RIF Framework for Logic Dialects

W3C Editor's Draft 1723 July 2008

This version:

http://www.w3.org/2005/rules/wg/draft/ED-rif-fld-20080717/http://www.w3.org/2005/rules/wg/draft/ED-rif-fld-20080723/

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-20080609/http://www.w3.org/2005/rules/wg/draft/ED-rif-fld-20080717/ (color-coded diff)

Editors:

Harold Boley, National Research Council, Canada

Michael Kifer, State University of New York at Stony Brook, USA

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

Abstract

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May Be Superseded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/.

Set of Documents

This document is being published as one of a set of 6 documents:

1. RIF Use Cases and Requirements
2. RIF Basic Logic Dialect
3. RIF Framework for Logic Dialects (this document)
4. RIF RDF and OWL Compatibility
5. RIF Production Rule Dialect
6. RIF Datatypes and Built-Ins 1.0

Please Comment By 2008-07-212008-07-28

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.

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The RIF Framework for Logic-based Dialects (RIF-FLD) is a formalism for specifying all logic-based dialects of RIF, including the RIF Basic Logic Dialect [RIF-BLD]. It is a logic in which both syntax and semantics are described through a number of mechanisms that are commonly used for various logic languages, but are rarely brought all together. Amalgamation of several different mechanisms is required because the framework must be broad enough to accommodate several different types of logic languages and because various advanced mechanisms are needed to facilitate translation into a common framework. RIF-FLD gives precise definitions to these mechanisms, but allows certain details to vary. The design of RIF envisages that future standard logic dialects will be based on RIF-FLD. Therefore, any logic dialect being developed to become a stardard should either be a specialization of FLD or justify its deviations from (or extensions to) FLD.

The framework described in this document is very general and captures most of the popular logic-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. All logic-based RIF dialects are required to be derived from RIF-FLD by specialization, as explained in Sections Syntax of a RIF Dialect as a Specialization of RIF-FLD and Semantics of a RIF Dialect as a Specialization of RIF-FLD. In addition to specialization, to lower the barrier of entry for their intended audiences, a dialect designer may choose to specify the syntax and semantics in a direct, but equivalent, way, which does not require familiarity with RIF-FLD. For instance, the RIF Basic Logic Dialect [RIF-BLD] is specified bothby specialization from RIF-FLD and also directly, without relying on the framework. Thus, the reader who is interested in [RIF-BLD ]only can proceed directly to that document.

RIF-FLD has the following main components:

• Syntactic framework. This framework defines the mechanisms for specifying the formal presentation syntax of RIF logic dialects by specializing the presentation syntax of the framework. The presentation syntax is used in RIF to define the semantics of the dialects and to illustrate the main ideas with examples. This syntax is not intended to be a concrete syntax for the dialects; it leaves out details such as the delimiters of the various syntactic components, parenthesizing, precedence of operators, and the like. Since RIF is an interchange format, it uses XML as its concrete syntax.
• Semantic framework. The semantic framework describes the mechanisms that are used for specifying the models of RIF logic-based dialects.
• XML serialization framework. This framework defines the general principles that logic-based dialects are to use in specifying their concrete XML-based syntaxes. For each dialect, its concrete XML syntax is a derivative of the dialect's presentation syntax. It can be seen as a serialization of that syntax.

Syntactic framework. The syntactic framework defines six types of RIF terms:

• Constants and variables. These terms are common to most logic languages.
• Positional terms. These terms are commonly used in first-order logic. RIF-FLD defines positional terms in a slightly more general way in order to enable dialects with higher-order syntax, such as HiLog [CKW93].
• Terms with named arguments. These are like positional terms except that each argument of a term is named and the order of the arguments is immaterial. Terms with named arguments generalize the notion of rows in relational tables, where column headings correspond to argument names.
• Frames. A frame term represents an assertion about an object and its properties. These terms correspond to molecules of F-logic [KLW95]. There is syntactic similarity between terms with named arguments and frames, since object properties resemble named arguments. However, the semantics of these terms are different.
• Classification. These terms are used to define the subclass and class membership relationships. Like frames, they are also borrowed from F-logic [KLW95].
• Equality. These terms are used to equate other terms.

Terms are then used to define several types of RIF-BLD formulas. RIF dialects can choose to supportpermit all or some of the aforesaid categories of terms. The syntactic framework also defines the following mechanisms for specializing these terms:specialization mechanisms:

• Symbol spaces.

  Symbol spaces partition the set of allnon-logical symbols (symbols used as variables,that correspond to individual constants, predicates, and functions)functions, and each partition is then given its own semantics. A symbol space has an identifier and a lexical space, which defines the "shape" of the symbols in that symbol space. Some symbol spaces in RIF are used to identify Web entities and their lexical space consists of strings that syntactically look like internationalized resource identifiers [RFC-3987], or IRIs (e.g., rif:irihttp://www.w3.org/2007/rif#iri). Other symbol spaces are used to represent the datatypes required by RIF (for example, xs:integer ).[DTB#xs-integer-space|http://www.w3.org/2001/XMLSchema#integer]).

• Signatures.

  Signatures determine which terms and formulas are well-formed. It is a generalization of the notion of a sort in classical first-order logic [Enderton01]. Each nonlogical symbol (and some logical symbols, like =) has an associated signature. A signature defines, in a precise way, the syntactic contexts in which the symbol is allowed to occur.

