RIF Production Rule Dialect

W3C Working Draft 18 December 2008

This version:

http://www.w3.org/TR/2008/WD-rif-prd-20081218/

Latest version:

http://www.w3.org/TR/rif-prd/

Previous version:

http://www.w3.org/TR/2008/WD-rif-prd-20080730/

Editors:

Christian de Sainte Marie, ILOG

Adrian Paschke, Free University Berlin

Gary Hallmark, Oracle

This document is also available in these non-normative formats: PDF version.

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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May Be Superseded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 5 documents:

1. RIF Use Cases and Requirements
2. RIF Core
3. RIF Datatypes and Built-Ins 1.0
4. RIF Production Rule Dialect (this document)
5. RIF Test Cases

Summary of Changes

The document has been reorganized to expose the abstract syntax, which is intended to support features that are shared with many concrete production rule languages. Each abstract construct is associated with normative semantics and a normative XML concrete syntax.

Also, the definition of the operational semantics of rules and rulesets has been completed by adding a specification of a conflict resolution strategy. This strategy is proposed to be the default for rulesets interchanged using RIF-PRD. The WG is particularly interested in feedback regarding the operational semantics of rules and rulesets.

Please Comment By 23 January 2009

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.

In the section Operational semantics of rules and rule sets, the semantics for rules and rule sets is specified, accordingly, as a labeled terminal transition system (PLO04), where state stransitions result from executing the action part of instantiated rules. When several rules are deemed able to be executed during the rule execution process, a conflict resolution strategy is used to determine the order of rules to execute Sub-section Instance Selection specifies how an intended conflict resolution strategy can be attached to a rule set interchanged with RIF-PRD, and defines a default conflict resolution strategy.

However, as a RIF dialect, RIF-PRD has also been designed to allow interoperability between rule languages over the World Wide Web. In RIF, this is achieved by sharing the same syntax for constructs that have the same semantics across multiple dialects. As a consequence, RIF-PRD shares most of the syntax for rule conditions with RIF-BLD [RIF-BLD], and the semantics associated to the syntactic constructs used for representing the condition part of rules in RIF-PRD is specified, in Section Semantics of condition formulas, in terms of a model theory, as it is in the specification of RIF-BLD as well. In addition to exploiting similarities between the two dialects, it allows them to share the same RIF definitions for data types and built-ins [RIF-DTB].

In the section Operational semantics of actions, the semantics associated with the constructs used to represent the action part of rules in RIF-PRD is specified in terms of a transition relation between successive states of the data source, as defined by the condition formulas that they entail, thus making the link between the model-theoretic semantics of conditions and the operational semantics of rules and rulesets.

The abstract syntax is specified in mathematical english, and the abstract syntactic constructs defined in the sections Abstract Syntax of Conditions, Abstract Syntax of Actions and Abstract Syntax of Rules and Rulesets, are mapped one to one onto the concrete XML syntax in Section XML syntax. A lightweight notation is also defined along with the abstract syntax, to allow for a human-friendlier specification of the semantics. A more complete presentation syntax is specified using an EBNF in Section Presentation Syntax. However, only the XML syntax and the associated semantics are normative. A normative XML schema will also be provided in future versions of the document.

Example 1.2. In RIF-PRD presentation syntax, the first rule in example 1.1. might be represented as follows:

Prefix(ex1 http://example.com/2008/prd#)
(* ex1:rule_1 *)
Forall ?customer ?purchasesYTD (
 If      And( ?customer#ex1:Customer
                ?customer[ex1:purchasesYTD->?purchasesYTD]
                External(pred:numeric-greater-than(?purchasesYTD 5000)))
 Then ex1:Gold(?customer) )

The condition languages of RIF-PRD and RIF-BLD have much in common, including much of their semantics. Although their abstract syntax and rule semantics are different, due to the operational nature of the actions in production rules, there is a subset for which they are equivalent: essentially, rules with no negation and no uninterpreted functions in the condition, and with only assertions in the action part. For that subset, the XML syntax is the same, so many XML documents are valid in both dialects and have the same meaning. The correspondence between RIF-PRD and RIF-BLD is detailed in Appendix Compatibility with RIF-BLD.

This document is mostly intended for the designers and developers of RIF-PRD implementations, that is, applications that serialize production rules as RIF-PRD XML (producer applications) and/or that deserialize RIF-PRD XML documents into production rules (consumer applications).

Note to the reader: this section depends on Section Constants, Symbol Spaces, and Datatypes of RIF data types and builtins [RIF-DTB].

Notice that the production rule systems for which RIF-PRD aims to provide a common XML serialization use only externally specified functions, e.g. builtins. RIF-BLD specifies, in addition, a construct to denote uninterpreted function symbols, which RIF-PRD does not require: this is one of two differences between the alphabets used in the condition languages of RIF-PRD and RIF-BLD.

The second point of difference is that RIF-PRD does support a form of negation. RIF-BLD does not support negation because logic rule languages use many different kinds of negations, none of them prevalent enough to justify inclusion in the basic logic dialect of RIF (see also the RIF framework for logic dialects).

Note that atomic formulas are sometimes also called terms, e.g. in the realm of logic languages: the specification of RIF-BLD, in particular, follows that usage. The abstract syntactic elements that are called terms in this specification, are called basic terms in the specification of RIF-BLD.

• If s occurs as a function symbol in a term of the form s(...) with arity α then s occurs in the context of an external function symbol with arity α (or simply the context of a function symbol with arity α, since RIF-PRD knows only external functions);
• If s occurs as a predicate in an atomic subformula of the form s(...) with arity α then s occurs in the context of a predicate symbol with arity α;
• If s occurs as a predicate in an atomic subformula External(s(...)) with arity α then s occurs in the context of an external predicate symbol with arity α;
• If s occurs in any other context (in a frame: s[...], ...[s->...], or ...[...->s]; or in a positional/named argument term: p(...s...), q(...->s...)), it is said to occur as an individual.   ☐

Definition (Well-formed formula). A formula φ is well-formed iff:

• every constant symbol mentioned in φ occurs in exactly one context.
• Whenever a formula contains a function term t or an external atomic formula External(t), t must be an instance of the coherent set of external schemas (Section Schemas for Externally Defined Terms of RIF data types and builtins [RIF-DTB]) associated with the language of RIF-PRD.
• If t is an instance of the coherent set of external schemas associated with the language then t can occur only as a function term t or as an external atomic formula External(t), i.e., as an external term or atomic formula.   ☐

Definition (RIF-PRD condition language). The RIF-PRD condition language consists of the set of all well-formed formulas.   ☐

For compatibility with other RIF specifications (in particular, RIF data types and builtins), and to make the interoperability with RIF logic dialects (in particular RIF Core [RIF-Core] and RIF-BLD), the intended semantics for RIF-PRD condition formulas is specified in terms of a model theory.

Definition (Semantic structure). A semantic structure, I, is a tuple of the form <TV, DTS, D, D_ind, D_func, I_C, I_V, I_F, I_frame, I_NF, I_sub, I_isa, I₌, I_external, I_truth>. Here D is a non-empty set of elements called the Herbrand domain of 'I, i.e., the set of all ground terms which can be formed by using the elements of Const. D_ind, D_func are nonempty subsets of D. D_ind is used to interpret the elements of Const that are individuals and D_func is used to interpret the elements of Const that are function symbols. Const denotes the set of all constant symbols and Var the set of all variable symbols. TV denotes the set of truth values that the semantic structure uses and DTS is a set of identifiers for primitive datatypes (please refer to Section Datatypes of RIF data types and builtins [RIF-DTB] for the semantics of datatypes). The set of all ground (positional|named|frame|external) formulas which can be formed by using the function symbols with the ground terms in the Herbrand domain is the Herbrand base, H_B. A semantic structure I is a Herbrand interpretation, I_H, if the corresponding subset of H_B is the set of all ground formulas which are true with respect to I.   ☐

As far as the assignment of a standard meaning to formulas in the RIF-PRD condition language is concerned, the set TV of truth values consists of just two values, t and f.

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

1. I_C maps Const to D.

   This mapping interprets constant symbols. In addition:

   • If a constant, c ∈ Const, is an individual then it is required that I_C(c) ∈ D_ind.
   • If c ∈ Const, is a function symbol (positional or with named arguments) then it is required that I_C(c) ∈ D_func.
2. I_V maps Var to D_ind.

   This mapping interprets variable symbols.

