This document, developed by the Rule Interchange Format (RIF) Working Group, specifies the production rule dialect of the W3C rule interchange format (RIF-PRD), a standard XML serialization format for production rule languages.
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 13 documents:
This document has been reviewed by W3C Members, by software developers, and by other W3C groups and interested parties, and is endorsed by the Director as a W3C Recommendation. It is a stable document and may be used as reference material or cited from another document. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.
This document specifies the production rule dialect of the W3C rule interchange format (RIF-PRD), a standard XML serialization format for production rule languages.
The production rule dialect is one of a set of rule interchange dialects that also includes the RIF Core dialect ([RIF-Core]) and the RIF basic logic dialect ([RIF-BLD]).
RIF-Core, the core dialect of the W3C rule interchange format, is designed to support the interchange of definite Horn rules without function symbols ("Datalog"). RIF-Core has both a standard first-order semantics and an operational semantics. Syntactically, RIF-Core has a number of extensions of Datalog:
RIF-Core is based on a rich set of datatypes and built-ins that are aligned with Web-aware rule system implementations [RIF-DTB]. In addition, the RIF RDF and OWL Compatibility document [RIF-RDF-OWL] specifies the syntax and semantics of combinations of RIF-Core, RDF, and OWL documents.
RIF-Core is intended to be the common core of all RIF dialects, and it has been designed,
in particular, to be a useful common subset of RIF-BLD and RIF-PRD. RIF-PRD includes and extends RIF-Core, and, therefore, RIF-PRD inherits all RIF-Core features. These features make RIF-PRD a Web-aware (even a semantic Web-aware) language. However, it should be kept in mind that RIF is designed to enable interoperability among rule languages in general, and its uses are not limited to the Web.
This document targets designers and developers of RIF-PRD implementations. A RIF-PRD implementation is a software application that serializes production rules as RIF-PRD XML (producer application) and/or that deserializes RIF-PRD XML documents into production rules (consumer application).
1.1 Production rule interchange
Production rules have an if part, or condition, and a then part, or action. The condition is like the condition part of logic rules (as covered by RIF-Core and its basic logic dialect extension, RIF-BLD). The then part contains actions. An action can assert facts, modify facts, retract facts, and have other side-effects. In general, an action is different from the conclusion of a logic rule, which contains only a logical statement. However, the conclusion of rules interchanged using RIF-Core can be interpreted, according to RIF-PRD operational semantics, as actions that assert facts in the knowledge base.
Example 1.1. The following are examples of production rules:
A customer becomes a "Gold" customer when his cumulative purchases during the current year reach $5000.
Customers that become "Gold" customers must be notified immediately, and a golden customer card will be printed and sent to them within one week.
For shopping carts worth more than $1000, "Gold" customers receive an additional discount of 10% of the total amount. ☐
Because RIF-PRD is a production rule interchange format, it specifies
an abstract syntax that shares features with concrete
production rule languages, and it associates the abstract constructs with normative semantics and a normative XML concrete syntax. Annotations (e.g. rule author) are the only constructs in RIF-PRD without a formal semantics.
The abstract syntax is specified in mathematical English, and the abstract syntactic constructs that are defined in the sections Abstract Syntax of Conditions, Abstract Syntax of Actions and Abstract Syntax of Rules and Rulesets, are mapped into the concrete XML constructs in the section XML syntax. A lightweight notation is used, instead of the XML syntax, to tie the abstract syntax to the specification of the semantics. A more complete presentation syntax is specified using an EBNF in Presentation Syntax. However, only the XML syntax and the associated semantics are normative. The normative XML schema is included in Appendix: XML Schema.
Example 1.2. In RIF-PRD presentation syntax, the first rule in example 1.1. can be represented as follows:
Production rules are statements of programming logic that specify the execution of one or more actions when their conditions are satisfied. Production rules have an operational semantics, that the OMG Production Rule Representation specification [OMG-PRR] summarizes as follows:
Match: the rules are instantiated based on the definition of the rule conditions and the current state of the data source;
Conflict resolution: a decision algorithm, often called the conflict resolution strategy, is applied to select which rule instance will be executed;
Act: the state of the data source is changed, by executing the selected rule instance's actions. If a terminal state has not been reached, the control loops back to the first step (Match).
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 transitions result from executing the action part of instantiated rules. When several rules are found to be executable at the same time, during
the rule execution process, a conflict resolution strategy is used to
select the rule to execute. The section Conflict resolution specifies how a conflict resolution strategy can be attached to a rule set. RIF-PRD defines a default conflict resolution strategy.
In the section Semantics of condition formulas, the semantics of the condition part of rules in RIF-PRD is specified operationally, in terms of matching substitutions. To emphasize the overlap between the rule conditions of RIF-BLD and RIF-PRD, and to share the same RIF definitions for datatypes and built-ins [RIF-DTB], an alternative, and equivalent, specification of the semantics of rule conditions in RIF-PRD, using a model theory, is provided in the appendix Model-theoretic semantics of RIF-PRD condition formulas.
The semantics of condition formulas and the semantics of rules and rule sets make no assumption regarding how condition formulas are evaluated. In particular, they do not require that condition formula be evaluated using pattern matching. However, RIF-PRD conformance, as defined in the section Conformance and interoperability, requires only support for safe rules, that is, forward-chaining rules where the conditions can be evaluated based on pattern matching only.
In the section Operational semantics of actions, the semantics of the action part of rules in RIF-PRD is specified using a transition relation between successive states of the data source, represented by ground condition formulas, thus making the link between the model-theoretic semantics of conditions and the operational semantics of rules and rule sets.
The abstract syntax of RIF-PRD documents, and the semantics of the combination of multiple RIF-PRD documents, is specified in the section Document and imports.
In addition to externally specified functions and predicates, and in particular, in addition to the functions and predicates built-ins defined in [RIF-DTB], RIF-PRD supports externally specified actions, and defines one action built-in, as specified in the section Built-in functions, predicates and actions.
1.2 Running example
The same example rules will be used throughout the document to illustrate the syntax and the semantics of RIF-PRD.
The rules are about the status of customers at a shop, and the discount awarded to them. The rule set contains four rules, to be applied when a customer checks out:
Gold rule: A "Silver" customer with a shopping cart worth at least $2,000 is awarded the "Gold" status.
Discount rule: "Silver" and "Gold" customers are awarded a 5% discount on the total worth of their shopping cart.
New customer and widget rule: A "New" customer who buys a widget is awarded a 10% discount on the total worth of her shopping cart, but she looses any voucher she may have been awarded.
Unknown status rule: A message must be printed, identifying any customer whose status is unknown (that is, neither "New", "Bronze", "Silver" or "Gold"), and the customer must be assigned the status: "New".
The Gold rule must be applied first; that is, e.g., a customer with "Silver" status and a shopping cart worth exactly $2,000 should be promoted to "Gold" status, before being given the 5% discount that would disallow the application of the Gold rule (since the total worth of his shopping cart would then be only $1,900).
In the remainder of this document, the prefix ex1 stands for the fictitious namespace of this example: http://example.com/2009/prd2#.
2 Conditions
This section specifies the syntax and semantics of the condition language of RIF-PRD.
The RIF-PRD condition language specification depends on Section Constants, Symbol Spaces, and Datatypes, in the RIF data types and builtins specification [RIF-DTB].
2.1 Abstract syntax
The alphabet of the RIF-PRD condition language consists of:
a countably infinite set of constant symbolsConst,
a countably infinite set of variable symbolsVar (disjoint from Const),
and syntactic constructs to denote:
lists,
function calls,
relations, including equality, class membership and subclass relations
conjunction, disjunction and negation,
and existential conditions.
For the sake of readability and simplicity, this specification introduces a notation for these constructs. The notation is not intended to be a concrete syntax, so it leaves out many details. The only concrete syntax for RIF-PRD is the XML syntax.
RIF-PRD supports externally defined functions only (including the built-in functions specified in [RIF-DTB]). RIF-PRD, unlike RIF-BLD, does not support uninterpreted function symbols (sometimes called logically defined functions).
RIF-PRD supports a form of negation. Neither RIF-Core nor RIF-BLD support negation, because logic rule languages use many different and incompatible kinds of negation. See also the RIF framework for logic dialects [RIF-FLD].
2.1.1 Terms
The most basic construct in the RIF-PRD condition language is the term. RIF-PRD defines several kinds of term: constants, variables, lists and positional terms.
Definition (Term).
Constants and variables. If t ∈ Const or t ∈ Var then t is a simple term;
List terms. A list has the form List(t1 ... tm), where m≥0 and t1, ..., tm are ground terms, i.e. without variables. A list of the form List() (i.e., a list in which m=0) is called the empty list;
Positional terms. If t ∈ Const and t1, ..., tn, n≥0, are terms then t(t1 ... tn) is a positional term. Here, the constant t represents a function and t1, ..., tn represent argument values. ☐
To emphasize interoperability with RIF-BLD, positional terms may also be written: External(t(t1...tn)).