  For instance, the signature associated with a symbol p might allow p to appear in a term of the form f(p), but disallow it to occur in a term like p(a,b). The signature for f, on the other hand, might allow that symbol to appear in f(p) and f(p,q), but disallow f(p,q,r) and f(f). In this way, it is possible to control which symbols are used for predicates and which for functions, where variables can occur, and so on.

• Restriction.

  A dialect might impose further restrictions on the form of a particular kind of terms or formulas.

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

• Truth values. RIF-FLD is designed to accommodate dialects that support reasoning with inconsistent and uncertain information. Most of the logics that are designed to deal with these situations are multi-valued. Consequently, RIF-FLD postulates that there is a set of truth values, TV, which includes the values t (true) and f (false) and possibly others. For example, RIF Basic Logic Dialect [RIF-BLD] is two-valued, but other dialects can have additional truth values.
• Data types.Datatypes. Some symbol spaces that are part of the RIF syntactic framework have fixed interpretations. For instance, symbols in the symbol space xs:string[DTB#xs-string-space|http://www.w3.org/2001/XMLSchema#string] are always interpreted as sequences of unicode characters, and a ≠ b for any pair of distinct symbols. A symbol space whose symbols have a fixed interpretation in any semantic structure is called a datatype.
• Entailment. This notion is fundamental to logic-based dialects. Given a set of formulas (e.g., facts and rules) G, entailment determines which other formulas necessarily follow from G. Entailment is the main mechanism underlying query answering in databases, logic programming, and the various reasoning tasks in Description Logic.Logics.

  A set of formulas G logically entails another formula g if for every semantic structure I in some set S, if G is true in I then g is also true in I. Almost all logics define entailment this way. The difference lies in which set S they use. For instance, logics that are based on the classical first-order predicate calculus, such as most Description Logic,Logics, assume that S is the set of all semantic structures. In contrast, most logic programming languages, whichlanguages use default negation , assume that. Accordingly, the set S contains only the so-called "minimal" Herbrand models of G and, furthermore, only the minimal models of a special kind. See [Shoham87] for a more detailed exposition of this subject.

XML serialization framework. This framework defines the general principles for mapping the presentation syntax of RIF-FLD to the concrete XML interchange format. This includes:

• A specification of the XML syntax for RIF-FLD, including the associated XML Schema document.
• A specification of a one-to-one mapping from the presentation syntax of RIF-FLD to its XML syntax. This mapping must map any well-formed group formula of RIF-FLD to an XML document that is valid with respect to the aforesaid XML Schema document.

This document is the latest draft of the RIF-FLD specification. Each RIF dialect that is derived from RIF-FLD will be described in its own document. The first of such dialects, RIF Basic Logic Dialect, is described in [RIF-BLD].

1. The alphabet of RIF-FLD can be restricted.restricted (by omitting symbols).
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 signatures are assigned depends on the dialect. An assignment can be explicit or implicit (for instance, derived from the context in which each symbol is used).

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 these terms or just a subset. For instance, some dialects might not support terms with named arguments or frame terms.terms or certain forms of external terms (e.g., external frames).

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

7. Symbol spaces determine the syntax of the constant symbols that are allowed in the dialect.

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

9. RIF-FLD allows formulas of the following kind:

   • Atomic
   • Conjunction
   • Disjunction
   • Classical negation
   • Default negation (as in logic programming)
   • Rule (as in logic programming as opposed to the classical material implication)
   • 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/or premises of implications might be restricted (e.g., [RIF-BLD] essentially allows Horn rules only), certain types of quantification might be prohibited (e.g., [RIF-BLD] disallows existential quantification in the rule head), classical or default negation (or both) might not be allowed (as in RIF-BLD), etc. A subdialect of RIF-BLD might disallow equality formulas in the conclusions of the rules.

Note that although the presentation syntax of a RIF logic-based dialect is normative, since semantics is defined in terms of that syntax, the presentation syntax is not intended as a concrete syntax, and conformant systems are not required to implement it.

The set of connective symbols, quantifiers, =, etc., is disjoint from Const and Var. Variables are written as Unicode strings preceded withby the symbol "?". The argument names in ArgNames are written as Unicode strings that do not start with a "?". The syntax for constant symbols is given in Section Symbol Spaces.

The symbols =, #, and ## are used in formulas that define equality, class membership, and subclass relationships. The symbol -> is used in terms that have named arguments and in frame terms. The symbol External indicates that an atomic formula or a function term is defined externally (e.g., a builtin), Dialect is a directive used to indicate the dialect of a RIF document (for those dialects that require this), the symbols Base and Prefix are used in abridged representations of IRIs, and Import is an import directive.

Finally, the symbol Document is used for specifying RIF-FLD documents and the symbol Group is used to organize RIF-FLD formulas into collections.   ☐

These and other abbreviations will be used as prefixes in the compact URI notation [CURIE], a notation for succinct representation of IRIs. The precise meaning of this notation in RIF is defined in [RIF-DTB].

The set of all constant symbols in a RIF dialect is partitioned into a number of subsets, called symbol spaces, which are used to represent XML Schema datatypes, datatypes defined in other W3C specifications, such as rdf:XMLLiteral, and to distinguish other sets of constants. All constant symbols have a syntax (and sometimes also semantics) imposed by the symbol space to which they belong.

Definition (Symbol space). A symbol space is a named subset of the set of all constants, Const. The semantic aspects of symbol spaces will be described in Section Semantic Framework. Each symbol in Const belongs to exactly one symbol space.

Each symbol space has an associated lexical space and a unique identifier. More precisely,

• The lexical space of a symbol space is a non-empty set of Unicode character strings.
• The identifier of a symbol space is a sequence of Unicode characters that form an absolute IRI.
• Different symbol spaces cannot share the same identifier.