3. I_F maps D to functions D*_ind → D (here D*_ind is a set of all sequences of any finite length over the domain D_ind).

   This mapping interprets positional terms and gives meaning to positional predicate function.

4. I_NF maps D to the set of total functions of the form SetOfFiniteSets(ArgNames × D_ind) → D.

   This mapping interprets function symbols with named arguments and gives meaning to named argument functions. This is analogous to the interpretation of positional terms with two differences:

   • Each pair <s,v> ∈ ArgNames × D_ind represents an argument/value pair instead of just a value in the case of a positional term.
   • The arguments of a term with named arguments constitute a finite set of argument/value pairs rather than a finite ordered sequence of simple elements. So, the order of the arguments does not matter.
5. I_frame maps D_ind to total functions of the form SetOfFiniteBags(D_ind × D_ind) → D.

   This mapping interprets frame terms and gives meaning to frame functions. An argument, d ∈ D_ind, to I_frame represents an object and the finite bag {<a1,v1>, ..., <ak,vk>} represents a bag of attribute-value pairs for d. I_frame is used to determine the truth valuation of frame terms.

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

6. I_sub gives meaning to the subclass relationship. It is a mapping of the form D_ind × D_ind → D.

   The operator ## is required to be transitive, i.e., c1 ## c2 and c2 ## c3 must imply c1 ## c3. TThis is ensured by a restriction in Section Interpretation of condition formulas;

7. I_isa gives meaning to class membership. It is a mapping of the form D_ind × D_ind → D.

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

8. I₌ is a mapping of the form D_ind × D_ind → D.

   It gives meaning to the equality operator.

9. I_truth is a mapping of the form D → TV.

   It is used to define truth valuation for formulas.

10. 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 external 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 built-in predicate or function, I_external(σ) is specified in [RIF-DTB] so that:

    • If σ is a schema of a built-in function then I_external(σ) must be the function defined in the aforesaid document.
    • If σ is a schema of a built-in 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 from terms to D:

• I(k) = I_C(k), if k is a symbol in Const;
• I(?v) = I_V(?v), if ?v is a variable in Var;
• I(p(t₁ ... t_n)) = I_P(I(p))(I(t₁),...,I(t_n));
• I(p(s₁->v₁ ... s_n->v_n)) = I_NF(I(p))({<s₁,I(v₁)>,...,<s_n,I(v_n)>})
  Here we use {...} to denote a set of argument/value pairs;
• I(o[a₁->v₁ ... a_k->v_k]) = I_frame(I(o))({<I(a₁),I(v₁)>, ..., <I(a_n),I(v_n)>})
  Here {...} denotes a bag of attribute/value pairs.
• I(c1##c2) = I_sub(I(c1), I(c2));
• I(o#c) = I_isa(I(o), I(c));
• I(x=y) = I₌(I(x), I(y));
• 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/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.

The effect of datatypes. The set DTS must include the datatypes described in Section Primitive Datatypes of RIF data types and builtins [RIF-DTB].

The datatype identifiers in DTS impose the following restrictions. Given dt ∈ DTS, let LS_dt denote the lexical space of dt, VS_dt denote its value space, and L_dt: LS_dt → VS_dt the lexical-to-value-space mapping (for the definitions of these concepts, see Section Primitive Datatypes of RIF data types and builtins [RIF-DTB]. Then the following must hold:

• VS_dt ⊆ D_ind; and
• For each constant "lit"^^dt such that lit ∈ LS_dt, I_C("lit"^^dt) = L_dt(lit).

That is, I_C must map the constants of a datatype dt in accordance with L_dt.

RIF-PRD does not impose restrictions on I_C for constants in symbol spaces that are not datatypes included in DTS.

• Positional atomic formulas: TVal_I(r(t₁ ... t_n)) = I_truth(I(r(t₁ ... t_n)));
• Atomic formulas with named arguments: TVal_I(p(s₁->v₁ ... s_k->v_k)) = I_truth(I(p(s₁->v₁ ... s_k->v_k)));
• Equality: TVal_I(x = y) = I_truth(I(x = y)).
  To ensure that equality has precisely the expected properties, it is required that:
  • I_truth(I(x = y)) = t if I(x) = I(y) and that I_truth(I(x = y)) = f otherwise. This is tantamount to saying that TVal_I(x = y) = t iff I(x) = I(y);
• Subclass: TVal_I(sc ## cl) = I_truth(I(sc ## cl)).
  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,   if TVal_I(c1 ## c2) = TVal_I(c2 ## c3) = t   then TVal_I(c1 ## c3) = t;
• Membership: TVal_I(o # cl) = I_truth(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,   if TVal_I(o # cl) = TVal_I(cl ## scl) = t   then   TVal_I(o # scl) = t;
• Frame: TVal_I(o[a₁->v₁ ... a_k->v_k]) = I_truth(I(o[a₁->v₁ ... a_k->v_k])).
  Since the bag of attribute/value pairs represents the conjunctions of all the pairs, the following is required, if k > 0:
  • TVal_I(o[a₁->v₁ ... a_k->v_k]) = t if and only if TVal_I(o[a₁->v₁]) = ... = TVal_I(o[a_k->v_k]) = t;
• Externally defined atomic formula: TVal_I(External(t)) = I_truth(I_external(σ)(I(s₁), ..., I(s_n))), if t is an atomic formula that is an instance of the external schema σ = (?X₁ ... ?X_n; τ) by substitution ?X₁/s₁ ... ?X_n/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;
• Conjunction: TVal_I(And(c₁ ... c_n)) = t if and only if TVal_I(c₁) = ... = TVal_I(c_n) = t. Otherwise, TVal_I(And(c₁ ... c_n)) = f.
  The empty conjunction is treated as a tautology, so TVal_I(And()) = t;
• Disjunction: TVal_I(Or(c₁ ... c_n)) = f if and only if TVal_I(c₁) = ... = TVal_I(c_n) = f. Otherwise, TVal_I(Or(c₁ ... c_n)) = t.
  The empty disjunction is treated as a contradiction, so TVal_I(Or()) = f;
• Negation: TVal_I(Not(c)) = f if and only if TVal_I(c) = t. Otherwise, TVal_I(Not(c)) = t;
• Existence: TVal_I(Exists ?v₁ ... ?v_n (φ)) = t if and only if for some I*, described below, TVal_I*(φ) = t.
  Here I* is a semantic structure of the form <TV, DTS, D, D_ind, D_pred, I_C, I*_V, I_P, I_frame, I_NP, I_sub, I_isa, I₌, I_external, I_truth>, which is exactly like I, except that the mapping I*_V, is used instead of I_V.   I*_V is defined to coincide with I_V on all variables except, possibly, on ?v₁,...,?v_n.   ☐

Let Term be the set of the terms in the RIF-PRD condition language (as defined in section Terms).

Definition (Substitution). A substitution is a finitely non-identical assignment of terms to variables; i.e., a function σ from Var to Term such that the set {x ∈ Var | x ≠ σ(x)} is finite. This set is called the domain of σ and denoted by Dom(σ). Such a substitution is also written as a set such as σ = {t_i/x_i}_i=0..n where Dom(σ) = {x_i}_i=0..n and σ(x_i) = t_i, i = 0..,n.   ☐

Definition (Ground Substitution). A ground substitution is a substitution σ that assigns only ground terms to the variables in Dom(σ): ∀ x ∈ Dom(σ), Var(σ(x)) = ∅   ☐

Notice that since RIF-PRD covers only externally defined interpreted functions, a ground term can always be replaced by a constant. In the remainder of this document, it will always be assumed that a ground substitution assigns only constants to the variables in its domain.

Definition (Matching Substitution). Let ψ be a condition formula, and φ be a set of ground formulas that satisfies ψ. We say that ψ matches φ with substitution σ : Var -> Terms if and only if there is a syntactic interpretation I such that for all ?x_i in Var(σ), I(?x_i) = I(σ(?x_i)).   ☐

Editor's Note: These actions may seem foreign to a reader who is familiar with typical production rule language actions, such as assert an object, modify/update a field of an object, and retract/remove an object. As noted in Section Atomic formulas, in this draft, objects are modeled using frame, membership, and subclass formulas that are re-used from RIF-BLD and RIF-Core. Therefore, the object-oriented actions are defined to act upon frame, membership, and subclass relations. Mappings from a typical Java-like object model to RIF-PRD maps "instanceof" to membership, "extends" and "implements" to subclass, and object properties/fields to frame slots. To assert an object requires asserting both its class membership and its frame slots. To modify a slot, e.g. change color from red to blue, requires retracting the old frame slot and asserting the new frame slot. Indeed, frame slots are multi-valued and, therefore, merely asserting a frame slot does not overwrite a prior value, it adds to the set of values. Frame slots do not inherently constrain either the datatype or the cardinality of their values. Future drafts will address the issue of object representation in RIF-PRD. We are open to suggestions.