Example 2.1.
List("New" "Bronze" "Silver" "Gold") is a term that denotes the list of the values for a customer's status that are known to the system. The elements of the list, "New", "Bronze", "Silver" and "Gold" are terms denoting string constants;
func:numeric-multiply(?value, 0.90) is a positional term that denotes the product of the value assigned to the variable ?value and the constant 0.90. That positional term can be used, for instance, to represent the new value, taking the discount into account, to be assigned a customer's shopping cart, in the rule New customer and widget rule. An alternative notation is to mark explicitly the positional term as externally defined, by wrapping it with the External indication: External(func:numeric-multiply(?value, 0.90)) ☐
2.1.2 Atomic formulas
Atomic formulas are the basic tests of the RIF-PRD condition language.
Definition (Atomic formula).
An atomic formula can have several different forms and is defined as follows:
Positional atomic formulas. If t ∈ Const and t1, ..., tn, n≥0, are terms then t(t1 ... tn) is a positional atomic formula (or simply an atom)
Equality atomic formulas. t = s is an equality atomic formula (or simply an equality), if t and s are terms
Class membership atomic formulas. t#s is a membership atomic formula (or simply membership) if t and s are terms. The term t is the object and the term s is the class
Subclass atomic formulas. t##s is a subclass atomic formula (or simply a subclass) if t and s are terms
Frame atomic formulas. t[p1->v1 ... pn->vn] is a frame atomic formula (or simply a frame) if t, p1, ..., pn, v1, ..., vn, n ≥ 0, are terms. The term t is the object of the frame; the pi are the property or attribute names; and the vi are the property or attribute values. In this document, an attribute/value pair is sometimes called a slot
Externally defined atomic formulas. If t is a positional atomic formula then External(t) is an externally defined atomic formula. ☐
Class membership, subclass, and frame atomic formulas are used to represent classifications, class hierarchies and object-attribute-value relations.
Externally defined atomic formulas are used, in particular, for representing built-in predicates.
In the RIF-BLD specification, as is common practice in logic languages,
atomic formulas are also called terms.
Example 2.2.
The membership formula ?customer # ex1:Customer tests whether the individual bound to the variable ?customer is a member of the class denoted by ex1:Customer.
The atom ex1:Gold(?customer) tests whether the customer represented by the variable ?customer has the "Gold" status.
Alternatively, gold status can be tested in a way that is closer to an object-oriented representation using the frame formula ?customer[ex1:status->"Gold"].
The following atom uses the built-in predicate pred:list-contains to validate the status of a customer against a list of allowed customer statuses: External(pred:list-contains(List("New", "Bronze", "Silver", "Gold"), ?status)). ☐
2.1.3 Formulas
Composite truth-valued constructs are called formulas, in RIF-PRD.
Note that terms (constants, variables, lists and functions) are not formulas.
More general formulas are constructed out of atomic formulas with the help of logical connectives.
Definition (Condition formula).
A condition formula can have several different forms and is defined as follows:
Atomic formula: If φ is an atomic formula then it is also a condition formula.
Conjunction: If φ1, ..., φn, n ≥ 0, are condition formulas then so is And(φ1 ... φn), called a conjunctive formula. As a special case, And() is allowed and is treated as a tautology, i.e., a formula that is always true.
Disjunction: If φ1, ..., φn, n ≥ 0, are condition formulas then so is Or(φ1 ... φn), called a disjunctive formula. As a special case, Or() is permitted and is treated as a contradiction, i.e., a formula that is always false.
Negation: If φ is a condition formula, then so is Not(φ), called a negative formula.
Existentials: If φ is a condition formula and ?V1, ..., ?Vn, n>0, are variables then Exists ?V1 ... ?Vn(φ) is an existential formula. ☐
In the definition of a formula, the component formulas φ and φi are said to be subformulas of the respective condition formulas that are built using these components.
Example 2.3.
The condition of the New customer and widget rule: A "New" customer who buys a widget, can be represented by the following RIF-PRD condition formula:
The function Var, that maps a term, an atomic formula or a condition formula to the set of its free variables is defined as follows:
if e ∈ Const, then Var(e) = ∅;
if e ∈ Var, then Var(e) = {e};
if e is a list term, then Var(e) = ∅;
if f(arg1...argn), n ≥ 0, is a positional term, then, Var(f(arg1...argn) = ∪i=1...nVar(argi);
if p(arg1...argn), n ≥ 0, is an atom, then, Var(p(arg1...argn) = Var(External(p(arg1...argn)) = ∪i=1...nVar(argi);
if t1 and t2 are terms, then Var(t1 [=|#|##] t2) = Var(t1) ∪ Var(t2);
if o', ki, i = 1...n, and vi, i = 1...n, n ≥ 1, are terms, then Var(o[k1->v1 ... kn->vn]) = Var(o) ∪i=1...nVar(ki) ∪i=1...nVar(vi).
if fi, i = 0...n, n ≥ 0, are condition formulas, then Var([And|Or](f1...fn)) = ∪i=0...nVar(fi);
if f is a condition formula, then Var(Not(f)) = Var(f);
if f is a condition formula and xi ∈ Var for i = 1...n, n ≥ 1, then, Var(Exists x1 ... xn (f)) = Var(f) - {x1...xn}.
Definition (Ground formula). A condition formula φ is a ground formula if and only if Varφ = ∅ and φ does not contain any existential subformula. ☐
In other words, a ground formula does not contain any variable term.
2.1.4 Well-formed formulas
Not all formulas are well-formed in RIF-PRD: it is required that no constant appear in more than one context. What this means precisely is explained below.
The set of all constant symbols, Const, is partitioned into the following subsets:
A subset of individuals. The symbols in Const that belong to the primitive datatypes are required to be individuals;
A subset for external function symbols;
A subset of plain predicate symbols;
A subset for external predicate symbols.
As seen from the following definitions, these subsets are not specified explicitly but, rather, are inferred from the occurrences of the symbols.
Definition (Context of a symbol).
The context of an occurrence of a symbol, s∈Const, in a formula, φ, is determined as follows:
If s occurs as a predicate in an atomic subformula of the form s(...) then s occurs in the context of a (plain) predicate symbol;
If s occurs as a predicate in an atomic subformula External(s(...)) then s occurs in the context of an external predicate symbol;
If s occurs as a function in a term (which is not a subformula) s(...) (or External(s(...))) then s occurs in the context of an (external) function symbol;
If s occurs in any other context (e.g. in a frame: s[...], ...[s->...], or ...[...->s]; or in a positional atom: p(...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 positional term, t (or External(t)), or an external atomic formula, External(t), t must be an instance of a schema in the coherent set of external schemas (Section Schemas for Externally Defined Terms in [RIF-DTB]) associated with the language of RIF-PRD;
if t is an instance of a schema in the coherent set of external schemas associated with the language then t can occur only 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. ☐
2.2 Operational semantics of condition formulas
This section specifies the semantics of the condition formulas in a RIF-PRD document.
Informally, a condition formula is evaluated with respect to a state of facts and it is satisfied, or true, if and only if:
it is an atomic condition formula and its variables are bound to individuals such that, when these constants are substituted for the variables, either
it matches a fact, or
it is implied by some background knowledge, or
it is an externally defined predicate, and its evaluation yelds true, or
it is a compound condition formula: conjunction, disjunction, negation or existential; and it is evaluated as expected, based on the truth value of its atomic components.
The semantics is specified in terms of matching substitutions in the sections below. The specification makes no assumption regarding how matching substitutions are determined. In particular, it does not require from well-formed condition formulas that they can be evaluated using pattern matching only. However, RIF-PRD requires safeness from well-formed rules, which implies that all the variables in the left-hand side can be bound by pattern matching.
For compatibility with other RIF specifications (in particular, RIF data types and built-ins [RIF-DTB] and RIF RDF and OWL compatibility [RIF-RDF-OWL]), and to make explicit the interoperability with RIF logic dialects (in particular RIF Core [RIF-Core] and RIF-BLD [RIF-BLD]), the semantics of RIF-PRD condition formulas is also specified using model theory, in appendix Model theoretic semantics of RIF-PRD condition formulas.
The two specifications are equivalent and normative.
2.2.1 Matching substitution
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 = {ti/xi}i=1..n where Dom(σ) = {xi}i=1..n and σ(xi) = ti, i = 1..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)) = ∅ ☐
Because RIF-PRD covers only externally defined interpreted functions, a ground positional term can always be replaced by the (non-positional) ground term to which it evaluates. As a consequence, a ground RIF-PRD formula can always be restricted, without loss of generality, to contain no positional term; that is, to be such that any ground positional terms have been replaced with the non-positional ground terms to which they evaluate. In the remainder of this document, it will always be assumed that a ground condition formula never contains any positional term. As a consequence, a ground substitution never assigns a ground positional term to the variables in its domain.