The identifiers for symbol spaces are not themselves constant symbols in RIF.   ☐

To simplify the language, we will often use symbol space identifiers to refer to the actual symbol spaces (for instance, we may use "symbol space xs:string" instead of "symbol space identified by xs:string").

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

     "literal"^^symspace

where literal is called the lexical part of the symbol, and symspace is an identifier of the symbol space. Here literal is a sequence of Unicode characters that must be an element in the lexical space of the symbol space symspace. For instance, "1.2"^^xs:decimal and "1"^^xs:decimal are legal symbolssyntactically valid constants because 1.2 and 1 are members of the lexical space of the XML Schema datatype xs:decimal. On the other hand, "a+2"^^xs:decimal is not a legalsyntactically valid symbol, since a+2 is not part of the lexical space of xs:decimal.

The set of all symbol spaces that partition Const is considered to be part of the logic language of RIF-FLD.

RIF requires that all dialects include the symbol spaces listed and described in Section Constants and Symbol Spaces of [RIF-DTB ].] as part of their language. These symbol spaces include constants that belong to several important XML Schema datatypes, certain RDF datatypes, and constant symbols specific to RIF. The latter include the symbol spaces rif:iri and rif:local, which are used to represent internationalized resource identifiers (IRIs) and constant symbols that are not visible outside of the RIF document in which they occur, respectively. Rule setsDocuments that are exchanged through RIF can use additional symbol spaces.

1. Constants and variables. If t ∈ Const or t ∈ Var then t is a simple term.
2. Positional terms. If t and t₁, ..., t_n are terms then t(t₁ ... t_n) 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, as in ?X(?Y ?Z(?V "12"^^xs:integer)), where ?X, ?Y, ?Z, and ?V are variables. Even ?X("abc"^^xs:string ?W)(?Y ?Z(?V "33"^^xs:integer)) is a positional term (as in HiLog [CKW93]).

3. Terms with named arguments. A term with named arguments is of the form t(s₁->v₁ ... s_n->v_n), where t, v₁, ..., v_n are terms, and s₁, ..., s_n are (not necessarily distinct) symbols from the set ArgNames.

   The term t here represents a predicate or a function; s₁, ..., s_n represent argument names; and v₁, ..., v_n represent argument values. Terms with named arguments are like regular positional terms except that the arguments are named and their order is immaterial. Note that a term with no arguments, like f(), isis, trivially, both a positional term and alsoa term with named arguments.

   For instance, "person"^^xs:string(name->?Y address->?Z), ?X("123"^^xs:integer ?W)(arg->?Y arg2->?Z(?V)), and "Closure"^^rif:local(relation->"http://example.com/Flight"^^rif:iri)(from->?X to->?Y) are terms with named arguments. The second of these terms has a positional term ?X(abc,?W), which occurs in the position of a function, and the third term's function is represented by a named arguments term.

4. Equality terms. An equality term has the form t = s, where t and s are terms.
5. 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 terms.
   • t##s is a subclass term if t and s are terms.

   Classification terms are used to describe class hierarchies.

6. Frame terms. t[p₁->v₁ ... p_n->v_n] is a frame term (or simply a frame) if t, p₁, ..., p_n, v₁, ..., v_n, n ≥ 0, are terms.

   Frame terms are used to describe properties of objects. As in the case of the terms with named arguments, the order of the properties p_i->v_i in a frame is immaterial.

7. Externally defined terms. If t is a constant, positional term, a term with named arguments, or a frame term then External(t) is an externally defined term.

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

   Note that the above syntax allows very general interfaces to access externally defined data sources: not only predicates can be used,and functions, but also frames.frames can be used. In this way, externally defined objects can be accessed using the more natural frame-based interface. For instance, External("http://example.com/acme"^^rif:iri["http://example.com/mycompany/president"^^rif:iri(?Year) -> ?Pres]) could be an interface provided to access an externally defined method "http://example.com/mycompany/president"^^rif:iri of an external object "http://example.com/acme"^^rif:iri.   ☐

The above definitions are very general. They make no distinction between constant symbols that represent individuals, predicates, and function symbols. The same symbol can occur in multiple contexts at the same time. For instance, if p, a, and b are symbols then p(p(a) p(a p c)) is a term. Even variables and general terms are allowed to occur in the position of predicates and function symbols, so p(a)(?v(a c) p) is also a term.

Frame, classification, and other terms can be freely nested, as exemplified by p(?X  q#r[p(1,2)->s](d->e f->g)). Some language environments, like FLORA-2 [FL2], OO jDREW [OOjD], NxBRE [NxBRE], and CycL [CycL] support fairly large (partially overlapping) subsets of RIF-FLD terms, but most languages support much smaller subsets. RIF dialects are expected to carve out the appropriate subsets of RIF-FLD terms, and the general form of the RIF logic framework allows a considerable degree of freedom.

Observe that the argument names of frame terms, p₁, ..., p_n, are baseterms and, as a special case, can be variables. In contrast, terms with named arguments can use only the symbols from ArgNames to represent their argument names. They cannot be constants from Const or variables from Var. The reason for this restriction has to do with the complexity of unification, which is integral part of many inference rules underlying first-order logic. We are not aware of any rule language where terms with named arguments use anything more general than what is defined here.

Dialects can restrict the contexts in which the various terms are allowed by using the mechanism of signatures. The RIF-FLD language associates a signature with each symbol (both constant and variable symbols) and uses signatures to define well-formed terms. Each RIF dialect is expected to select appropriate signatures for the symbols in its alphabet, and only the terms that are well-formed according to the selected signatures are allowed in that particular dialect.