The specification fo RIF-PRD does not assign a standard meaning to all the action blocks that can be standardized using its concrete XML syntax. Action blocks that can be meaningfully serialized are called well-formed. The notion of well-formedness, already defined for condition formulas, is extended to atomic actions, action variable declarations and action blocks.

The main restrictions are that one and only one action variable bindings can assign a value to each action variable binding, and that the assertion of a membership atomic formula is meaningful only if for a new frame object.

Definition (Well-formed atomic action). An atomic action is well-formed if and only if one of the following is true:

• it is an Assert and its target is a well-formed atom (positional or with named arguments), or a well-formed frame, membership or subclass atomic formula;
• it is a Retract and its target is a well-formed term or a well-formed atom (positional or with named arguments), or a well-formed frame atomic formula.   ☐

Definition (Well-formed action variable declaration). An action variable declaration (?v p) is well-formed if and only if one of the following is true:

• the action variable binding, p, is the declaration of a new frame object: p = New(?v), and its argument is the action variable that is declared in the same action variable declaration, ?v;
• the action variable binding, p, is a well formed frame atomic formula, p = o[a₁->t₁...a_n->t_n], n ≥ 1, and the action variable, v occurs in the position of a slot value, and nowhere else, that is: v ∉ Var(o) ∪ Var(a₁) ∪ ... ∪ Var(a_n) and v ∈ {t₁ ... t_n}.   ☐

For the definition of a well-formed action block, the function Var(f), that has been defined for condition formulas, is extended to atomic actions and frame object declarations as follows:

• if f is an atomic action with target t, then Var(f) = Var(t);
• if f is a frame object declaration, New(?v), then Var(f) = {?v}.

Definition (Well-formed action block). An action block is well-formed if and only if all of the following is true:

• all the action variable declarations, if any, are well-formed;
• each action variable, if any, is assigned a value by one and only one action binding, that is: if b₁ = (v₁ p₁) and b₂ = (v₂ p₂) are two action variable declarations in the action block, then v₂ ∉ Var(p₁) if v₁ ∈ Var(p₂), and, reciprocally, v₁ ∉ Var(p₂) if v₂ ∈ Var(p₁);
• all the actions in the action block are well-formed atomic actions;
• if an atomic action in the action block, a, asserts a membership atomic formula, a = Assert(t₁ # t₂), then the object term in the membership atomic formula, t₁, is an action variable that is declared in the action block and the action variable binding is the declaration of a new frame object.   ☐

Definition (RIF-PRD action language). The RIF-PRD action language consists of the set of all the well-formed action blocks.   ☐

Since the satisfaction of condition formulas is defined with respect to the Herbrand interpretation of ground formulas (Section Satisfaction of a condition), we will assume in the following that the states of the fact base are represented by such sets, for the purpose of specifying the intended operational semantics of atomic actions, or rules and of rule sets serialized using RIF-PRD.

Definition (RIF-PRD transition relation). The intended semantics of RIF-PRD atomic actions is completely specified by the transition relation →_RIF-PRD ⊆ W × L × W. (w, α, w') ∈ →_RIF-PRD if and only if w ∈ W, w' ∈ W, α is a ground atomic action, and one of the following is true:

1. α is Assert(φ), where φ is a ground atomic formula, and w' = w + φ;
2. α is Retract(φ), where φ is a ground atomic formula, and w' = w - φ;
3. α is Retract(o), where o is a constant, and w' = w - {o[s->v] | for all the values of terms s and v} - {o#c | for all the values of term c}.   ☐

Rule 1 says that all the condition formulas that were satisfied before an assertion will be satisfied after, and that, in addition, the condition formulas that are satisfied by the asserted ground formula will be satisfied after the assertion.

Rule 2 says that all the condition formulas that were satisfied before a retraction will be satisfied after, except if they are satisfied only by the retracted fact.

Rule 3 says that all the condition formulas that were satisfied before the removal of a frame object will be satisfied after, except if they are satisfied only by one of the frame or membership formulas about the removed object or a conjunction of such formulas.

As was already mentioned in Section Overview, production rules have an operational semantics that can be described in terms of matching rules against states of the fact base, selecting rule instances to be executed, and executing rule instances' actions to transition to new states of the fact base.

When production rules are interchanged, the intended rule instance selection strategy, often called the conflict resolution strategy, need be interchanged along with the rules : in RIF-PRD, the group is the construct that is used to group sets of rules and to associate them with a conflict resolution strategy. Many production rule systems use priorities associated with rules as part of their conflict resolution strategy: in RIF-PRD, the group is also used to carry the priority information that may be associated to the interchanged rules.

Definition (Group). If strategy is an IRI that denotes a conflict resolution strategy, if priority is an integer, and if each rg_j, 0 ≤ j ≤ n, is either a rule or a group, then any of the following is a group:

• Group rg₀ ... rg_n, n ≥ 0;
• Group strategy rg₀ ... rg_n, n ≥ 0;
• Group priority rg₀ ... rg_n, n ≥ 0;
• Group strategy priority rg₀ ... rg_n, n ≥ 0.   ☐

The function Var(f), that has been defined for condition formulas and extended to actions, is further extended to rules, as follows:

• if f is an action block that declares action variables ?v₁ ... ?v_n, n ≥ 0, and that contains actions a₁ ... a_m, m ≥ 1, then Var(f) = U_{1 ≤ i ≤ m} Var(a_i) - {?v₁ ... ?v_n};
• if f is an conditional action where c is the condition formula and a is the action, then Var(f) = Var(c) ∪ Var(a);
• if f is a quantified rule where ?v₁ ... ?v_n, n > 0, are the declared variables; p₁ ... p_m, m ≥ 0, are the binding patterns, and r is the rule, then Var(f) = (Var(r) ∪ Var(p₁) ∪ ... ∪ Var(p_m)) - {?v₁ ... ?v_n}.

Definition (Well-formed rule). A rule, r, is a well-formed rule if and only if it contains no free variable, that is, Var(r) = ∅, and either:

• it is an unconditional well-formed action block, a;
• or it is a conditional action where the condition formula, c, is a well-formed condition formula, and the action block, a, as a well-formed action block, and no atomic action in a has a subclass atomic formula as its target;
• or it is a quantified rule, Forall V (P) (r), and the quantified rule, r is a well-formed rule, and each of the declared variables in V = {?v_i}_{0 ≤ i ≤ n} is free in some of the binding patterns in P = {p_j}_{0 ≤ j ≤ m} or in the quantified rule, r; that is, V ⊆ Var(r) ∪ Var(p₁) ∪ ... ∪ Var(p_m), m ≥ 0.   ☐

Definition (Well-formed group). A well-formed group is either a group that contains only well-formed rules and well-formed groups, or a group that contains no rule or group (an empty group).   ☐

The set of the well-formed groups contains all the production rulesets that can be meaningfully interchanged using RIF-PRD.

As already mentioned in Section Overview, the description of a production rule system as a transition system can be used to specify the intended semantics that is associated with production rules and rulesets interchanged using RIF-PRD.

The intuition of describing a production rule system as a transition system is that, given a set of production rules RS and a fact base w₀, the rules in RS that are satisfied, in some sense, in w₀ determine an action a₁, whose execution results in a new fact base w₁; the rules in RS that are satisfied in w₁ determine an action a₂ to execute in w₁, and so on, until the system reaches a final state and stops. The result is the fact base w_n when the system stops.