If t is a term or a condition formula, and if σ is a ground substitution such that Var(t) ∈ Dom(σ), σ(t) denotes the ground term or the ground condition formula obtained by substituting, in t:
σ(x) for all x ∈ Var(t), and
the externally defined results of interpreting a function with ground arguments, for all externally defined terms.
Definition (Matching substitution). Let ψ be a RIF-PRD condition formula; let σ be a ground substitution such that Var(ψ) ⊆ Dom(σ); and let Φ be a set of ground RIF-PRD atomic formulas.
We say that the ground substitution σmatchesψ to Φ if and only if one of the following is true:
ψ is an atomic formula and either
σ(ψ) ∈ Φ, or
ψ is a frame with multiple slots, o[s1->v1...sn->vn], n > 1, and there is one i, 1≤i≤n, such that σ matches the conjunction And(o[si->vi] o[s1->v1...si-1->vi-1 si+1->vi+1...sn->vn] to Φ; or
ψ is an equality formula, t1 = t2, and either
σ(t1) and σ(t2) are the same ground term;
or the ground terms σ(t1) and σ(t2) are list terms with the same length n≥0 and, for all i, 0≤i≤n-1, such that l1i and l2i are the ground terms of rank i in σ(t1) and σ(t2), respectively, either l1i and l2i are both constants in symbol spaces that are data types and they have the same value, or l1i = l2i ∈ Φ,
or the ground terms σ(t1) and σ(t2) are constants in symbol spaces that are data types and they have the same value; or
ψ is a membership formula o # c, and there is a ground term c' such that σ matches the conjunction And(o#c' c'##c) to Φ, or
ψ is an external atomic formula and the external definition maps σ(ψ) to t (or true),
ψ is Not(f) and σ does not match the condition formula f to Φ,
ψ is And(f1 ... fn) and either n = 0 or ∀ i, 1 ≤ i ≤ n, σ matches fi to Φ,
ψ is Or(f1 ... fn) and n> 0 and ∃ i, 1 ≤ i ≤ n, such that σ matches fi to Φ, or
ψ is Exists ?v1 ... ?vn (f), and there is a substitution σ' that extends σ in such a way that σ' agrees with σ where σ is defined, and Var(f) ⊆ Dom(σ'); and σ' matches f to Φ. ☐
2.2.2 Condition satisfaction
We define, now, what it means for a state of the fact base to satisfy a condition formula. The satisfaction of condition formulas in a state of the fact base provides formal underpinning to the operational semantics of rule sets interchanged using RIF-PRD.
Definition (State of the fact base). A state of the fact base, wΦ, is associated to every set of ground atomic formulas, Φ, that contains no frame with multiple slots and that satisfies all the following conditions:
for every equality formula t1 = t2 ∈ Φ, if t1 and t2 are, both, constants in symbol spaces that are data types, then they have the same value;
for every equality formula t1 = t2 ∈ Φ, either t1 is not a constant in a symbol space that is a data type, or t2 is not a list term;
for every pair of constants c1 and c2, if c1 = c2 ∈ Φ, then c2 = c1 ∈ Φ;
for every triple of constants c1, c2 and c3, if c1 = c2 ∈ Φ and c2 = c3 ∈ Φ, then c1 = c3 ∈ Φ;
for all triple of constants c1, c2, c3, if c1##c2 ∈ Φ and c2##c3 ∈ Φ, then c1##c3 ∈ Φ.
We say that wΦ is represented by Φ; or, equivalently, by the conjunction of all the ground atomic formulas in Φ. ☐
Each ground atomic formula in Φ represents a single fact, and, often, the ground atomic formulas, themselves, are called facts, as well. Notice that the restriction that Φ can contain only single slot frames, in the definition of a state of the fact base is not a limitation: given the definition of a matching substitution, a frame with multiple slots is only syntactic shorthand for the semantically equivalent conjunction of single slot frames.
Definition (Condition satisfaction). A RIF-PRD condition formula ψ is satisfied in a state of the fact base, w, if and only if w is represented by a set of ground atomic formulas Φ, and there is a ground substitution σ that matches ψ to Φ. ☐
Alternative, but equivalent, definitions of a state of the fact base and of the satisfaction of a condition are given in the appendix Model theoretic semantics of RIF-PRD condition formulas: they provide the formal link between the model theory of RIF-PRD condition formulas and the operational semantics of RIF-PRD documents.
3 Actions
This section specifies the syntax and semantics of the RIF-PRD action language. The conclusion of a production rule is often called the action part, the then part, or the right-hand side, or RHS.
The RIF-PRD action language is used to add, delete and modify facts in the fact base.
As a rule interchange format, RIF-PRD does not make any assumption regarding the nature of the data sources that the producer or the consumer of a RIF-PRD document uses (e.g. a rule engine's working memory, an external data base, etc). As a consequence, the syntax of the actions that RIF-PRD supports are defined with respect to the RIF-PRD condition formulas that represent the facts that the actions affect.
In the same way, the semantics of the actions is specified in terms of how their execution affects the evaluation of rule conditions.
3.1 Abstract syntax
The alphabet of the RIF-PRD action language includes symbols to denote:
the assertion of a fact represented by a positional atom, a frame, or a membership atomic formula,
the retraction of a fact represented by a positional atom or a frame,
the retraction of all the facts about the values of a given slot of a given frame object,
the addition of a new frame object,
the removal of a frame object and the retraction of all the facts about it, represented by the corresponding frame and class membership atomic formulas,
the replacement of all the values of an object's attribute by a single, new value,
the execution of an externally defined action, and
a sequence of these actions, including the declaration of local variables and a mechanism to bind a local variable to a frame slot value or a new frame object.
3.1.1 Actions
The RIF-PRD action language includes constructs for actions that are atomic, from a transactional point of view, and constructs that represent compounds of atomic actions. Action constructs take constructs from the RIF-PRD condition language as their arguments.
Definition (Atomic action). An atomic action is a construct that represents an atomic transaction. An atomic action can have several different forms and is defined as follows:
Assert simple fact: If φ is a positional atom, a single slot frame or a membership atomic formula in the RIF-PRD condition language, then Assert(φ) is an atomic action. φ is called the target of the action.
Retract simple fact: If φ is a positional atom or a single slot frame in the RIF-PRD condition language, then Retract(φ) is an atomic action. φ is called the target of the action.
Retract all slot values: If o and s are terms in the RIF-PRD condition language, then Retract(o s) is an atomic action. The pair (o, s) is called the target of the action.
Retract object: If t is a term in the RIF-PRD condition language, then Retract(t) is an atomic action. t is called the target of the action.
Execute: if φ is a positional atom in the RIF-PRD condition language, then Execute(φ) is an atomic action. φ is called the target of the action. ☐
Definition (Compound action). A compound action is a construct that can be replaced equivalently by a pre-defined, and fixed, sequence of atomic actions. In RIF-PRD, a compound action can have three different forms, defined as follows:
Assert compound fact: If φ is a frame with multiple slots: φ = o[s1->v1...sn->vn], n > 1; then Assert(φ) is a compound action, defined by the sequence Assert(o[s1->v1]) ... Assert(o[sn->vn]). φ is called the target of the action.
Retract compound fact: If φ is a frame with multiple slots: φ = o[s1->v1...sn->vn], n > 1; then Retract(φ) is a compound action, defined by the sequence Retract(o[s1->v1]) ... Retract(o[sn->vn]). φ is called the target of the action.
Modify fact: if φ is a frame in the RIF-PRD condition language: φ = o[s1->v1...sn->vn], n > 0; then Modify(φ) is a compound action, defined by the sequence: Retract(o s1) ... Retract(o sn), followed by Assert(φ). φ is called the target of the action. ☐
Definition (Ground action). An action with target t is a ground action if and only if
t is an atomic formula and Var(t) = ∅;
or t = (o, s) is a pair of terms and Var(o) = Var(s) = ∅.
☐
Example 3.1.
Assert( ?customer[ex1:voucher->?voucher] ) and Retract( ?customer[ex1:voucher->?voucher] ) denote two atomic actions with the frame ?customer[ex1:voucher->?voucher] as their target,
Retract( ?customer ex1:voucher ) denotes an atomic action with the pair of terms (?customer, ex1:voucher) as its target,
Modify(?customer[ex1:voucher->?voucher]) denotes a compound action with the frame ?customer[ex1:voucher->?voucher] as its target. Modify(?customer[ex1:voucher->?voucher]) can always be equivalently replaced by the sequence: Retract( ?customer ex1:voucher ) then Assert( ?customer[ex1:voucher->?voucher] );
Retract( ?voucher ) denotes an atomic action whose target is the individual bound to the variable ?voucher,
Execute( act:print("Hello, world!") ) denotes an atomic action whose target is the externally defined action act:print. ☐
3.1.2 Action blocks
The action block is the top level construct to represent the conclusions of RIF-PRD rules. An action block contains a non-empty sequence of actions. It may also include action variable declarations.