This section introduces the notion of external schemas, which serve as templates for externally defined terms. These schemas determine which externally defined terms are acceptable in a RIF dialect. Externally defined terms include RIF builtins, which are specified in [RIF-DTB], but are more general. They are designed to accommodate the ideas of procedural attachments and querying of external data sources. Because of the need to accommodate many difference possibilities, the RIF logical framework supports a very general notion of an externally defined term. Such a term is not necessarily a function or a predicate -- it can be a frame, a classification term, and so on.

Definition (Schema for external term). An external schema is a statement of the form (?X₁ ... ?X_n; τ) where

• τ is a term , as defined earlier, with the exception that it is not permitted to be a variable or an externally defined term.of one of these kinds: constant, positional, named-argument, frame.
• ?X₁ ... ?X_n is a list of all distinct variables that occur in τ

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

A term t is an instance of an external schema (?X₁ ... ?X_n; τ) iff t can be obtained from τ by a simultaneous substitution ?X₁/s₁ ... ?X_n/s_n of the variables ?X₁ ... ?X_n with terms s₁ ... s_n, respectively. Some of the terms s_i can be variables themselves. For example, ?Z[foo->f(a ?P)] is an instance of (?X ?Y; ?X[foo->?Y]) by the substitution ?X/?Z  ?Y/f(a ?P).    ☐

Observe that a variable cannot be an instance of an external schema, since τ in the above definition cannot be a variable. It will be seen later that this implies that a term of the form External(?X) is not well-formed in RIF.

The intuition behind the notion of an external schema, such as (?X ?Y;  ?X["foo"^^xs:string->?Y]) or (?V;  "pred:isTime"^^rif:iri(?V)), is that ?X["foo"^^xs:string->?Y] or "pred:isTime"^^rif:iri(?V) are invocation patterns for querying external sources, and instances of those schemas correspond to concrete invocations. Thus, External("http://foo.bar.com"^^rif:iri["foo"^^xs:string->"123"^^xs:integer]) and External("pred:isTime"^^rif:iri("22:33:44"^^xs:time) are examples of invocations of external terms -- one querying an external source and another invoking a builtin.

Definition (Coherent set of external schemas). A set of external schemas is coherent if there can beis no term, t, that is an instance of two distinct schemas.schemas in the set.    ☐

The intuition behind this notion is to ensure that any use of an external term is associated with at most one external schema. This assumption is relied upon in the definition of the semantics of externally defined terms. Note that the coherence condition is easy to verify syntactically and that it implies that schemas like (?X ?Y;  ?X[foo->?Y]) and (?Y ?X;  ?X[foo->?Y]), which differ only in the order of their variables, cannot be in the same coherent set.

It is important to understandkeep in mind that external schemas are not part of the language in RIF, since they do not appear anywhere in RIF statements. Instead, like signatures, which are defined below, they are best thought of as part of the grammar of the language. In particular, they will be used to determine which external terms, i.e., the terms of the form External(t) are well-formed.

The above example provides intuition behind the use of signatures in RIF-FLD. Much of the development, below, is inspired by [CK95]. It should be kept in mind that signatures are not part of the logic language in RIF, since they do not appear anywhere in RIF-FLD formulas. Instead they are part of the grammar: they are used to determine which sequences of tokens are in the language and which are not. The actual way by which signatures are assigned to the symbols of the language may vary from dialect to dialect. In some dialects (for example [RIF-BLD]), this assignment is derived from the context in which each symbol occurs and no separate language for signatures is used. Other dialects may choose to assign signatures explicitly. In that case, they would require a concrete language for signatures (which would be separate from the language for specifying the logic formulas of the dialect).

Definition (Signature name). Let SigNames be a non-empty, partially-ordered finite or countably infinite set of symbols, called signature names. Since signatures are not part of the logic language, their names do not have to be disjoint from Const, Var, and ArgNames. We require that this set includes at least the following signature names:

• atomic -- used to represent 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 may introduce additional signature names. For instance, RIF Basic Logic Dialect [RIF-BLD] introduces one other signature name, individual. The partial order on SigNames is dialect-specific; it is used in the definition of well-formed terms below.

We use the symbol < to represent the partial order on SigNames. Informally, α < β means that terms with signature α can be used wherever terms with signature β are allowed. We will write α ≤ β if either α = β or α < β.

Definition (Signature). A signature is a statement of the form η{e₁, ..., e_n, ...} where η ∈ SigNames is the name of the signature and {e₁, ..., e_n, ...} is a countable set of arrow expressions. Such a set can thus be infinite, finite, or even empty. In RIF-BLD, signatures can have at most one arrow expression. Other dialects (such as HiLog [CKW93], for example) may require polymorphic symbols and thus allow signatures with more than one arrow expression in them.

An arrow expression is defined as follows:

• If κ, κ₁, ..., κ_n ∈ SigNames, n≥0, are signature names then (κ₁ ... κ_n) ⇒ κ is a positional arrow expression.

  For instance, () ⇒ term and (term) ⇒ term are positional arrow expressions, if term is a signature name.

• If κ, κ₁, ..., κ_n ∈ SigNames, n≥0, are signature names and p₁, ..., p_n ∈ ArgNames are argument names then (p₁->κ₁ ... p_n->κ_n) => 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.   ☐

RIF dialects are always associated with sets of coherent signatures, defined next. The overall idea is that a coherent set of signatures must include all the predefined signatures (such as signatures for equality and classification terms) and the signatures included in a coherent set should not conflict with each other. For instance, two different signatures should not have identical names and if one signature is said to extend another then the arrow expressions of the supersignature should be included among the arrow expressions of the subsignature (a kind of an arrow expression "inheritance").