Example 3.1. Judicael, a chicken and potato farmer, uses a rule based system to decide on the daily grain allowance for each of her chicken. Currently, Judicael's rule base contains one single rule, the chicken and mashed potatoes rule:

(* ex:ChickenAndMashedPotatoes *)
Forall ?chicken ?potato ?weight
      (And(?chicken#ex:Chicken
           (Exists ?age
                   And(?chicken[ex:age->?age]
                       External(pred:numeric-greater-than(?age, 8)))))
       And(?potato#ex:Potato          
           ex:owns(?chicken ?potato)
           (Exists ?weight
                   And(?potato[ex:weight->?weight]
                       External(pred:numeric-greater-than(?weight External(func:numeric-divide(?age 2)))))))
  If And(External(pred:string-not-equals(External(ex:today()), "Tuesday"))
         Not(External(ex:foxAlarm())))
  Then Do( (?allowance ?chicken[ex:allowance->?allowance])
           Execute(ex:mash(?potato))
           Retract(?potato)
           Retract(ex:owns(?chicken ?potato))
           Retract(?chicken[ex:allowance->?allowance])
           Assert(?chicken[ex:allowance->External(func:numeric-multiply(?allowance 1.1))]))

Judicael has four chickens, Jim, Jack, Joe and Julia, that own three potatoes (BigPotato, SmallPotato, UglyPotato) among them:

• Jim (daily grain allowance = 10) is 12 months old, Jack (daily grain allowance = 12) is 9 months old, Joe (daily grain allowance = 6) is 6 months old and Julia (daily grain allowance = 14) is 10 months old;
• BigPotato weights 70g, SmallPotato weights 10g, UglyPotato weights 50g;
• Jim owns BigPotato, Jack and Woof own SmallPotato jointly (Woof is the farm's dog. It inherited joint ownership of SmallPotato from its aunt Georgette) and Joe owns UglyPotato.

That is the initial set of facts w₀.

When the rule is applied to w₀:

• the first pattern selects {Jim/?chicken, Jack/?chicken, Julia/?chicken} as possible values for variable ?chicken (Joe is too young);
• the second pattern selects {(Jim/?chicken, BigPotato/?potato)} as the only possible substitution for the variables ?chicken and ?potato (UglyPotato does not belong to either Joe, Jack or Julia and SmallPotato is too small);

Suppose that Judicael's implementation of today() returns Monday and that the foxAlarm() is false when the Chicken and mashed potatoes rule is applied: the condition is satisfied, and the actions in the conclusion are executed with BigPotato substituted for ?potato, Jim substituted for ?chicken, and 10 substituted for ?allowance. This results in the following changes in the set of facts:

• BigPotato is mashed (and removed from the list of potatoes to be considered in future applications of the Chicken and mashed potatoes rule);
• the daily grain allowance of Jim is now 11;
• Jim does not own a potato anymore.

The resulting set of facts w₁ is thus:

• Jim (daily grain allowance = 11) is 12 months old, Jack (daily grain allowance = 12) is 9 months old, Joe (daily grain allowance = 6) is 6 months old and Julia (daily grain allowance = 14) is 10 months old;
• SmallPotato weights 10g, UglyPotato weights 50g;
• Jack and Woof own SmallPotato jointly and Joe owns UglyPotato.

When the Chicken and mashed potatoes rule in applied to w₁, the first pattern still selects {Jim/?chicken, Jack/?chicken, Julia/?chicken} as possible values for variable ?chicken, but the second pattern does not select any possible substitution for the couple (?chicken, ?potato) anymore: the rule cannot be satisfied, and the system, having detected a final state, stops.

The result of the execution of the system is w₁.   ☐

More precisely, a production rule system is defined as a labeled terminal transition system (e.g. PLO04), for the purpose of specifying the intended semantics of a RIF-PRD rule or group of rules.

Definition (labeled terminal transition system): A labeled terminal transition system is a structure {C, L, →, T}, where

• C is a set of elements, c, called configurations, or states;
• L is a set of elements, a, called labels, or actions;
• → ⊆ C × L × C is the transition relation, that is: (c, a, c' ) ∈ → iff there is a transition labeled a from the state c to the state c' or, more appropriately in the case of a production rule system, the execution of action a in the state c causes a transition to state c' ;
• T ⊆ C is the set of final states, that is, the set of all the states c from which there are no transitions: T = {c ∈ C | ∀ a ∈ L, ∀ c' ∈ C, (c, a, c') ∉ →}.   ☐

For many purposes, a representation of the states of the fact base is an appropriate representation of the states a production rule system seen as a transition system. However, the most widely used conflict resolution strategies require information about the history of the system, in particular with respect to the rule instances that have been selected for execution in previous states. Therefore, each state of the transition system used to represent a production rule system must keep a memory of the previous states and the rule instances that where selected and triggered the transition in those states.

To avoid the confusion between the states of the fact base and the states of the transition system, the latter will be called production rule system states.

Definition (Production rule system state). A production rule system state (or, simply, a system state), s, is characterized by

• a state of the fact base, facts(s);
• if s is not the initial state: a previous system state, previous(s), such that, given two system states s₁ and s₂, s₁ = previous(s₂) if and only if the production rule system the sequential execution of the action parts of the rule instances in picked(s₁) transitioned the system from system state s₁ to system state s₂;
• if s is not the current state: the ordered set of rule instances, picked(s), that the conflict resolution strategy picked, among the all the rule instances that matched facts(s).   ☐

In the following, we will write previous(s) = NIL to denote that a system state s is the initial state.

Here, a rule instance is defined as the result of the substitution of constants for all the rule variables in a rule.

Let R denote the set of all the rules in the rule language under consideration.

Definition (Rule instance). Given a rule, r ∈ R and a ground substitution, σ, such that Var(r) ⊆ Dom(σ), where Var(r) denotes the set of the rule variables in r, the result, ri = σ(r), of the substitution of the constant σ(?x) for each variable ?x ∈ Var(r) is a rule instance (or, simply, an instance) of r.  ☐

Given a rule instance ri, let rule(ri) identify the rule from which ri is derived by substitution of constants for the rule variables, and let substitution(ri) denote the substitution by which ri is derived from rule(ri).

In the following, two rule instances ri₁ and ri₂ of a same rule r will be considered different if and only if substitution(ri₁) and substitution(ri₂) substitute a different constant for at least one of the rule variables in Var(r).

In the definition of a production rule system state, a rule instance, ri, is said to match a state of a fact base, w, if its defining substitution, substitution(ri), matches the RIF-PRD condition formula that represents the condition of the instantiated rule, rule(ri), to the ground formula that represents the state of facts w.

Let W denote the set of all the possible states of a fact base.

Definition (Matching rule instance). Given a rule, ri, and a state of the fact base, w ∈ W, ri is said to match w if and only if one of the following is true:

• rule(ri) is an unconditional action block;
• rule(ri) is a conditional action block: If condition, Then action, and substitution(ri) matches the condition formula condition to the ground condition formula that represents w;
• rule(ri) is a rule with bound variables: Forall ?v₁...?v_n (p₁...p_n) (r'), n ≥ 0, m ≥ 0, and substitution(ri) matches each of the condition formulas p_i, 0 ≤ i ≤ m, to the ground condition formula that represents w, and the rule instance ri' matches w, where ri' is the instance of rule r' such that substitution(ri') = substitution(ri).   ☐

Definition (Conflict set). Given a rule set, RS ⊆ R, and a system, s, the set, conflictSet(RS, s) of all the different instances of the rules in RS that match the state of the fact base, facts(s) ∈ W is called the conflict set determined by RS in s.   ☐

In each non-final state, s, of a production rule system, a subset, picked(s), of the rule instances in the conflict set is selected and ordered; their action parts are instantiated, and they are executed. This is sometimes called: firing the selected instances.

Definition (Action instance). Given a system state, s; given a rule instance, ri, of a rule in a rule set, RS; and given the action block in the action part of the rule rule(ri): Do((v₁ p₁)...(v_n p_n) a₁...a_m), n ≥ 0, m ≥ 1, where the (v₁ p₁), 0 ≤ i ≤ n, represent the action variable declarations and the a_j, 1 ≤ j ≤ m, represent the sequence of atomic actions in the action block; if ri is a matching instance in the conflict set determined by RS in system state s: ri ∈ conflictSet(RS, s), the substitution σ = substitution(ri) is extended to the action variables v₁...v_n, n ≥ 0, in the following way:

• if v_i is assigned the identifier of a new frame by the action variable declaration: (v_i New(v_i), then σ(v_i) = c_new, where c_new is a constant of type rif:IRI that does not occur in any subformula of the ground formula that represents the state of the fact base that is associated to s, facts(s);
• if v_i is assigned the value of a frame's slot by the action variable declaration: (v_i o[s->v_i]), then σ(v_i) is a constant such that the frame formula o[s->v_i]) matches the state of the fact base facts(s) with subtitution σ.