The action variable declaration construct is used to declare variables that are local to the action block, called action variables, and to assign them a value within the action block.
Definition (Action variable declaration). An action variable declaration is a pair, (v p) made of an action variable, v, and an action variable binding (or, simply, binding), p, where p has one of two forms:
frame object declaration: if the action variable, v, is to be assigned the identifier of a new frame, then the action variable binding is a frame object declaration: New(). In that case, the notation for the action variable declaration is: (?o New());
frame slot value: if the action variable, v, is to be assigned the value of a slot of a ground frame, then the action variable binding is a frame: p = o[s->v], where o is a term that represents the identifier of the ground frame and s is a term that represents the name of the slot. The associated notation is: (?value o[s->?value]). ☐
Definition (Action block). If (v1 p1), ..., (vn pn), n ≥ 0, are action variable declarations, and if a1, ..., am, m ≥ 1, are actions, then Do((v1 p1) ... (vn pn) a1 ... am) denotes an action block. ☐
Example 3.2. In the following action block, a local variable ?oldValue is bound to a value of the attribute value of the object bound to the variable ?shoppingCart. The ?oldValue is then used to compute a new value, and the Modify action is used to overwrite the old value with the new value in the fact base:
one and only one action variable binding can assign a value to each action variable, and
the assertion of a membership atomic formula is meaningful only if it is about a frame object that is created in the same action block.
The notion of well-formedness, already defined for condition formulas, is extended to actions, action variable declarations and action blocks.
Definition (Well-formed action). An actionα is well-formed if and only if one of the following is true:
α is an Assert and its target is a well-formed atom, a well-formed frame or a well-formed membership atomic formula,
α is a Retract with one single argument and its target is a well-formed term or a well-formed atom or a well-formed frame atomic formula,
α is a Retract with two arguments: o and s, and both are well-formed terms,
α is a Modify and its target is a well-formed frame, or
α is an Execute and its content is an instance of the coherent set of external schemas (Section Schemas for Externally Defined Terms in RIF data types and builtins [RIF-DTB]) associated with the RIF-PRD language (section Built-in functions, predicates and actions). ☐
Definition (Well-formed action variable declaration). An action variable declaration (?vp) 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(), or
the action variable binding, p, is a well formed frame atomic formula, p = o[a1->t1...an->tn], n ≥ 1, and the action variable, v occurs in the position of a slot value, and nowhere else, that is: v ∈ {t1 ... tn} and v ∉ Var(o) ∪ Var(a1) ∪ ... ∪ Var(an) and ∀ ti, either v = ti or v ∉ Var(ti). ☐
For the definition of a well-formed action block, the function Var(f), that has been defined for condition formulas, is extended to actions and frame object declarations as follows:
if f is an action with target t and t is an atomic formula, then Var(f) = Var(t);
if f is an action with target t and t is a pair, (o, s) of terms, then Var(f) = Var(o) ∪ Var(s);
if f is a frame object declaration, New(), then Var(f) = ∅.
Definition (Well-formed action block). An action block is well-formed if and only if all of the following are 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 variable binding, that is: if b1 = (v1 p1) and b2 = (v2 p2) are two action variable declarations in the action block with different bindings: p1 ≠ p2, then v1 ≠ v2,
in addition, the action variable declarations, if any, are partially ordered by the ordering defined as follows: if b1 = (v1 p1) and b2 = (v2 p2) are two action variable declarations in the action block, then b1 < b2 if and only if v1 ∈ Var(p2),
all the actions in the action block are well-formed actions, and
if an action in the action block asserts a membership atomic formula, 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 a frame object declaration. ☐
Definition (RIF-PRD action language). The RIF-PRD action language consists of the set of all the well-formed action blocks. ☐
3.2 Operational semantics of atomic actions
This section specifies the semantics of the atomic actions in a RIF-PRD document.
The effect of the ground atomic actions in the RIF-PRD action language is to modify the state of the fact base, in such a way that it changes the set of conditions that are satisfied before and after each atomic action is performed.
As a consequence, the semantics of the ground atomic actions in the RIF-PRD action language determines a relation, called the RIF-PRD transition relation: →RIF-PRD ⊆ W × L × W, where W denotes the set of all the states of the fact base, and where L denotes the set of all the ground atomic actions in the RIF-PRD action language.
The semantics of a compound action follows directly from the semantics of the atomic actions that compose it.
Individual states of the fact base are represented by sets of ground atomic formulas (Section Satisfaction of a condition). In the following, the operational semantics of RIF-PRD actions, rules, and rule sets is specified by describing the changes they induce in the fact base.
Definition (RIF-PRD transition relation). The semantics of RIF-PRD atomic actions is 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, where Φ is a set of ground atomic formulas that represents w and Φ' is a set of ground atomic formulas that represent w':
α is Assert(φ), where φ is a ground atomic formula, and Φ' = Φ ∪ {φ};
α is Retract(φ), where φ is a ground atomic formula, and Φ' = Φ \ {φ};
α is Retract(o s), where o and s are constants, and Φ' = (Φ \ {o[s->v] | for all the values of v});
α is Retract(o), where o is a constant, and Φ' = Φ \ {o[s->v] | for all the values of terms s and v} - {o#c | for all the values of term c};
α is Execute(φ), where φ is a ground atomic builtin action, and Φ' = Φ. ☐
Rule 1 says that all the atomic condition formulas that were satisfied before an assertion will be satisfied after, and that, in addition, the atomic condition formulas that are satisfied by the asserted ground formula will be satisfied after the assertion. No other atomic condition formula will be satisfied after the execution of the action.
Rule 2 says that all the atomic condition formulas that were satisfied before a retraction will be satisfied after, except if they are satisfied only by the retracted fact. No other atomic condition formula will be satisfied after the execution of the action.
Rule 3 says that all the condition formulas that were satisfied before the retraction of all the values of a given slot of a given object will be satisfied after, except if they are satisfied only by one of the frame formulas about the object and the slot that are the target of the action, or a conjunction of such formulas. No other condition formula will be satisfied after the execution of the action.
Rule 4 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. No other condition formula will be satisfied after the execution of the action.
Rule 5 says that all the condition formulas that were satisfied before the execution of an action builtin will be satisfied after. No other condition formula will be satisfied after the execution of the action.
Example 3.3. Assume an initial state of the fact base that is represented by the following set, w0, of ground atomic formulas, where _c1, _v1 and _s1 denote individuals and where ex1:Customer, ex1:Voucher and ex1:ShoppingCart represent classes:
Assert( _c1[ex1:status->"New"] ) denotes an atomic action that adds to the fact base, a fact that is represented by the ground atomic formula: _c1[ex1:status->"New"]. After the action is executed, the new state of the fact base is represented by
Retract( _c1[ex1:voucher->_v1] ) denotes an atomic action that removes from the fact base, the fact that is represented by the ground atomic formula _c1[ex1:voucher->_v1]. After the action, the new state of the fact base is represenetd by:
Retract( _v1 ) denotes an atomic action that removes the individual denoted by the constant _v1 from the fact base. All the class membership and the object-attribute-value facts where _v1 is the object are removed. After the action, the new state of the fact base is represenetd by:
Retract( _s1 ex1:value ) denotes an atomic action that removes all the object-attribute-value facts that assign a ex1:value to the ex1:ShoppingCart_s1. After the action, the new state of the fact base is represented by
Assert( _s1[ex1:value->450] ) adds in the fact base_the single fact that is represented by the ground frame: <tt>_s1[ex1:value->450]. After the action, the new state of the fact base is represented by:
Execute( act:print(func:concat("New customer: " _c1)) ) denotes an action that does not impact the state of the fact base, but that prints a string to an output stream. After the action, the new state of the fact base is represented by:
Notice that steps 4 and 5 can be equivalently replaced by the single compound action:
Modify( _s1[ex1:value->450] ), which denotes an action that replaces all the object-attribute-value facts that assign a ex1:value to the ex1:ShoppingCart_s1 by the single fact that is represented by the ground frame: _s1[ex1:value->450].
☐
4 Production rules and rule sets
This section specifies the syntax and semantics of RIF-PRD rules and rule sets.
combining a condition and an action block into a rule,
declaring (some) variables that are free in a rule R, specifying their bindings, and combining them with R into a new rule R' (with fewer free variables),
grouping rules and associating specific operational semantics to groups of rules.