Definition (Coherent signature set). A set Σ of signatures is coherent iff

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

   All arrow expressions e_i here have the form (κ κ) ⇒ γ (the arguments in an equation must be compatible) and at least one of these expressions must have the form (κ κ) ⇒ atomic (i.e., equation terms are also atomic formulas). Dialects may further specialize this signature.

3. Σ contains the signature #{e₁, ..., e_n...}.

   Here all arrow expressions e_i are binary (have two arguments) and at least one has the form (κ γ) ⇒ atomic. Dialects may further specialize this signature.

4. Σ contains the signature ##{e₁, ..., e_n...}.

   Here all arrow expressions e_i have the form (κ κ) ⇒ γ (the arguments must be compatible) and at least one of these arrow expressions has the form (κ κ) ⇒ atomic. Dialects may further specialize this signature.

5. Σ contains the signature ->{e₁, ..., e_n...}.
   Here all arrow expressions e_i 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.
6. Σ has at most one signature for any given signature name.
7. Whenever Σ contains a pair 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 a signature named κ with the set of expressions B. The requirement that B⊆A ensures that symbols that have signature η can be used wherever the symbols with signature κ are allowed.   ☐

The requirement that coherent sets of signatures must include the signatures for =, #, ->, and so on is just a technicality needed to simplify the definitions. Some of these signatures may go "unused" in a dialect even though, technically speaking, they must be present in the signature set associated with that dialect. If a dialect disallows equality, classification terms, or frames in its syntax then the corresponding signatures will remain unused. Such restrictions can be imposed by specializing RIF-FLD -- see Section Syntax of a RIF Dialect as a Specialization of RIF-FLD.

An incoherent set of signatures would be one that includes signatures mysig{() ⇒ atomic} and mysig{atomic ⇒ atomic} because it has two different signatures with the same name. Likewise, if this set contains mysig₁{() ⇒ atomic} and mysig₂{atomic ⇒ atomic} and mysig₁ < mysig₁ then it is incoherent because the set of arrow expressions of mysig₁ does not contain the set of arrow expressions of mysig₂.

The presentation language of a RIF dialect is a set of allwell-formed formulas, as defined in the next section. The language is determined by the following parameters (see Syntax of a RIF Dialect as a Specialization of RIF-FLD):

• An alphabet.
• A set of symbol spaces.
• An assignment of signatures from a coherent set of signatures to the symbols in Var and Const:

  Each variable symbol is associated with exactly one signature from a coherent set of signatures. A constant symbol can have one or more signatures, and different symbols can be associated with the same signature. (If variables were allowed to have multiple signatures then well-formed terms would not be closed under substitutions. For instance, a term like f(?X,?X) could be well-formed, but f(a,a) could be ill-formed.)

• Restrictions on the classes of terms allowed in the language of the dialect.
• Restrictions on the classes of formulas allowed in the language of the dialect.
• A coherent set of external schemas.

We have already seen how the alphabet and the symbol spaces are used to define RIF terms. The next section shows how signatures and external schemas are used to further specialize this notion to define well-formed RIF-FLD terms.

1. A constant or variable symbol with signature η is a well-formed term with signature η.
2. A positional term t(t₁ ... t_n), 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 ... σ_n) ⇒ σ; and
   • Each t_i is a well-formed term whose signature is γ_i ,such that γ_i, ≤ σ_i.

   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.

3. A term with named arguments t(p₁->t₁ ... p_n->t_n), 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 ... p_n->σ_n) ⇒ σ; and
   • Each t_i is a well-formed term whose signature is γ_i, such that γ_i ≤ σ_i.

   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.

4. 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
   • t_i and t₂ are well-formed terms with signatures γ₁ and γ₂, respectively, such that γ_i ≤ σ, i=1,2.
5. 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
   • t₁ and t₂ are well-formed terms with signatures γ₁ and γ₂, respectively, such that γ_i ≤ σ_i, i=1,2.
6. 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
   • t₁ and t₂ are well-formed terms with signatures γ₁ and γ₂, respectively, such that γ_i ≤ σ, i=1,2.
7. A frame term of the form t[s₁->v₁ ... s_n->v_n] is well-formed and has a signature κ iff
   • The signature -> has arrow expressions (σ σ₁₁ σ₁₂) ⇒ 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.
8. An externally defined term, External(t), is well-formed and has signature κ iff
   • t is well-formed and has signature κ.
   • t is an instance of an external schema from a coherent set of external schemas of the language.

     Note that, according to the definition of coherent sets of schemas, a term can be an instance of at most one external schema.    ☐

Note that, like constant symbols, well-formed terms can have more than one signature. Also note that, according to the above definition, f() and f are distinct terms.

Definition (Well-formed formula). A well-formed term is also a well-formed atomic formula iff one of its signatures is atomic or is ≤ atomic. Note that equality, membership, subclass, and frame terms are atomic formulas, since atomic is one of their signatures.

More general formulas are constructed out of atomic formulas with the help of logical connectives. A well-formed formula is a statement that can have one of the forms (1) -- (9) below. The Group and Document formulas, defined in (8) and (9), are aggregate formulas while the formulas in (1) -- (7) are non-aggregate. This distinction manifests itself in that Group and Document cannot be part of non-aggregate formulas, and Document cannot be part of a group.