The sequence of ground atomic actions that is the result of substituting a constant for each variable in the atomic actions of the action block of the rule instance, ri, according to the extended substitution, is the action instance associated to ri.   ☐

Let actions(ri) denote the action instance that is associated to a rule instance ri. By extension, given an ordered set of rule instances, ori, actions(ori) denotes the sequence of ground atomic actions that is the concatenation, preserving the order in ori, of the action instances associated to the rule instances in ori.

All the elements that are required to define a production rule system as a labeled terminal transition system have now been defined.

Definition (RIF-PRD Production Rule System). A RIF-PRD production rule system is defined as a labeled terminal transition system PRS = {S, A, →_PRS, T}, where :

• S is a set of system states;
• A is a set of transition labels, where each transition label is a sequence of ground RIF-PRD atomic actions;
• The transition relation →_PRS ⊆ S × A × S, is defined as follows:
  ∀ (s, a, s' ) ∈ S × A × S, (s, a, s' ) ∈ →_PRS if and only if all of the following hold:
  1. (facts(s), a, facts(s')) ∈ →^*_RIF-PRD, where →^*_RIF-PRD denotes the transitive closure of the transition relation →_RIF-PRD that is determined by the specification of the semantics of the atomic actions supported by RIF-PRD;
  2. a = actions(picked(s));
• T ⊆ S, a set of final system states.   ☐

Intuitively, the first condition in the definition of the transition relation →_PRS states that a production rule system can transition from one system state to another only if the state of facts in the latter system state can be reached from the state of facts in the former by performing a sequence of ground atomic actions supported by RIF-PRD, according to the semantics of the atomic actions.

The second condition states that the allowed paths out of any given system state are determined only by how rules instances are picked from the conflict set for execution by the conflict resolution strategy.

Therefore, the exact behavior of a RIF-PRD production rule system depends on:

1. the conflict resolution strategy, that is, how rule instances are, precisely, selected for execution from the rule instances that match a given state of the fact base;
2. and how the set T of final system states is, precisely, defined.

The process of selecting one or more rule instances from the conflict set for firing is often called: conflict resolution.

In RIF-PRD the conflict resolution algorithm (or conflict resolution strategy) that is intended for a set of rules is denoted by a keyword or a set of keywords that is attached to the rule set. In this version of the RIF-PRD specification, a single conflict resolution strategy is specified normatively: it is denoted by the keyword rif:forwardChaining (a constant of type rif:IRI), for it accounts for a common conflict resolution strategy used in most forward-chaining production rule systems.

Future versions of the RIF-PRD specification may specify normatively the intended conflict resolution strategies to be attached to additional keywords. In addition, RIF-PRD documents may include non-standard keywords: it is the responsability of the producers and consumers of such document to agree on the intended conflict resolution strategies that are denoted by such non-standard keywords.

Conflict resolution strategy: rif:forwardChaining

Most existing production rule systems implement conflict resolution algorithms that are a combination of the following elements (under these or other, idiosyncratic names; and possibly combined with additional, idiosyncratic rules):

• Refraction. The essential idea of refraction is that a given instance of a rule must not be fired more than once as long as the reasons that made it eligible for firing hold. In other terms, if an instance has been fired in a given state of the system, it is no longer eligible for firing as long as it satisfies the states of facts associated to all the subsequent system states;
• Priority. The rule instances are ordered by priority of the instantiated rules, and only the rule instances with the highest priority are eligible for firing;
• Recency. the rule instances are ordered by how long a rule instance has been continuously satisfied in the states of facts associated to previous system states, and only the most recent ones are eligible for firing.

The RIF-PRD keyword rif:forwardChaining denotes the common conflict resolution strategy that can be summarized as follows: given a conflict set

1. Refraction is applied to the conflict set, that is, all the refracted rule instances are removed from the conflict set;
2. The remaining rule instances are ordered by decreasing priority, and only the rule instances with the highest priority are kept in the conflict set;
3. The remaining rule instances are ordered by decreasing recency, and only the most recent rule instances are kept in the conflict set;
4. Any remaining tie is broken arbitrarily, and a single rule instance is kept for firing.

As specified earlier, picked(s) denotes the ordered list of the rule instances that were picked in a system state, s. Under the conflict resolution strategy denoted by rif:forwardChaining, the list denoted by picked(s) contains a single rule instance, for any given system state, s.

Given a system state, s, a rule set, RS, and a rule instance, ri ∈ conflictSet(RS, s), let recency(ri, s) denote the number of system states before s, in which ri has been continuously a matching instance: if s is the current system state, recency(ri, s) provides a measure of the recency of the rule instance ri. recency(ri, s) is specified recursively as follows:

• if previous(s) = NIL, then recency(ri, s) = 1;
• else if ri ∈ conflictSet(RS, previous(s)), then recency(ri, s) = 1 + recency(ri, previous(s));
• else, recency(ri, s) = 1.

In the same way, given an rule instance, ri, and a system state, s, let lastPicked(ri, s) denote the number of system states before s, since ri has been last fired. lastPicked(ri, s) is specified recursively as follows:

• if previous(s) = NIL, then lastPicked(ri, s) = 1;
• else if ri ∈ picked(previous(s)), then lastPicked(ri, s) = 1;
• else, lastPicked(ri, s) = 1 + lastPicked(ri, previous(s)).

Finally, given a rule instance, ri, let priority(ri) denote the priority that is associated to rule(ri), or zero, if no priority is associated to rule(ri). If rule(ri) is inside nested Groups, priority(ri) denotes the priority that is associated with the innermost Group to which a priority is explicitely associated, or zero.

Given a conflict set, cs, the conflict resolution strategy rif:forwardChaining can now be described with the help of four rules, where ri and ri' are rule instances:

1. Refraction rule: if ri ∈ cs and lastPicked(ri, s) ≤ recency(ri, s), then cs = cs - ri;
2. Priority rule: if ri ∈ cs and ri' ∈ cs and priority(ri) < priority(ri'), then cs = cs - ri;
3. Recency rule: if ri ∈ cs and ri' ∈ cs and recency(ri, s) > recency(ri', s), then cs = cs - ri;
4. Tie-break rule: if ri ∈ cs, then cs = {ri}.

The refraction rule removes the instances that have been in the conflict set in all the system states at least since they were last fired, that is, it removes the refracted instances from the current conflict set; the priority rule removes the instances such that there is at least one instance with a higher priority; the recency rule removes the instances such that there is at least one instance that is more recent; and the tie-break rule keeps one rule from the set, arbitrarily.

To select the singleton rule instance, picked(s), to be fired in a system state, s, given a rule set, RS, the conflict resolution strategy denoted by the keyword rif:forwardChaining consists in the following sequence of steps:

1. start with the conflict set, cs, that a rule set RS determines in a system state s: cs = conflictSet(RS, s);
2. apply the refraction rule to all the rule instances in cs;
3. then apply the priority rule to all the remaining instances in cs;
4. then apply the recency rule to all the remaining instances in cs;
5. then apply the tie-break rule.

Editor's Note: This section is still under discussion (see ISSUE-65). This version specifies a single, default halting test: future version of this draft may specify additional halting tests, and/or a different default. The Working Group seeks feedback on which halting tests and which combinations of tests should be supported by RIF-PRD and/or required from RIF-PRD implementations; and which halting test should be the default, if any.

By default, a system state is final, given a rule set, RS, and a conflict resolution strategy, LS, if there is no rule instance available for firing after application of the conflict resolution strategy.

For the conflict resolution strategy identified by the RIF-PRD keyword rif:forwardChaining, a system state, s, is final given a rule set, RS if and only if the remaining conflict set is empty after application of the refraction rule to all the rule instances in conflictSet(RS, s). In particular, all the system states, s, such that conflictSet(RS, s) = ∅ are final.

Syntax such as xsd:string should be understood as a compact URI (CURIE) -- a macro that expands to a concatenation of the character sequence denoted by the prefix xsd and the string string. The compact URI notation is used for brevity only, and xsd:string should be understood, in this document, as an abbreviation for http://www.w3.org/2001/XMLSchema#string.

<!-- sample pseudo-schema -->
    <defined_element
          required_attribute_of_type_string="xs:string"
          optional_attribute_of_type_int="xs:int"? >
      <required_element />
      <optional_element />?
      <one_or_more_of_these_elements />+
      [ <choice_1 /> | <choice_2 /> ]*
    </defined_element>

This section specifies the XML constructs that are used in RIF-PRD to serialize condition formulas.