4.1.1 Rules
Definition (Rule). A rule can be one of:
an unconditional action block,
a conditional action block: if condition is a formula in the RIF-PRD condition language, and if action is a well-formed action block, then If condition, Then action is a rule,
a rule with variable declaration: if ?v1 ... ?vn, n ≥ 1, are variables; p1 ... pm, m ≥ 1, are condition formulas (called patterns), and rule is a rule, then Forall ?v1...?vn such that (p1...pm) (rule) is a rule. ☐
Example 4.1. The Gold rule, from the running example: A "Silver" customer with a shopping cart worth at least $2,000 is awarded the "Gold" status, can be represented using the following rule with variable declaration:
Forall ?customer such that And( ?customer # ex1:Customer
?customer[ex1:status->"Silver"] )
(Forall ?shoppingCart such that And( ?shoppingCart # ex1:ShoppingCart
?customer[ex1:shoppingCart->?shoppingCart] )
(If Exists ?value (And( ?shoppingCart[ex1:value->?value]
pred:numeric-greater-than-or-equal(?value 2000))
Then Do( Modify( ?customer[ex1:status->"Gold"] ) ) )
if f is an action block that declares action variables ?v1 ... ?vn, n ≥ 0, and that contains actions a1 ... am, m ≥ 1, then Var(f) = ∪1 ≤ i ≤ m Var(ai) \ {?v1 ... ?vn};
if f is a conditional action block where c is the condition formula and a is the action block, then Var(f) = Var(c) ∪ Var(a);
if f is a quantified rule where ?v1 ... ?vn, n > 0, are the declared variables; p1 ... pm, m ≥ 0, are the patterns, and r is the rule, then Var(f) = (Var(r) ∪ Var(p1) ∪ ... ∪ Var(pm)) \ {?v1 ... ?vn}.
4.1.2 Groups
As was already mentioned in the 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, needs to be interchanged along with the rules. In RIF-PRD, the group construct 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 with the interchanged rules.
Definition (Group). A group consists of a, possibly empty, set of rules and groups, associated with a conflict resolution strategy and, a priority. If strategy is an IRI that identifies a conflict resolution strategy, if priority is an integer, and if each rgj, 0 ≤ j ≤ n, is either a rule or a group, then any of the following represents a group:
Group (rg0 ... rgn), n ≥ 0;
Group strategy (rg0 ... rgn), n ≥ 0;
Group priority (rg0 ... rgn), n ≥ 0;
Group strategy priority (rg0 ... rgn), n ≥ 0.
If a conflict resolution strategy is not explicitly attached to a group, the strategy defaults to rif:forwardChaining (specified below, in section Conflict resolution). ☐
4.1.3 Safeness
The definitions in this section are unchanged from the definitions in the section Safeness in [RIF-Core], except for the definition of RIF-PRD rule safeness, that is extended from the definition of RIF-Core rule safeness. The definitions are reproduced for the reader's convenience.
Intuitively, safeness of rules guarantees that all the variables in a rule can be bound, using pattern matching only, before they are used, in a test or in an action.
To define safeness, we need to define, first, the notion of binding patterns for externally defined functions and predicates, as well as under what conditions variables are considered bound.
Definition (Binding pattern). (from [RIF-Core]) Binding patterns for externally defined functions and predicates are lists of the form (p1, ..., pn), such that pi=b or pi=u, for 1 ≤ i ≤ n: b stands for a "bound" and u stands for an "unbound" argument. ☐
Each external function or predicate has an associated list of valid binding patterns. We define here the binding patterns valid for the functions and predicates defined in [RIF-DTB].
Every function or predicate f defined in [RIF-DTB] has a valid binding pattern for each of its schemas with only the symbol b such that its length is the number of arguments in the schema. In addition,
the external predicate pred:iri-string has the valid binding patterns (b, u) and (u, b) and
the external predicate pred:list-contains has the valid binding pattern (b, u).
The functions and predicates defined in [RIF-DTB] have no other valid binding patterns.
To keep the definitions concise and intuitive, boundedness and safeness are defined, in [RIF-Core], for condition formulas in disjunctive normal form, that can be existentially quantified themselves, but that contain, otherwise, no existential sub-formula. The definitions apply to any valid RIF-Core condition formula, because they can always, in principle, be put in that form, by applying the following syntactic transforms, in sequence:
if f contains existential sub-formulas, all the quantified variables are renamed, if necessary, and given a name that is unique in f, and the scope of the quantifiers is extended to f. Assume, for instance, that f has an existential sub-formula, sf = Exists v1...vn (sf'), n ≥ 1, such that the names v1...vn do not occur in f outside of sf. After the transform, f becomes Exists v1...vn (f'), where f' is f with sf replaced by sf' . The transform is applied iteratively to all the existential sub-formulas in f;
the (possibly existentially quantified) resulting formula is rewritten in disjunctive normal form ([Mendelson97], p. 30).
In RIF-PRD, the definitions apply to conditions formulas in the same form as in [RIF-Core], with the exception that, in the disjunctive normal form, negated sub-formulas can be atomic formulas or existential formulas: in the latter case, the existentially quantified formula must be, itself, in disjunctive normal form, and contain no further existential sub-formulas. The definitions apply to any valid RIF-PRD condition formula, because they can always, in principle, be put in that form, by applying the above syntactic transform, modified as follows to take negation into account:
if the condition formula under consideration, f, contains negative sub-formulas, existential formulas that occur inside a negated formula are handled as if they were atomic formulas, with respect to the two processing steps. Extending the scope of an existential quantifier beyond a negation would require its transformation into an universal quantifier, and universal formulas are not part of RIF-PRD condition language;
in addition, the two pre-processing steps are applied, separately, to these existentially quantified formulas, to be able to determine the status of the existentially quantified variables with respect to boundedness.
Definition (Boundedness). (from [RIF-Core]) An external term External(f(t1,...,tn)) is bound in a condition formula, if and only if f has a valid binding pattern (p1, ..., pn) and, for all j, 1 ≤ j ≤ n, such that pj=b, tj is bound in the formula.
A variable, v, is bound in an atomic formula, a, if and only if
a is neither an equality nor an external predicate, and v occurs as an argument in a;
or v is bound in the conjunctive formula f = And(a).
A variable, v, is bound in a conjunction formula, f = And(c1...cn), n ≥ 1, if and only if, either
v is bound in at least one of the conjuncts;
or v occurs as the j-th argument in a conjunct, ci, that is an externally defined predicate, and the j-th position in a binding pattern that is associated with ci is u, and all the arguments that occur, in ci, in positions with value b in the same binding pattern are bound in f' = And(c1...ci-1 ci+1...cn);
or v occurs in a conjunct, ci, that is an equality formula, and v occurs as the term on one side of the equality, and the term on the other side of the equality is bound in f' = And(c1...ci-1 ci+1...cn).
A variable, v, is bound in a disjunction formula, if and only if v is bound in every disjunct where it occurs;
A variable, v, is bound in an existential formula, Exists v1,...,vn (f'), n ≥ 1, if and only if v is bound in f'. ☐
Notice that the variables, v1,...,vn, that are existentially quantified in an existential formula f = Exists v1,...,vn (f'), are bound in any formula, F, that contains f as a sub-formula, if and only if they are bound in f, since they do not exist outside of f.
Definition (Variable safeness). (from [RIF-Core]) A variable, v, is safe in a condition formula, f, if and only if
f is an atomic formula and f is not an equality formula in which both terms are variables and v occurs in f;
or f is a conjunction, , f = And(c1...cn), n ≥ 1, and v is safe in at least one conjunct in f, or v occurs in a conjunct, ci, that is an equality formula in which both terms are variables, and v occurs as the term on one side of the equality, and the variable on the other side of the equality is safe in f' = And(c1...ci-1 ci+1...cn);
or f is a disjunction, and v is safe in every disjunct;
or f is an existential formula, f = Exists v1,...,vn (f'), n ≥ 1, and v is safe in f' . ☐
Notice that the two definitions, above, are not extended for negation and, followingly, that an universally quantified (rule) variable is never bound or safe in a condition formula as a consequence of occurring in a negative formula.
The definition of rule safeness is replaced by the following one, that extends the one for RIF-Core rules.
Definition (RIF-PRD rule safeness). A RIF-PRD rule, r, is safe if and only if
r is an unconditional action block, and Var(r) = ∅;
or r is a conditional action block, If C Then A, and all the variables in Var(A) are safe in C, and all the variables in Var(r) are bound in C;
or r is a rule with variable declaration, ∀ v1...vn such that p1...pm (r'), n ≥ 1, m ≥ 0, and either
r' is an unconditional action block, A, and the conditional action block If And(p1...pm) Then A is safe;
or r' is a conditional action block, If C Then A, and the conditional action block If And(C p1...pm) Then A is safe;
or r' is a rule with variable declaration, ∀ v'1...v'n' such that p'1...p'm' (r"), n' ≥ 1, m' ≥ 0, and the rule with variable declaration ∀ v1...vn v'1...v'n' such that p1...pm p'1...p'm' (r"), is safe. ☐
Definition (Group safeness). (from [RIF-Core]) A group, Group (s1...sn), n ≥ 0, is safe if and only if
it is empty, that is, n = 0;
or s1 and ... and sn are safe. ☐
4.1.4 Well-formed rules and groups
If f is a rule, Var(f) is the set of the free variables in f.