1. Atomic: If φ is a well-formed atomic formula then it is also a well-formed formula.
2. Conjunction: If φ₁, ..., φ_n, n ≥ 0, are non-aggregate well-formed formulas then so is And(φ₁ ... φ_n).

3. As a special case, And() is allowed and is treated as a tautology, i.e., a formula that is always true.

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

5. As a special case, Or() is treated as a contradiction, i.e., a formula that is always false.

6. Classical negation: If φ is a non-aggregate well-formed formula then Neg φ is also a well-formed formula.
7. Default negation: If φ is a non-aggregate well-formed formula then Naf φ is also a well-formed formula.
8. Rule implication: If φ and ψ are non-aggregate well-formed formulas then φ :- ψ is also a well-formed formula.
9. Quantification: If φ is a non-aggregate well-formed formula and ?V₁, ..., ?V_n are variables then the following formulas are also well-formed:
   • Exists ?V₁ ... ?V_n(φ)
   • Forall ?V₁ ... ?V_n(φ)
10. Group: If φ₁, ..., φ_n are well-formed non-Document formulas or group formulasthen Group (φGroup(φ₁ ... φ_n) is a well-formed RIF-FLD group formula (or simply a group formula when context is clear).

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

11. Document: An expression of the form Document(directive₁ ... directive_n Γ) is a well-formed RIF-FLD document formula (or simply a document formula if no ambiguity arises), if
    • Γ is an optional well-formed group formula that makesencompasses the actuallogical content of the document.
    • directive₁, ..., directive_n areis an optional sequence of directives. A directive can be a dialect directive, a base directive, a prefix directive, or an import directive.
    • • A dialect directive has the form Dialect(D), where D is an alphanumerica Unicode string that specifies the name of a dialect. This directive specifies the dialect of a RIF document. Some dialects may require this directive in all of its documents, while others (notably, RIF-BLD) may not allow it and instead may entirely rely on other syntax. (Purely syntactic identification may not always be possible for dialects that are syntactically identical but semantically different, such as deductive databases with stable model semantics [GL88] and with well-founded semantics [GRS91]. These two dialects are examples where the Dialect directive might be necessary.)
      • A base directive has the form Base(iri), where iri is a unicode string in the form of an IRI.

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

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

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

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

        RIF-FLD defines the semantics for the directive Import(t) only. The directive Import(t p) is reserved for RIF dialects, which might use it to import non-RIF logical entities, such as RDF data and OWL ontologies. The profile might specify what kind of entity is being imported and under what semantics (for instance, the various RDF entailment regimes).

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

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

Observe that the restrictions in (1) -- (8) above imply that groups and documents cannot be nested inside non-aggregate formulas and documents cannot be nested inside groups.

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

We illustrate the above definitions with the following examples. In addition to atomic, let there be another signature, term{ }, which is intended here to represent the context of the arguments to positional terms or atomic formulas.

Consider the term p(p(a) p(a b c)). If p has the (polymorphic) signature mysig{(term)⇒term, (term term)⇒term, (term term term)⇒term} and a, b, c each has the signature term{ } then p(p(a) p(a b c)) is a well-formed term with signature term{ }. If instead p had the signature mysig2{(term term)⇒term, (term term term)⇒term} then p(p(a) p(a b c)) would not be a well-formed term since then p(a) would not be well-formed (in this case, p would have no arrow expression which allows p to take just one argument).

For a more complex example, let r have the signature mysig3{(term)⇒atomic, (atomic term)⇒term, (term term term)⇒term}. Then r(r(a) r(a b c)) is well-formed. The interesting twist here is that r(a) is an atomic formula that occurs as an argument to a function symbol. However, this is allowed by the arrow expression (atomic term)⇒ term, which is part of r's signature. If r's signature were mysig4{(term)⇒atomic, (atomic term)⇒atomic, (term term term)⇒term} instead, then r(r(a) r(a b c)) would be not only a well-formed term, but also a well-formed atomic formula.

An even more interesting example arises when the right-hand side of an arrow expression is something other than term or atomic. For instance, let John, Mary, NewYork, and Boston have signatures term{ }; flight and parent have signature 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.   ☐

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

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

A convention is used to avoid a syntactic ambiguity in the above definition. For instance, in (* id φ *) t[w -> v] the annotation can be attributed to the term t or to the entire frame t[w -> v]. Similarly, for an annotated HiLog-like term of the form (* id φ *) f(a)(b,c), the annotation can be attributed to the entire term f(a)(b,c) or to just f(a). The convention adopted in RIF-FLD is that any annotation is syntactically associated with the largest RIF-FLD term/formula that appears to the right of that annotation. Therefore, in our examples the annotation (* id φ *) is considered to be attached to the entire frame t[w -> v] and to the entire term f(a)(b,c).

Example 2 (A RIF-FLD document with nested groups and annotations).

We illustrate formulas, including documents and groups, with the following complete example (with apologies to Shakespeare for the imperfect rendering of the intended meaning in logic). For better readability, we use the shortcut notation defined in [RIF-DTB]. The example also illustrates attachment of annotations.

