Atom>
                     </formula>
                     <formula>
                       <Member>
                         <lower><Var>D</Var></lower>
                         <upper><Const type="rif:iri">ex:desire</Const></upper>
                       </Member>
                     </formula>
                   </And>
                 </Exists>
               </formula>
             </And>
           </then>
         </Implies>
       </formula>
     </Forall>
   </formula>
   <formula>
     <Group>
       <meta>
         <Frame>
           <object>
             <Const type="rif:iri">hamlet:facts</Const>
           </object>
         </Frame>
       </meta>
       <formula>
         <Member>
           <lower><Const type="rif:iri">hamlet:Yorick</Const></lower>
           <upper><Const type="rif:iri">ex:poor</Const></upper>
         </Member>
       </formula>
       <formula>
         <Member>
           <lower><Const type="rif:iri">hamlet:Hamlet</Const></lower>
           <upper><Const type="rif:iri">ex:prince</Const></upper>
         </Member>
       </formula>
     </Group>
    </formula>
   </Group>


We say that R entails φ, written as R   |=  φ, if and only ifnow serialize the syntax of Section EBNF Grammar for every intended semantic structure Ithe Presentation Syntax of R and every ψ ∈ R , it isRIF-FLD by defining a mapping from the case that TVal I (ψ) ≤ TVal I (φ).presentation syntax to XML.

This general notion of entailment covers both first-order logic and non-monotonic logics that underlie many rule-based languages [ Shoham87 ]. 4 XML Serialization Frameworkmapping will be given in the next draft.

5 References

5.1 Normative References

[RDF-CONCEPTS]
Resource Description Framework (RDF): Concepts and Abstract Syntax, Klyne G., Carroll J. (Editors), W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/.http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/. Latest version available at http://www.w3.org/TR/rdf-concepts/.http://www.w3.org/TR/rdf-concepts/.

[RDF-SEMANTICS]
RDF Semantics, Patrick Hayes, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-mt-20040210/.http://www.w3.org/TR/2004/REC-rdf-mt-20040210/. Latest version available at http://www.w3.org/TR/rdf-mt/.http://www.w3.org/TR/rdf-mt/.

[RDF-SCHEMA]
RDF Vocabulary Description Language 1.0: RDF Schema, Brian McBride ,McBride, Editor, W3C Recommendation 10 February 2004, http://www.w3.org/TR/rdf-schema/.http://www.w3.org/TR/rdf-schema/.

[RFC-3066]
RFC 3066 - Tags for the Identification of Languages, H. Alvestrand, IETF, January 2001. This document is http://www.isi.edu/in-notes/rfc3066.txt.at http://www.isi.edu/in-notes/rfc3066.txt.

[RFC-3987]
RFC 3987 - Internationalized Resource Identifiers (IRIs), M. Duerst and M. Suignard, IETF, January 2005. This document is http://www.ietf.org/rfc/rfc3987.txt.at http://www.ietf.org/rfc/rfc3987.txt.

[XML-SCHEMA2]
XML Schema Part 2: Datatypes, W3C Recommendation, World Wide Web Consortium, 2 May 2001. This version is http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/.http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/. The latest version is available at http://www.w3.org/TR/xmlschema-2/.http://www.w3.org/TR/xmlschema-2/.

5.2 Informational References

[ANF01]
Normal Form Conventions for XML Representations of Structured Data, Henry S. Thompson. October 2001.

[KLW95]
Logical foundations of object-oriented and frame-based languages, M. Kifer, G. Lausen, J. Wu. Journal of ACM, July 1995, pp. 741--843.

[CKW93]
HiLog :HiLog: A Foundation for higher-order logic programming, W. Chen, M. Kifer, D.S. Warren. Journal of Logic Programming, vol. 15, no. 3, February 1993, pp. 187--230.

[CK95]
Sorted HiLog :HiLog: Sorts in Higher-Order Logic Data Languages, W. Chen, M. Kifer. Sixth Intl. Conference on Database Theory, Prague, Czech Republic, January 1995, Lecture Notes in Computer Science 893, Springer Verlag, pp. 252--265.

[RDFSYN04]
RDF/XML Syntax Specification (Revised), Dave Beckett, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-syntax-grammar-20040210/.http://www.w3.org/TR/2004/REC-rdf-syntax-grammar-20040210/. Latest version available at http://www.w3.org/TR/rdf-syntax-grammar/.http://www.w3.org/TR/rdf-syntax-grammar/.

[Shoham87]
Nonmonotonic logics: meaning and utility, Y. Shoham. Proc. 10th International Joint Conference on Artificial Intelligence, Morgan Kaufmann, pp. 388--393, 1987.

[CURIE]
CURIE Syntax 1.0: A compact syntax for expressing URIs, Mark Birbeck. Draft, 2005. Available at http://www.w3.org/2001/sw/BestPractices/HTML/2005-10-27-CURIE.http://www.w3.org/2001/sw/BestPractices/HTML/2005-10-27-CURIE.

[CycL]
The Syntax of CycL, Web site. Available at http://www.cyc.com/cycdoc/ref/cycl-syntax.html.

[FL2]
FLORA-2: An Object-Oriented Knowledge Base Language, M. Kifer. Web site. Available at http://flora.sourceforge.net.http://flora.sourceforge.net.

[OOjD]
Object-Oriented jDREW, Web site. Available at http://www.jdrew.org/oojdrew/.http://www.jdrew.org/oojdrew/.

[GRS91]
The Well-Founded Semantics for General Logic Programs, A. Van Gelder, K.A. Ross, J.S. Schlipf. Journal of ACM, 38:3, pages 620-650, 1991.

[GL88]
The Stable Model Semantics for Logic Programming, M. Gelfond and V. Lifschitz. Logic Programming: Proceedings of the Fifth Conference and Symposium, pages 1070-1080, 1988.

[vEK76]
The semantics of predicate logic as a programming language, M. van Emden and R. Kowalski. Journal of the ACM 23 (1976), 733-742.

[Enderton01]
A Mathematical Introduction to Logic, Second Edition, H. B. Enderton. Academic Press, 2001.

[Mendelson97]
Introduction to Mathematical Logic, Fourth Edition, E. Mendelson. Chapman & Hall, 1997.