Definition (Well-formed rule). A rule, r, is a well-formed rule if and only if either
and the quantified rule, r', is a well-formed rule. ☐
Definition (Well-formed group). A group is well-formed group if and only if it is safe and it contains only well-formed groups, g1...gn, n ≥ 0, and well-formed rules, r1...rm, m ≥ 0, such that Var(ri) = ∅ for all i, 0 ≤ i ≤ m. ☐
The variables that are universally quantified in a rule are sometimes called rule variables in the remainder of this document, to distinguish them from the action variables and from the existentially quantified variables. The function CVar, that maps a rule to the set of its rule variables is defined as follows:
if r is a conditional or unconditional action block, CVar(r) = ∅
if r is a rule with variable declaration, Forall ?v1...?vn (r'), CVar(r) = CVar(r') ∪ {?v1...?vn}.
The set of the well-formed groups contains all the production rule sets that can be meaningfully interchanged using RIF-PRD.
4.2 Operational semantics of rules and rule sets
4.2.1 Motivation and example
As mentioned in the Overview, the description of a production rule system as a transition system is used to specify the semantics of production rules and rule sets 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 wn when the system stops.
Example 4.2. The Rif Shop, Inc. is a rif-raf retail chain, with brick and mortar shops all over the world and virtual storefronts in many on-line shops. The Rif Shop, Inc. maintains its customer fidelity management policies in the form of production rule sets. The customer management department uses RIF-PRD to publish rule sets to all the shops and licensees so that everyone uses the latest version of the rules, even though several different rule engines are in use (in fact, some of the smallest shops actually run the rules by hand).
Here is a small rule set that governs discounts and customer status updates at checkout time (to keep the example short, this is a subset of the rules described in the running example):
(* ex1:CheckoutRuleset *)
Group rif:forwardChaining (
(* ex1:GoldRule *)
Group 10 (
Forall ?customer such that (And( ?customer # ex1:Customer
?customer[ex1:status->"Silver"] ) )
(Forall ?shoppingCart such that (?customer[ex1:shoppingCart->?shoppingCart])
(If Exists ?value (And( ?shoppingCart[ex1:value->?value]
pred:numeric-greater-than-or-equal(?value 2000))
Then Do( Modify( ?customer[ex1:status->"Gold"] ) ) ) )
(* ex1:DiscountRule *)
Group (
Forall ?customer such that (And( ?customer # ex1:Customer ) )
(If Or (?customer[ex1:status->"Silver"]
?customer[ex1:status->"Gold"] )
Then Do( (?s ?customer[ex1:shoppingCart->?s])
(?v ?s[ex1:value->?v])
Modify( ?s[ex1:value->func:numeric-multiply(?v 0.95)] ) ) ) )
To see how the rule set works, consider the case of a shop where the checkout processing of customer John is about to start. The initial state of the fact base can be represented as follows:
When instantiated against w0, the first pattern in the "Gold rule", And( ?customer#ex1:Customer ?customer[ex1:status->"Silver"] ), yields the single matching substitution: {(_john/?customer)}. The second pattern in the same rule also yields a single matching substitution: {(_john/?customer)(_s1/?shoppingCart)}, for which the existential condition is satisfied.
Likewise, the instantiation of the "Discount rule" yields a single matching substitution that satisfies the condition: {(_john/?customer)}. The conflict set is: {ex1:GoldRule/{(_john/?customer)(_s1/?shoppingCart)}, ex1:DiscountRule/{(_john/?customer)}}
The instance ex1:GoldRule/{(_john/?customer)(_s1/?shoppingCart)} is selected because of its higher priority. The ground compound action: Modify(_john[ex1:status->"Gold"]), is executed, resulting in a new state of the fact base, represented as follows:
In the next cycle, there is no substitution for the rule variable ?customer that matches the pattern to the state of the fact base, and the only matching rule instance is: ex1:DiscountRule/{(_john/?customer)}, which is selected for execution. The action variables ?s and ?v are bound, based on the state of the fact base, to _s1 and 200, respectively, and the ground compound action, Modify(_s1[ex1:value->1900]), is executed, resulting in a new state of the fact base:
In w2, the only matching rule instance is, again: ex1:DiscountRule/{(_john/?customer)}. However, that same instance has already been selected and the corresponding action has been executed. Nothing has changed in the state of the fact base that would justify that the rule instance be selected gain. The principle of refraction applies, and the rule instance is removed from consideration.
This leaves the conflict set empty, and the system, having detected a final state, stops.
The result of the execution of the system is w2. ☐
4.2.2 Rules normalization
A rule, R, whose condition, rewritten in disjunctive normal form as described in section Safeness, consists of more than one disjunct, is equivalent, logically as well as operationally, to a set (or conjunction) of rules that have, all, the same conclusion as R, and each rule has one of the disjuncts as its condition: the rule R: If C1 Or ... Or Cn Then A is equivalent to the set of rules {ri=0..n| ri: If Ci Then A}.
Without loss of generality, and to keep the specification as simple and intuitive as possible, the operational semantics of production rules and rule sets is specified, in the following sections, for rules and rule sets that have been normalized as follows:
All the rules are rewritten in disjunctive normal form as described in section Safeness;
that contains as many rules as the condition of the original rule in disjunctive normal form contains disjuncts,
where the condition, in each rule in the group is one of the disjunct in the condition of the original rule,
and where all the rules in the group have a different condition and the same action part as the original rule.
In the same way, without loss of generality, and to keep the specification as simple and intuitive as possible, the operational semantics of production rules and rule sets is specified, in the following sections, for rules and rule sets where all the compound actions have been replaced by the equivalent sequences of atomic actions.
4.2.3 Definitions and notational conventions
Formally, a production rule system is defined as a labeled terminal transition system (e.g. PLO04), for the purpose of specifying the 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' . In the case of a production rule system: in the state c of the fact base, the execution of action a causes a transition to state the c' of the fact base;
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 of 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 of the rule instances that where selected and that triggered the transition in those states.
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 CVar(r) ⊆ Dom(σ), where CVar(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∈ CVar(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 CVar(r).
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 set of ground atomic formulas 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 instance, ri, and a state of the fact base, w ∈ W, ri is said to matchw if and only if one of the following is true:
rule(ri) is a conditional action block: If condition, Then action, and substitution(ri)matches the condition formula condition to the set of ground atomic condition formulas that represents w;
rule(ri) is a rule with variable declaration: Forall ?v1...?vn (p1...pm) (r'), n ≥ 0, m ≥ 0, and substitution(ri)matches each of the condition formulas pi, 0 ≤ i ≤ m, to the set of ground atomic condition formulas that represents w, and the rule instance ri' matches w, where rule(ri') = r' and substitution(ri') = substitution(ri). ☐
Definition (Action instance). Given a state of the fact base, w ∈ W, 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)...(vn pn) a1...am), n ≥ 0, m ≥ 1, where the (vi pi), 0 ≤ i ≤ n, represent the action variable declarations and the aj, 1 ≤ j ≤ m, represent the sequence of atomic actions in the action block; if ri matches w, the substitution σ = substitution(ri) is extended to the action variables v1...vn, n ≥ 0, in the following way:
if the binding, pi, associated to vi, in the action variable declaration, is the declaration of a new frame object: (viNew()), then σ(vi) = cnew, where cnew is a constant of type rif:IRI that does not occur in any of the ground atomic formulas in w;
if vi is assigned the value of a frame's slot by the action variable declaration: (vio[s->vi]), then σ(vi) is a ground term such that the substitution σ matches the frame formula o[s->vi] to w.
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.
Notice that RIF-PRD does not specify semantics for the case where there is no matching substitution for the binding frame formula o[s->vi] in an action variable declaration (vi o[s->vi]). Indeed, although the rule might be valid from an interchange viewpoint, applying it in a context where object o has no value for
attribute s is applying it outside the domain where it is meaningful, and the specification of the context where an otherwise valid RIF-PRD rule is validly applicable is out of the scope of RIF-PRD.
The components of the states of a production rule system seen as a transition system can now be defined more precisely. To avoid 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) is either a system cycle state or a system transitional state. Every production rule system state, s, cycle or transitional, is characterized by
a state of the fact base, facts(s);
if s is not the current state, an ordered set of rule instances, picked(s), defined as follows:
if s is a system cycle state, picked(s) is the ordered set of rule instances picked by the conflict resolution strategy, among the set of all the rule instances that matched facts(s);
if s is a system transitional state, picked(s) is the empty set;
if s is not the initial state, a previous system state, previous(s), defined as follows: given a system cycle state, sc, and given the sequence of system transitional states, s1,...,sn, n ≥ 0, such that the execution of the first ground atomic action in action(picked(sc)) transitioned the system from sc to s1 and ... and the n-th ground atomic action in action(picked(sc)) transitioned the system from sn-1 to sn, then previous(s) = sn if and only if the (n+1)-th ground atomic action in action(picked(sc)) transitioned the system from sn to s. ☐
In the following, we will write previous(s) = NIL to denote that a system state s is the initial state.