 Document(
   Prefix(dc     http://http://purl.org/dc/terms/)
   Prefix(ex     http://example.org/ontology#)
   Prefix(hamlet http://www.shakespeare-literature.com/Hamlet/)
   
   (* hamlet:assertions hamlet:assertions[dc:title->"Hamlet" dc:creator->"Shakespeare"] *)
   Group(
      Exists ?X (And(?X # ex:RottenThing
                     ex:partof(?X <http://www.denmark.dk>)))
      Forall ?X (Or(hamlet:tobe(?X)  Naf hamlet:tobe(?X)))
      Forall ?X (And(Exists ?B (And(ex:has(?X ?B) ?B # ex:business))
                     Exists ?D (And(ex:has(?X ?D) ?D # ex:desire)))
                   :- ?X # ex:man)
      (* hamlet:facts *)
      Group(
         hamlet:Yorick # ex:poor
         hamlet:Hamlet # ex:prince
      )
   )
 )

Observe that the above set of formulas has a nested subset with its own annotation, hamlet:facts, which contains only a global IRI.   ☐

• The syntax of RIF-FLD relies on the signature mechanism and is not context-free, so EBNF does not capture this syntax precisely. As a result, the EBNF grammar defines a strict superset of RIF-FLD (not all formulas that are derivable using the EBNF grammar are well-formed).
• The EBNF syntax is not a concrete syntax: it does not address the details of how constants (defined in [RIF-DTB]) and variables are represented, and it is not sufficiently precise about the delimiters and escape symbols. Instead,White space is informally used as a delimiter, and white spaceis implied in productions that use Kleene star. For instance, TERM* is to be understood as TERM TERM ... TERM, where each ' ' abstracts from one or more blanks, tabs, newlines, etc. This is done intentionally since RIF's presentation syntax is used as a tool for specifying the semantics and for illustration of the main RIF concepts through examples.
• RIF defines a concrete syntax only for exchanging rules, and that syntax is XML-based, obtained as a refinement and serialization of the EBNF syntax via the presentation-syntax-to-XML mapping for RIF-FLD.

In view of the above, the EBNF grammar can be viewed as just an intermediary between the mathematical English and the XML. However, it also gives a succinct overview of the syntax of RIF-FLD and as such can be useful for dialect designers and users alike.

  Document       ::= IRIMETA? 'Document' '(' Dialect? Base? Prefix* Import* Group? ')'
  Dialect        ::= 'Dialect' '(' Name ')'
  Base           ::= 'Base' '(' IRI ')'  
  Prefix         ::= 'Prefix' '(' Name IRI ')'
  Import         ::= IRIMETA? 'Import' '(' IRICONST PROFILE? ')'
  Group          ::= IRIMETA? 'Group' '(' (FORMULA | Group)* ')'
  Implies        ::= IRIMETA? FORMULA ':-' FORMULA
  FORMULA        ::= IRIMETA? 'And' '(' FORMULA* ')' |
                     IRIMETA? 'Or' '(' FORMULA* ')' |
                     Implies |
                     IRIMETA? 'Exists' Var* '(' FORMULA ')' |
                     IRIMETA? 'Forall' Var* '(' FORMULA ')' |
                     IRIMETA? 'Neg' FORMULA |
                     IRIMETA? 'Naf' FORMULA |
                     FORM
  FORM           ::= IRIMETA? (Var | ATOMIC | 'External' '(' ATOMIC ')')
  ATOMIC         ::= Const | Atom | Equal | Member | Subclass | Frame
  Atom           ::= UNITERM
  UNITERM        ::= TERM '(' (TERM* | (Name '->' TERM)*) ')'
  Equal          ::= TERM '=' TERM
  Member         ::= TERM '#' TERM
  Subclass       ::= TERM '##' TERM
  Frame          ::= TERM '[' (TERM '->' TERM)* ']'
  TERM           ::= IRIMETA? (Var | EXPRIC | 'External' '(' EXPRIC ')')
  EXPRIC         ::= Const | Expr | Equal | Member | Subclass | Frame
  Expr           ::= UNITERM
  Const          ::= '"' UNICODESTRING '"^^' SYMSPACE | CONSTSHORT
   IRICONST ::= '"' IRI '"^^' 'rif:iri'PROFILE        ::= TERM
  Name           ::= UNICODESTRING
  Var            ::= '?' UNICODESTRING
  SYMSPACE       ::= ANGLEBRACKIRI | CURIE
  
  IRIMETA        ::= '(*' IRICONST? (Frame | 'And' '(' Frame* ')')? '*)'

RIF-FLD formulas and terms can be prefixed with optional annotations, IRIMETA , for identification and metadata. IRIMETA is represented using (*...*)-brackets that contain an optional IRI constant as identifier followed by an optional Frame or conjunction of Frame s as metadata. An IRI has the form of an internationalized resource identifier as defined by [ RFC-3987 ].The RIF-FLD presentation syntax does not commit to any particular vocabulary and permits arbitrary Unicode strings in constant symbols, argument names, and variables. Constant symbols can have this form: "UNICODESTRING"^^SYMSPACE, where SYMSPACE is a ANGLEBRACKIRI or CURIE that represents an identifier of the symbol space of the constant, and UNICODESTRING is a Unicode string from the lexical space of that symbol space. ANGLEBRACKIRI and CURIE are defined in Section Shortcuts for Constants in RIF's Presentation Syntax of [RIF-DTB]. Constant symbols can also have several shortcut forms, which are represented by the non-terminal CONSTSHORT. These shortcuts are also defined in the same section of [RIF-DTB]. One of themof them is the CURIE shortcut, which is used in the examples in this document. Names are Unicode character sequences. Variables are composed of UNICODESTRING symbols prefixed with a ?-sign.