The TERM class of constructs is used to serialize terms, be they simple terms, that is, constants and variables; or positional terms or terms with named arguments, both being, per the definition of a well-formed formula, representations of externally defined functions.

As an abstract class, TERM is not associated with specific XML markup in RIF-PRD instance documents.

    [ Const | Var | External ]

In RIF, the Const element is used to serialize a constant.

The Const element has a required type attribute and an optional xml:lang attribute:

• The value of the type attribute is the identifier of the Const symbol space. It must belong to the type xsd:anyURI. In the RIF data types and builtins document, the section about [DTB#Constants.2C_Symbol_Spaces.2C_and_Datatypes|Constants, Symbol spaces and Datatypes]] lists the builtin symbol spaces and data types that all implementations of RIF-PRD must support. Rule sets that are exchanged through RIF-PRD can use additional, user-defined symbol spaces;
• The xml:lang attribute, as defined by 2.12 Language Identification of XML 1.0 or its successor specifications in the W3C recommendation track, is optionally used to identify the language for the presentation of the Const to the user. It is allowed only in association with constants of the type rif:text. A compliant implementation MUST ignore the xml:lang attribute if the type of the Const is not rif:text.

The content of the Const element is the constant's litteral, which can be any Unicode character string.

    <Const type=xsd:anyURI [xml:lang=xsd:language]? >
        Any Unicode string
    </Const>

\{\{EdNote|text=The case of non-standard data types, that is, of constants that do not belong or cannot be cast in one of RIF builtin types for interchange purposes, is still under discussion in the WG. The WG seeks feedback on whether they should be allowed and why.\}\}

Example 2.1. In each of the examples below, a constant is first described, followed by its serialization in RIF-PRD XML syntax.

a. A constant with builtin type xsd:integer and value 123:

<Const type="xsd:integer">123</Const>

b. A constant which symbol today is defined in Joe the Hen Public's namespace http://example.com/2008/joe#. The type of the constant is rif:iri:

<Const type="rif:iri">
   http://example.com/2008/joe#today
</Const>

c. A constant with symbol BigPotato that is local to the set of rules where it appears (e.g. a RuleSet specific to Paula's farm). The type of the constant is rif:local:

<Const type="rif:local">BigPotato</Const>

d. A constant with non-builtin type xsd:int and value 123:

<Const type="xsd:int">123</Const>

In RIF, the Var element is used to serialize a variable.

The content of the Var element is the variable's name, serialized as an Unicode character string.

    <Var> any Unicode string </Var>

Example 2.2. The example below shows the XML serialization of a reference to a variable named: ?chicken.

<Var> chicken <Var>

As a TERM, the External element is used to serialize a positional term or a term with named arguments. In RIF-PRD, a positional or a named-argument term represents always a call to an externally specified function, e.g. a builtin, a user-defined function, a query to an external data source...

The External element contains one content element, which in turn contains one Expr element that contains one op element, followed zero or one args element or zero of more slot elements:

• The content and Expr element ensure compatibility with the RIF Basic Logic Dialect [RIF-BLD] that allows non-evaluated (that is, logic) functions to be serialized using an Expr element;
• The content of the op element must be a Const. When the External is a TERM, the content of the op element serializes a constant symbol of type rif:iri that must uniquely identify the evaluated function to be applied to the args TERMs. In the RIF data types and builtins document, the section List of RIF Builtin Predicates and Functions lists the builtin functions that all implementations of RIF-PRD must support. The content of the op element can also identify an user-defined function: it is the responsibility of the producers and consumers of RIF-PRD rulesets that reference non-builtin functions to agree on their semantics;
• The optional args element contains zero or more constructs from the TERM abstract class. The args element is used to serialize the arguments of a positional term. The order of the args sub-elements is, therefore, significant and MUST be preserved. This is emphasized by the required value "yes" of the required attribute rif:ordered;
• Each optional slot element contains one required Name sub-element, that contains an Unicode string that serializes the slot key, and a required TERM that serializes its value. The slot element is used to serialize an argument name-value pair in a term with named arguments. The order of the slot elements is, therefore, not significant;

    <External>
       <content>
          <Expr>
             <op> Const </op>
             [ <args rif:ordered="yes"> TERM* </args>?
               |
               <slot rif:ordered="yes">
                  <Name> Any Unicode string </Name>
                  TERM
               <slot>* ]
          </Expr>
       </content>
    </External>

Editor's Note: The slotted, or named arguments form of the External TERM construct is still under discussion (see also ISSUE-68). The working group seeks feedback on whether or not it should be included in PRD.

Example 2.3.

a. The first example below shows one way to serialize, in RIF-PRD, the sum of integer 1 and a variable ?x, where the addition conforms to the specification of the builtin fn:numeric-add.

The prefix fn is associated with the namespace http://www.w3.org/2007/rif-builtin-function#.

<External>
   <content>
      <Expr>    
         <op> <Const type="rif:iri"> fn:numeric-add </Const> </op>
         <args rif:ordered="yes"> 
            <Const type="xsd:integer"> 1 </Const>
            <Var> x </Var>
         </args>
      </Expr>
   </content>
</External>

b. Another example, that shows the RIF XML serialization of a call to the application-specific nullary function today(), which symbol is defined in the example's namespace http://example.com/2008/joe#:

<External>
   <content>
      <Expr>    
         <op>
            <Const type="rif:iri">
               http://example.com/2008/joe#today
            </Const>
         </op>
      </Expr>
   </content>
</External>

The ATOMIC class is used to serialize atomic formulas: positional and named-arguments atoms, equality, membership and subclass atomic formulas, frame atomic formulas and externally defined atomic formulas.

As an abstract class, ATOMIC is not associated with specific XML markup in RIF-PRD instance documents.

    [ Atom | Equal | Member | Subclass | Frame | External ]

In RIF, the Atom element is used to serialize a positional atomic formula or an atomic formula with named arguments.

The Atom element contains one op element, followed by zero or one args element or zero or more slot arguments:

• The content of the op element must be a Const. It serializes the predicate symbol (the name of a relation);
• The optional args element contains zero or more constructs from the TERM abstract class. The args element is used to serialize the arguments of a positional atomic formula. The order of the arg's sub-elements is, therefore, significant and MUST be preserved. This emphasized by the required value "yes" of the required attribute rif:ordered;
• Each optional slot element contains one required Name sub-element, that contains an Unicode string that serializes the slot key, and a required TERM that serializes its value. The slot element is used to serialize an argument name-value pair in an atomic formula with named arguments. The order of the slot elements is, therefore, not significant;

    <Atom>
       <op> Const </op>
       [ <args rif:ordered="yes"> TERM* </args>?
         |
         <slot rif:ordered="yes">
            <Name> Any Unicode string </Name>
            TERM
         <slot>* ]
    </Atom>

Editor's Note: The slotted, or named arguments form of the Atom construct is still under discussion (see also ISSUE-68). The working group seeks feedback on whether or not it should be included in PRD.

Example 2.4. The example below shows the RIF XML serialization of the positional atom owns(?c ?p), where the predicate symbol owns is defined in the example namespace http://example.com/2008/joe#.

<Atom>
   <op>
      <Const type="rif:iri">
         http://example.com/2008/joe#owns
      </Const>
   </op>
   <args rif:ordered="yes">
      <Var> c </Var>
      <Var> p </Var>
   </args>
</Atom>

In RIF, the Equal element is used to serialize equality atomic formulas.

The Equal element must contain one left sub-element and one right sub-element. The content of the left and right elements must be a construct from the TERM abstract class. The order of the sub-elements is not significant.

    <Equal>
       <left> TERM </left>
       <right> TERM </right>
    </Equal>

In RIF, the Member element is used to serialize membership atomic formulas.

The Member element contains two unordered sub-elements:

• the instance elements must be a construct from the TERM abstract class. It is required;
• the class element must be a construct from the TERM abstract class. It is required as well.

    <Member>
       <instance> TERM </instance>
       <class> TERM </class>
    </Member>

Example 2.5. The example below shows the RIF XML serialization of a boolean expression that tests whether the individual denoted by the variable ?c is a member of the class Chicken that is defined in the example namespace http://example.com/2008/joe#.