Definition (Conflict set). Given a rule set, RS ⊆ R, and a system state, s, the conflict set determined by RS in s is 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. ☐
The rule instances that are in the conflict set are, sometimes, said to be fireable.
In each non-final cycle 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 the resulting sequence of ground atomic actions is executed. This is sometimes called: firing the selected instances.
4.2.4 Operational semantics of a production rule system
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 :
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:
(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;
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 cycle state to another only if the state of facts in the latter system cycle 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 cycle state are determined only by how rule instances are picked for execution, from the conflict set, by the conflict resolution strategy.
Given a rule set RS ⊆ R, the associated conflict resolution strategy, LS, and halting test, H, and an initial state of the fact base, w ∈ W, the input function to a RIF-PRD production rule system is defined as:
Eval(RS, LS, H, w) →PRS s ∈ S, such that facts(s) = w and previous(s) = NIL.
Using →*PRS to denote the transitive closure of the transition relation →PRS, there are zero, one or more final states of the system, s' ∈ T, such that:
Eval(RS, LS, H, w) →*PRS s'.
The execution of a rule set, RS, in a state, w, of a fact base, may result in zero, one or more final state of the fact base, w' = facts(s'), depending on the conflict resolution strategy and the set of final system states.
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, and
how the set T of final system states is precisely defined.
4.2.5 Conflict resolution
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), because it accounts for a common conflict resolution strategy used in most forward-chaining production rule systems. That conflict resolution strategy selects a single rule instance for execution.
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 responsibility of the producers and consumers of such document to agree on the intended conflict resolution strategies that are denoted by such non-standard keywords. Future or non-standard conflict resolution strategies may select an ordered set of rule instances for execution, instead of a single one: the functions picked and actions, in the previous section, have been defined to take this case into account.
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 (cycle and transitional);
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 the number of consecutive system states, cycle and transitional, in which they have been in the conflict set, and only the most recently fireable ones are eligible for firing. Note that the recency rule, used alone, results in depth-first processing.
Many existing production rule systems implement also some kind of fire the most specific rule first strategy, in combination with the above. However, whereas they agree on the definition of refraction and the priority or recency ordering, existing production rule systems vary widely on the precise definition of the specificity ordering. As a consequence, rule instance specificity was not included in the basic conflict resolution strategy that RIF-PRD specifies normatively.
The RIF-PRD keyword rif:forwardChaining denotes the common conflict resolution strategy that can be summarized as follows:
given a conflict set
Refraction is applied to the conflict set, that is, all the refracted rule instances are removed from further consideration;
The remaining rule instances are ordered by decreasing priority, and only the rule instances with the highest priority are kept for further consideration;
The remaining rule instances are ordered by decreasing recency, and only the most recent rule instances are kept for further consideration;
Any remaining tie is broken is some way, 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, for any given system cycle state, s, the list denoted by picked(s) contains a single rule instance. By definition, if s is a system transitional state, picked(s) is the empty set.
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 a 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;
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 explicitly associated, or zero.
Example 4.3. Consider the following RIF-PRD document:
No conflict resolution strategy is identified explicitly, so the default strategy rif:forwardChaining is used.
Because the ex2:ExampleRuleSet group does not specify a priority, the default priority 0 is used. Rule 1, not being in any other group, inherits its priority, 0, from the top-level group.
Rule 2 inherits its priority, 10, from the enclosing group, identified as ex2:HighPriorityRules. Rule 3 specifies its own, lower, priority: 9.
Since neither Rule 4 nor Rule 5 specify a priority, they inherit their priority from the enclosing group ex2:NoPriorityRules, which does not specify one either, and, thus, they inherit 0 from the top-level group, ex2:ExampleRuleSet. ☐
Given a set of rule instances, cs, the conflict resolution strategy rif:forwardChaining can now be described with the help of four rules, where ri and ri' are rule instances:
Refraction rule: if ri ∈ cs and lastPicked(ri, s) < recency(ri, s), then cs = cs - ri;
Priority rule: if ri ∈ cs and ri' ∈ cs and priority(ri) < priority(ri'), then cs = cs - ri;
Recency rule: if ri ∈ cs and ri' ∈ cs and recency(ri, s) > recency(ri', s), then cs = cs - ri;
Tie-break rule: if ri ∈ cs, then cs = {ri}. RIF-PRD does not specify the tie-break rule more precisely: how a single instance is selected from the remaining set is implementation specific.
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; 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.
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 of the following sequence of steps:
initialize picked(s) with the conflict set, that a rule set RS determines in a system state s: picked(s) = conflictSet(RS, s);
apply the refraction rule to all the rule instances in picked(s);
then apply the priority rule to all the remaining instances in picked(s);
then apply the recency rule to all the remaining instances in picked(s);
then apply the tie-break rule to the remaing instance in picked(s);
return picked(s).
Example 4.4. Consider, from example 4.2, the conflict set that the rule set ex1:CheckoutRuleset determines in the system state, s2, that corresponds to the state w2 = facts(s2) of the fact base, and use it to initialize the set of rule instance considered for firing, picked(s2):
The single rule instance in the conflict set, ri = ex1:DiscountRule/{(_john/?customer)}, did already belong to the conflict sets in the two previous states, conflictSet(ex1:CheckoutRuleset, s1) and conflictSet(ex1:CheckoutRuleset, s0); so that its recency in s2 is: recency(ri, s2) = 3.
On the other hand, that rule instance was fired in system state s1: picked(s1) = (ex1:DiscountRule/{(_john/?customer)}); so that, in s2, it has been last fired one cycle before: lastPicked(ri, s2) = 1.
Therefore, lastPicked(ri, s2) < recency(ri, s2), and ri is removed from picked(s2) by refraction, leaving picked(s2) empty. ☐
4.2.6 Halting test
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.
5 Document and imports
This section specifies the structure of a RIF-PRD document and its semantics when it includes import directives.
5.1 Abstract syntax
In addition to the language of conditions, actions, and rules, RIF-PRD provides a construct to denote the import of a RIF or non-RIF document. Import enables the modular interchange of RIF documents, and the interchange of combinations of multiple RIF and non-RIF documents.
5.1.1 Import directive
Definition (Import directive). An import directive consists of:
an IRI, the locator, that identifies and locates the document to be imported, and
an optional second IRI that identifies the profile of the import. ☐
RIF-PRD gives meaning to one-argument import directives only. Such directives can be used to import other RIF-PRD and RIF-Core documents. Two-argument import directives are provided to enable import of other types of documents, and their semantics is covered by other specifications. For example, the syntax and semantics of the import of RDF and OWL documents, and their combination with a RIF document, is specified in [RIF-RDF-OWL].
5.1.2 RIF-PRD document
Definition (RIF-PRD document). A RIF-PRD document consists of zero or more import directives, and zero or one group. ☐
Definition (Imported document). A document is said to be directly imported by a RIF document, D, if and only if it is identified by the locator IRI in an import directive in D. A document is said to be imported by a RIF document, D, if it is directly imported by D, or if it is imported, directly or not, by a RIF document that is directly imported by D. ☐
Definition (Document safeness). (from [RIF-Core]) A document is safe if and only if it
it contains a safe group, or no group at all,
and all the documents that it imports are safe. ☐
5.1.3 Well-formed documents
Definition (Conflict resolution strategy associated with a document). A conflict resolution strategy is associated with a RIF-PRD document, D, if and only if
it is explicitly or implicitly attached to the top-level group in D, or
it is explicitly or implicitly attached to the top-level group in a RIF-PRD document that is imported by D. ☐
Definition (Well-formed RIF-PRD document). A RIF-PRD document, D, is well-formed if and only if it satisfies all the following conditions:
the locator IRI provided by all the import directives in D, if any, identify well-formed RIF-PRD documents,
D contains a well-formed group or no group at all,
D has only one associated conflict resolution strategy (that is, all the conflict resolution strategies that can be associated with it are the same), and
every non-rif:local constant that occurs in D or in one of the documents imported by D, occurs in the same context in D and in all the documents imported by D. ☐
The last condition in the above definition makes the intent behind the rif:local constants clear: occurrences of such constants in different documents can be interpreted differently even if they have the same name. Therefore, each document can choose the names for the rif:local constants freely and without regard to the names of such constants used in the imported documents.
In addition to externally specified functions and predicates, and in particular, in addition to the functions and predicates built-ins defined in [RIF-DTB], RIF-PRD supports externally specified actions, and defines action built-ins.
The syntax and semantics of action built-ins are specified like for the other buit-ins, as described in the section Syntax and Semantics of Built-ins in [RIF-DTB]. However, their formal semantics is trivial: action built-ins behave like predicates that are always true, since action built-ins, in RIF-PRD, MUST NOT affect the semantics of the rules.
Although they must not affect the semantics of the rules, action built-ins may have other side effects.