RIF-FLD formulas and terms can be prefixed with optional annotations, IRIMETA, for identification and metadata. IRIMETA is represented using (*...*)-brackets that contain an optional IRI constant as identifier followed by an optional Frame or conjunction of Frames as metadata. An IRICONST is the special case of a Const with the symbol space rif:iri, again permitting the shortcut forms defined in [RIF-DTB]. One such specialization is '"' IRI '"^^' 'rif:iri' from the CURIE shortcut, whichConst production, where IRI is used in the examples in this document. Names are Unicode character sequences. Variables are composeda sequence of UNICODESTRING symbols prefixed with a ?-sign.Unicode characters that forms an internationalized resource identifier as defined by [RFC-3987].

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

1. The effect of the syntax.

2. The syntax of a dialect may limit the kinds of terms that are supported.allowed. For instance, if the dialect does not supporta dialect's syntax excludes frames or terms with named arguments then the parts of the semantic structures whose purpose is to interpret the unsupportedthose types of terms (I_frame and I_NF in this case) 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 typesDatatypes.

6. A datatype is a symbol space whose symbols have a fixed interpretation in any semantic structure. RIF-FLD defines a set of core datatypes that each dialect is required to support, butinclude as part of its semanticssyntax and semantics. However, RIF-FLD does not limit supportdialects to just the core types. RIF dialectstypes: they can introduce additional datatypes, and each dialect must define the exact set of datatypes that it supports.includes.

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 well-founded or stable models [GRS91, GL88].

These notions are defined in the remainder of this document.

Semantic structures are always defined with respect to a particular set of datatypes, denoted by DTS. In a concrete dialect, DTS always includes the datatypes supported by that dialect. All RIF dialects must support the primitive datatypes that are listed in Section Data TypesDatatypes of [RIF-DTB]. Their value spaces and the lexical-to-value-space mappings fot these datatypes are described in the same section.

Although the lexical and the value spaces might sometimes look similar, one should not confuse them. Lexical spaces define the syntax of the constant symbols in the RIF language. Value spaces define the meaning of the constants. The lexical and the value spaces are often not even isomorphic. For example, 1.2^^xs:decimal and 1.20^^xs:decimal are two legal -- and distinct -- constants in RIF because 1.2 and 1.20 belong to the lexical space of xs:decimal. However, these two constants are interpreted by the same element of the value space of the xs:decimal type. Therefore, 1.2^^xs:decimal = 1.20^^xs:decimal is a RIF tautology. Likewise, RIF semantics for datatypes implies certain inequalities. For instance, abc^^xs:string ≠ abcd^^xs:string is a tautology, since the lexical-to-value-space mapping of the xs:string type maps these two constants into distinct elements in the value space of xs:string.

The central step in specifying a model-theoretic semantics for a logic-based language is defining the notion of a semantic structure , 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, I_C, I_V, I_F, I_frame, I _SFNF, I_sub, I_isa, I₌, I_external, I_truth>. Here D is a non-empty set of elements called the domain of I. We will continue to use Const to refer to the set of all constant symbols and Var to refer to the set of all variable symbols. TV denotes the set of truth values that the semantic structure uses and DTS is a set of identifiers for primitive datatypes.

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

1. I_C maps Const to elements of D.

2. This mapping interprets constant symbols.

3. I_V maps Var to elements of D.

4. This mapping interprets variable symbols.

5. I_F 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. I _SFNF interprets terms with named arguments. It is a total mapping from D 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:

   • Each pair <s,v> ∈ ArgNames × D represents an argument/value pair instead of just a value in the case of a positional term.
   • The argument to a term with named arguments is a finite bag of argument/value pairs rather than a finite ordered sequence of simple elements.
   • Bags are used here because the order of the argument/value pairs in a term with named arguments is immaterial and the pairs may repeat: p(a->b a->b).

     To see why such repetition can occur, note that argument names may repeat: p(a->b a->c). This can be understood as treating a as a set-valued argument. Identical argument/value pairs can then arise as a result of a substitution. For instance, p(a->?A a->?B) becomes p(a->b a->b) if the variables ?A and ?B are both instantiated with the symbol b.

8. I_frame is a total mapping from D to total functions of the form SetOfFiniteBags(D × D) → D.

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

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

9. I_sub 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. I_isa gives meaning to class membership. It is a total function D × D → D.

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

13. I₌ is a total function D × D → D.

    It gives meaning to the equality operator.

14. I_truth is a total mapping D → TV.

    It is used to define truth valuation for formulas.

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

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

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

For convenience, we also define the following mapping I :

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

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

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

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

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

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

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

1. If sg < sg' then D_sg⊆D_sg'.
2. If k is a constant that has signature sg then I_C(k) ∈ D_sg.
3. If ?v is a variable that has signature sg then I_V(?v) ∈ D_sg.
4. If sg has an arrow expression of the form (s1 ... sn)⇒s then, for every d∈D_sg, I_F(d) must map D_s1× ... ×D_sn to D_s.
5. If sg has an arrow expression of the form (p1->s1 ... pn->sn)⇒s then, for every d∈D_sg, I _SFNF(d) must map the set {<p1,D_s1>, ..., <pn,D_sn>} to D_s.
6. If the signature -> has arrow expressions (sg,s₁,r₁)⇒k, ..., (sg,s_n,r_n)⇒k, then, for every d∈D_sg, I_frame(d) must map {<D_s1,D_r1>, ..., <D_sn,D_rn>} to D_k.
7. If the signature # has an arrow expression (s r)⇒k then I_isa must map D_s×D_r to D_k.
8. If the signature ## has an arrow expression (s s)⇒k then I_sub must map D_s×D_s to D_k.
9. If the signature = has an arrow expression (s s)⇒k then I₌ must map D_s×D_s to D_k.

The