<Member>
   <instance> <Var> c </Var> </instance>
   <class>
      <Const type="rif:iri">
         http://example.com/2008/joe#Chicken
      </Const>
   </class>
</Member>

In RIF, the Subclass element is used to serialize subclass atomic formulas.

The Subclass element contains two unordered sub-elements:

• the sub element must be a construct from the TERM abstract class. It is required;
• the super elements must be a construct from the TERM abstract class. It is required.

    <Subclass>
       <sub> TERM </sub>
       <super> TERM </super>
    </Subclass>

In RIF, the Frame element is used to serialize frame atomic formulas.

Accordingly, a Frame element must contain:

• an object element, that contains an element of the TERM abstract class, the content of which serializes the individual;
• zero to many slot elements, each containing a Prop element that serializes an attribute-value pair as a pair of elements of the TERM abstract class, the first one that serializes the name of the attribute (or property); the second that serializes the attribute's value. The order of the slot's sub-elements is significant and MUST be preserved. This is emphasized by the required value "yes" of the required attribute rif:ordered.

    <Frame>
       <object> TERM </object>
       <slot rif:ordered="yes"> TERM TERM </slot>*
    </Frame>

Example 2.6. The example below shows the RIF XML syntax that serializes an expression that states that the object denoted by the variable ?c has the value denoted by the variable ?a for the property Chicken/age that is defined in the example namespace http://example.com/2008/joe#.

<Frame>
   <object> <Var> c </Var> </object>
   <slot rif:ordered="yes"> 
      <Const type="rif:iri">
         ttp://example.com/2008/joe#Chicken/age
      </Const>
      <Var> a </Var>
   </slot>
</Frame>

Editor's Note: The example uses an XPath style for the key. How externally specified data models and their elements should be referenced is still under discussion (see ISSUE-37).

In RIF-PRD, the External element is also used to serialize an externally defined atomic formula.

When it is a ATOMIC (as opposed to a TERM; that is, in particular, when it appears in a place where an ATOMIC is expected, and not a TERM), the External element contains one content element that contains one Atom element. The Atom element serializes the externally defined atom properly said:

    <External>
       <content>
          Atom
       </content>
    </External>

The op Const in the Atom element must be a symbol of type rif:iri that must uniquely identify the externally defined predicate to be applied to the args TERMs. It can be one of the builtin predicates specified for RIF dialects, as listed in section List of RIF Builtin Predicates and Functions of the RIF data types and builtins document, or it can be application specific. In the latter case, it is up to the producers and consumers of RIF-PRD rulesets that reference non-builtin predicates to agree on their semantics.

Example 2.7. The example below shows the RIF XML serialization of an externally defined atomic formula that tests whether the value denoted by the variable named ?a (e.g. the age of a chicken) is greater than the integer value 8, where the test is intended to behave like the builtin predicate op:numeric-greater-than as specified in XQuery 1.0 and XPath 2.0 Functions and Operators.

In the example, the prefix op: is associated with the namespace http://www.w3.org/2007/rif-builtin-predicate#.

<External>
   <content>
      <Atom>    
         <op> <Const type="rif:iri"> op:numeric-greater-than </Const> </op>
         <args rif:ordered="yes">
            <Var> ?a </Var>
            <Const type="xsd:decimal"> 8 </Const>
         </args>
      </Atom>
   </content>
</External>

The FORMULA class is used to serialize condition formulas, that is, atomic formulas, conjunctions, disjunctions, negations and existentials.

As an abstract class, FORMULA is not associated with specific XML markup in RIF-PRD instance documents.

    [ ATOMIC | And | Or | NmNot | Exists ]

An atomic formula is serialized using a single ATOMIC statement. See specification of ATOMIC, above.

A conjunction is serialized using the And element.

The And element contains zero or more formula sub-elements, each containing an element of the FORMULA group.

    <And>
       <formula> FORMULA </formula>*
    </And>

A disjunction is serialized using the Or element.

The Or element contains zero or more formula sub-elements, each containing an element of the FORMULA group.

    <Or>
       <formula> FORMULA </formula>*
    </Or>

A negation is serialized using the NmNot element.

The NnNot element contains exactly one formula sub-element. The formula element contains an element of the FORMULA group, that serializes the negated statement.

    <NmNot>
       <formula> FORMULA </formula>
    </NmNot>

Editor's Note: The name of that construct may change, including the tag of the XML element.

An existentially quantified formula is serialized using the Exists element.

The Exists element contains:

• one or more declare sub-elements, each containing a Var element that serializes one of the existentially quantified variables;
• exactly one required formula sub-element that contains an element from the FORMULA abstract class: the FORMULA serializes the formula in the scope of the quantifier.

    <Exists>
       <declare> Var </declare>+
       <formula> FORMULA </formula>
    </Exists>

Example 2.8. The example below shows the RIF XML serialization of a boolean expression that tests whether the chicken denoted by variable ?c is older than 8 months, by testing the existence of a value, denoted by variable ?a, that is both the age of ?c, as serialized as a Frame element, as in example 2.6, and greater than 8, as serialized as an External ATOMIC, as in example 2.7.

<Exists>
   <declare> <Var> a </Var> </declare>
   <formula>
      <And>
         <Frame>
            <object> <Var> c </Var> </object>
            <slot rif:ordered="yes"> 
               <Const type="rif:iri">
                  http://example.com/2008/joe#Chicken/age
               </Const>
               <Var> a </Var>
            </slot>
         </Frame>
         <External>
            <content>
               <Atom>    
                  <op> <Const type="rif:iri"> op:numeric-greater-than </Const> </op>
                  <args rif:ordered="yes">
                     <Var> a </Var>
                     <Const type="xsd:decimal"> 8 </Const>
                  </args>
               </Atom>
            </content>
         </External>
      </And>
   </formula>
</Exists>

    [ Assert | Retract ]

    <Assert>
         <target> [ Atom | Frame | Member | Subclass ] </target>
    </Assert>

    <Retract>
       <target> [ Atom | Frame | TERM ] </target>
    </Retract>

Example 2.10. The example below shows the RIF XML representation of an action that updates the chicken-potato ownership table by removing the predicate that states that the chicken denoted by variable ?c owns the potato denoted by variable ?p. The predicate is represented as in example 2.4.

<Retract>
   <target>
      <Atom>
         <op>
            <Const type="rif:iri">
               http://example.com/2008/joe#owns
            </Const>
         </op>
         <args rif:ordered="yes">
            <Var> c </Var>
            <Var> p </Var>
         </args>
      </Atom>
   </target>
</Retract>

    [ New | Frame ]

    <New>
       <instance> Var </instance>?
    </New>

    <Frame>
       <object> TERM </object>
       <slot rif:ordered="yes">
          TERM
          Var
       </slot>
    </Frame>

If action variables are declared in the action part of a rule, or if some atomic actions are not assertions, the conclusion must be serialized as a full action block, using the Do element. However, simple action blocks that contain only one or more assert actions can be serialized like the conclusions of logic rules using RIF-Core or RIF-BLD, that is, as a single asserted ATOMIC or as a conjunction of the asserted ATOMICs.

As an abstract class, ACTION_BLOCK is not associated with specific XML markup in RIF-PRD instance documents.

    [ Do | And | ATOMIC ]

    <Do>
       <actionVar rif:ordered="yes">
          Var
          INITIALIZATION
       </actionVar>*
       <actions rif:ordered="yes">
          ATOMIC_ACTION+
       </actions>
    </Do>

<Do>
   <actionVar>
      <Var>p</Var>
      <New>
         <instance><Var>p</Var></instance>
      </New>
   </actionVar>
   <actions  rif:ordered="yes">
      <Assert>
         <target>
            <Member>
               <instance><Var>p</Var></instance>
               <class>
                  <Const type="rif:iri">http://example.com/2008/joe#Potato</Const>
               </class>
            </Member>
         </target>
      </Assert>
      <Assert>
         <target>
            <Frame>
               <object><Var>p</Var></object>
               <slot rif:ordered="yes">
                  <Const type="rif:iri">http://example.com/2008/joe#weight</Const>
                  <Const type="xsd:decimal">100</Const>
               </slot>
            </Frame>
         </target>
      </Assert>
   </actions>
</Do>

An action block that contains only assert atomic actions can be serialized using the And element, for compatibility with RIF-Core and RIF-BLD.

However, the atomic formulas allowed as conjuncts are restricted to atoms (positional or with named arguments), frames, and membership or subclass atomic formulas.

In that position, an And element must contain at least one sub-element.

    <And>
       <formula> [ Atom | Frame | Member | Subclass ] </formula>+
    </And>

For compatibility with RIF-Core and RIF-BLD, an action block that contains only a single assert atomic action can be serialzed as the ATOMIC that serializes the target of the assert action.

However, the only atomic formulas allowed in tat position are the ones that are allowed as targets to an atomic assert action: atoms (positional or with named arguments), frames, and membership or subclass atomic formulas.

    [ Atom | Frame | Member | Subclass ]

In RIF-PRD, the RULE class of constructs is used to serialize rules, that is, unconditional as well as conditional actions, or rules with bound variables.

As an abstract class, RULE is not associated with specific XML markup in RIF-PRD instance documents.

    [ Implies | Forall | ACTION_BLOCK ]

An unconditional action block is serialized, in RIF-PRD XML, using the ACTION_BLOCK class of construct.

Conditional actions are serialized, in RIF-PRD, using the XML element Implies.

The Implies element contains an optional if sub-element and a then sub-element:

• the optional if element contains an element from the FORMULA class of constructs, that serializes the condition of the rule;
• the required then element contains one element from the ACTION_BLOCK class of constructs, that serializes its conlusion.

    <Implies>
       <if> FORMULA </if>?
       <then> ACTION_BLOCK </then>
    </Implies>

The Forall construct is used, in RIF-PRD, to represent rules with bound variables.

The Forall element contains:

• one or more declare sub-elements, each containing a Var element that represents one of the universally quantified variable;
• zero or more pattern sub-elements, each containing an element from the FORMULA group of constructs, serializing one binding pattern;
• exactly one formula sub-element that serializes the formula in the scope of the variables binding, and that contains an element of the RULE group.

    <Forall>
       <declare> Var </declare>+
       <pattern> FORMULA </pattern>*
       <formula> RULE </formula>
    </Forall>

Editor's Note: Nested Foralls make explicit the scope of the declared variables, and, thus, impose an order on the evaluation of the pattern and if FORMULAs in a rule. That ordering and the use of patterns to constrain the binding of variables may be of practical significance for some production rule systems, but they are irrelevant with respect to the intended semantics of the rules being interchanged (although they would be relevant if RIF-PRD was to be extended to support some kind of "else" or "case" construct). In addition, RIF-BLD does not allow nested Forall and does not support the association of contraining patterns to declared variables. The working group seeks feedback regarding whether nested Forall and constrainting pattern should be supported in RIF-PRD, to the cost of reducing the interoperability with RIF-BLD.

Example 2.10. The example below shows how the CMP rule extract: "if a chicken owns a potato and ..." could be serialized using a binding pattern FORMULA:

<Forall>
   <declare><Var>chicken</Var></declare>
   <formula>
      <Forall>
         <declare><Var>potato</Var></declare>
         <pattern>
            <External>
               <content>
                  <Atom>
                     <Const type="rif:iri">
                        http://example.com/2008/joe#owns
                     </Const>
                     <Args rif:ordered="yes">
                        <Var>chicken</Var>
                        <Var>potato</Var>
                     </Args>
                  </Atom>
               </content>
            </External>
         </pattern>
         <formula>
            ...
         </formula>
      </Forall>
   </formula>
</Forall>

The Group construct is used to serialize a group.

The Group element has zero or one behavior sub-element and zero or more sentence sub-elements:

• the behavior element contains
  • zero or one ConflictResolution sub-element that contains exactly one IRI. The IRI identifies the conflict resolution strategy that is associated with the Group; and
  • zero or one Priority sub-element that contains exactly one signed integer between -10,000 and 10,000. The integer associates a priority with the Group's sentences;
• a sentence element contains either a Group element or an element of the RULE abstract class of constructs.

    <Group>
       <behavior>
          <ConflictResolution> xsd:anyURI </ConflictResolution>?
          <Priority> -10,000 ≤ xsd:int ≤ 10,000 </Priority>?
       </behavior>?
       <sentence> [ RULE | Group ] </sentence>*
    </Group>

    <Document>
       <payload> Group </payload>?
    </Document>

    CLASSELT = [ TERM | ATOMIC | FORMULA | ACTION | RULE | Group | Document ]

   <CLASSELT>
       <id> Const </id>?
       <meta>
          [ Frame 
            |
            <And>
               <formula> Frame </formula>*
            </And>
          ]
       </meta>?
       other CLASSELT content
    </CLASSELT>

It is suggested to use Dublin Core, RDFS, and OWL properties for metadata, along the lines of http://www.w3.org/TR/owl-ref/#Annotations -- specifically owl:versionInfo, rdfs:label, rdfs:comment, rdfs:seeAlso, rdfs:isDefinedBy, dc:creator, dc:description, dc:date, and foaf:maker.

Example 2.11. TBC

[CIR04]

Production Systems and Rete Algorithm Formalisation, Cirstea H., Kirchner C., Moossen M., Moreau P.-E. Rapport de recherche n° inria-00280938 - version 1 (2004).

[CURIE]

CURIE Syntax 1.0 - A compact syntax for expressing URIs, W3C note 27 October 2005, M. Birbeck (ed.).

[HAK07]

Data Models as Constraint Systems: A Key to the Semantic Web, Hassan Ait-Kaci, Constraint Programming Letters, 1:33--88, 2007.

[PLO04]

A Structural Approach to Operational Semantics, Gordon D. Plotkin, Journal of Logic and Algebraic Programming, Volumes 60-61, Pages 17-139 (July - December 2004).

[PRR07]

Production Rule Representation (PRR), OMG specification, version 1.0, 2007.

[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/. Latest version available at http://www.w3.org/TR/rdf-concepts/.

[RDF-SCHEMA]

RDF Vocabulary Description Language 1.0: RDF Schema, Brian McBride, Editor, W3C Recommendation 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-schema-20040210/. Latest version available at http://www.w3.org/TR/rdf-schema/.

[RFC-3066]

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

[RFC-3987]

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

[RIF-BLD]

RIF Basic Logic Dialect Harold Boley, Michael Kifer, eds. W3C Working Draft, 30 July 2008, http://www.w3.org/TR/2008/WD-rif-bld-20080730/. Latest version available at http://www.w3.org/TR/rif-bld/.

[RIF-Core]

RIF Core Harold Boley, Gary Hallmark, Michael Kifer, Adrian Paschke, Axel Polleres, Dave Reynolds, eds. W3C Working Draft, 18 December 2008, http://www.w3.org/TR/2008/WD-rif-core-20081218/. Latest version available at http://www.w3.org/TR/rif-core/.

[RIF-DTB]

RIF Datatypes and Built-Ins 1.0 Axel Polleres, Harold Boley, Michael Kifer, eds. W3C Working Draft, 18 December 2008, http://www.w3.org/TR/2008/WD-rif-dtb-20081218/. Latest version available at http://www.w3.org/TR/rif-dtb/.

[XDM]

XQuery 1.0 and XPath 2.0 Data Model (XDM), W3C Recommendation, World Wide Web Consortium, 23 January 2007. This version is http://www.w3.org/TR/2007/REC-xpath-datamodel-20070123/. Latest version available at http://www.w3.org/TR/xpath-datamodel/.

[XML-SCHEMA2]

XML Schema Part 2: Datatypes Second Edition, W3C Recommendation, World Wide Web Consortium, 28 October 2004, http://www.w3.org/TR/2004/REC-xmlschema-2-20041028/. Latest version available at http://www.w3.org/TR/xmlschema-2/.

[XPath-Functions]

XQuery 1.0 and XPath 2.0 Functions and Operators, W3C Recommendation, World Wide Web Consortium, 23 January 2007, http://www.w3.org/TR/2007/REC-xpath-functions-20070123/. Latest version available at http://www.w3.org/TR/xpath-functions/.

Editor's Note: RIF-PRD and RIF-BLD [RIF-BLD]] share essentially the same presentation syntax and XML syntax. Future versions of this, or another, RIF document will include a complete, construct by construct, comparison table of RIF-PRD and RIF-BLD presentation and XML syntaxes.

The intended semantics of any RIF XML document which is both a syntactically valid RIF-PRD document and a syntactically valid RIF-BLD document is the same whether it is considered a RIF-PRD or a RIF-BLD document. For any input set of facts, the set of rules contained in the document must produce the same output set of facts whether it is consumed as a RIF-PRD or a RIF-BLD document.

Proof. TBC