RIF action built-ins are defined in the namespace: http://www.w3.org/2007/rif-builtin-action#. In this document, we will use the prefix: act: to denote the RIF action built-ins namespace.
6.1 Built-in actions
6.1.1 act:print
Schema:
(?arg; act:print(?arg))
Domains:
The value space of the single argument is xs:string.
Mapping:
When s belongs to its domain, Itruth ο IExternal( ?arg; act:print(?arg) )(s) = t.
If an argument value is outside of its domain, the truth value of the function is left unspecified.
Side effects:
The value of the argument MUST be printed to an output stream, to be determined by the user implementation.
7 Conformance and interoperability
7.1 Semantics-preserving transformations
RIF-PRD conformance is described partially in terms of semantics-preserving transformations.
The intuitive idea is that, for any initial state of facts, the conformant consumer of a conformant RIF-PRD document must reach at least one of the final state of facts intended by the conformant producer of the document, and that it must never reach any final state of facts that was not intended by the producer. That is:
a conformant RIF-PRD producer, P, must translate any rule set from its own rule language, LP, into RIF-PRD, in such a way that, for any possible initial state of the fact base, the RIF-PRD translation of the rule set must never produce, according to the semantics specified in this document, a final state of the fact base that would not be a possible result of the execution of the rule set according to the semantics of LP (where the state of the facts base are meant to be represented in LP or in RIF-PRD as appropriate), and
a conformant RIF-PRD consumer, C, must translate any rule set from a RIF-PRD document into a rule set in its own language, LC, in such a way that, for any possible initial state of the fact base, the translation in LC of the rule set, must never produce, according to the semantics of LC, a final state of the fact base that would not be a possible result of the execution of the rule set according to the semantics specified in this document (where the state of the facts base are meant to be represented in LC or in RIF-PRD as appropriate).
Let Τ be a set of datatypes and symbol spaces that includes the datatypes specified in [RIF-DTB] and the symbol spaces rif:iri and rif:local. Suppose also that Ε is a set of external predicates and functions that includes the built-ins listed in [RIF-DTB] and in the section Built-in actions. We say that a rule r is a RIF-PRDΤ,Ε rule if and only if
all the datatypes and symbol spaces used in r are in Τ, and
all the externally defined functions and predicates used in r are in Ε.
Suppose, further, that C is a set of conflict resolution strategies that includes the one specified in section Conflict resolution, and that H is a set of halting tests that includes the one specified in section Halting test: we say that a rule set , R, is a RIF-PRDΤ,Ε,C,H rule set if and only if
R contains only RIF-PRDΤ,Ε rules,
the conflict resolution strategy that is associated to R is in C, and
the halting test that is associated to R is in H.
Given a RIF-PRDΤ,Ε,C,H rule set, R, an initial state of the fact base, w, a conflict resolution strategy c ∈ C and a halting test h ∈ H, let FR,w,c,h denote the set of all the sets, f, of RIF-PRD ground atomic formulas that represent final states of the fact base, w' , according to the operational semantics of a RIF-PRD production rule system, that is: f ∈ FR,w,c,h if and only if there is a state, s' , of the system, such that Eval(R, c, h, w) →*PRS s' and w' = facts(s') and f is a representation of w' .
In addition, given a rule language, L, a rule set expressed in L, RL, a conflict resolution strategy, c, a halting test, h, and an initial state of the fact base, w, let FL,RL, c, h, w denote the set of all the formulas in L that represent a final state of the fact base that an L processor can possibly reach.
Definition (Semantics preserving mapping).
A mapping from a RIF-PRDΤ,Ε,C,H, R, to a rule set, RL, expressed in a language L, is semantics-preserving if and only if, for any initial state of the fact base, w, conflict resolution strategy, c, and halting test, h, it also maps each L formula in FL,RL, c, h, w onto a set of RIF-PRD ground formulas in FR,w,c,h;
A mapping from a rule set, RL, expressed in a language L, to a RIF-PRDΤ,Ε,C,H, R, is semantics-preserving if an only if, for any initial state of the fact base, w, conflict resolution strategy, c, and halting test, h, it also maps each set of ground RIF-PRD atomic formulas in FR,w,c,h onto an L formula in FL,RL, c, h, w. ☐
7.2 Conformance Clauses
Definition (RIF-PRD conformance).
A RIF processor is a conformantRIF-PRDΤ,Ε,C,Hconsumer iff it implements a semantics-preserving mapping from the set of all safeRIF-PRDΤ,Ε,C,H rule sets to the language L of the processor;
A RIF processor is a conformantRIF-PRDΤ,Ε,C,Hproducer iff it implements a semantics-preserving mapping from a subset of the language L of the processor to a set of safeRIF-PRDΤ,Ε,C,H rule sets;
An admissible document is an XML document that conforms to all the syntactic constraints of RIF-PRD, including ones that cannot be checked by an XML Schema validator;
A conformant RIF-PRD consumer is a conformant RIF-PRDΤ,Ε,C,H consumer in which Τ consists only of the symbol spaces and datatypes, Ε consists only of the externally defined functions and predicates, C consists only of the conflict resolution strategies, and H consists only of halting tests that are required by RIF-PRD. The required symbol spaces are rif:iri and rif:local, and the datatypes and externally defined terms (built-ins) are the ones specified in [RIF-DTB] and in the section Built-in actions. The required conflict resolution strategy is the one that is identified as rif:forwardChaining, as specified in section Conflict resolution; and the required halting test is the one specified in section Halting test. A conformant RIF-PRD consumer must reject any document containing features it does not support.
A conformant RIF-PRD producer is a conformant RIF-PRDΤ,Ε,C,H producer which produces documents that include only the symbol spaces, datatypes, externals, conflict resolution strategies and halting tests that are required by RIF-PRD. ☐
In addition, conformant RIF-PRD producers and consumers SHOULD preserve annotations.
7.3 Interoperability
[RIF-Core] is specified as a specialization of RIF-PRD: all valid [RIF-Core] documents are valid RIF-PRD documents and must be accepted by any conformant RIF-PRD consumer.
Conversely, it is desirable that any valid RIF-PRD document that uses only abstract syntax that is defined in [RIF-Core] be a valid [RIF-Core] document as well. For some abstract constructs that are defined in both RIF-Core and RIF-PRD, RIF-PRD defines alternative XML syntax that is not valid RIF-Core XML syntax. For example, an action block that contains no action variable declaration and only assert atomic actions can be expressed in RIF-PRD using the XML elements Do or And. Only the latter option is valid RIF-Core XML syntax.
To maximize interoperability with RIF-Core and its non-RIF-PRD extensions, a conformant RIF-PRD consumer SHOULD produce valid [RIF-Core] documents whenever possible. Specifically, a conformant RIF-PRD producer SHOULD use only valid [RIF-Core] XML syntax to serialize a rule set that satisfies all of the following:
the conflict resolution strategy is effectively equivalent to the stratagy that RIF-PRD identifies by the IRI rif:forwardChaining,
no condition formula contains a negation, in any rule in the rule set,
When processing a rule set that satisfies all the above conditions, a RIF-PRD producer is guaranteed to produce a valid [RIF-Core] XML document by applying the following rules recursively:
Remove redundant information. The behavior role element and all its sub-elements should be omitted in the RIF-PRD XML document;
Remove nested rule variable declarations. If the rule inside a rule with variable delcaration, r1, is also a rule with variable declaration, r2, all the rule variable delarations and all the patterns that occur in r1 should be added to the rule variable declarations and the patterns that occur in r2, and, after the transform, r1 should be replaced by r2, in the rule set. If the names of some variables declared in r1 are the same as the names of some variables declared in r2, the former names must be changed prior to the transform.;
if the rule inside r1 is a conditional action block, r2, the formula that represents the condition in r2 should be replaced by the conjunction of that formula and the formula that represents the pattern, and the pattern should be removed from r1;
Convert action blocks. The action block, in each rule, should be replaced by a conjunction, and, inside the conjunction, each assert action should be replaced by its target atomic formula.
Example 7.1. Consider the following rule, R, derived from the Gold rule, in the running example, to have only assertions in the action part:
R: Forall ?customer such that (And( ?customer # ex1:Customer
?customer[status->"Silver"] ) )
(Forall ?shoppingCart such that (?customer[shoppingCart->?shoppingCart])
(If Exists ?value (And( ?shoppingCart[value->?value]
pred:numeric-greater-than-or-equal(?value 2000))
Then Do( Assert(ex1:Foo(?customer))
Assert(ex1:Bar(?shoppingCart)) ) ) )
The serialization of R in the following RIF-Core conformant XML form does not impacts its semantics (see example 8.12 for another valid RIF-PRD XML serialization, that is not RIF-Core conformant):
This section specifies the concrete XML syntax of RIF-PRD. The concrete syntax is derived from the abstract syntax defined in sections 2.1, 3.1 and 4.1 using simple mappings. The semantics of the concrete syntax is the same as the semantics of the abstract syntax.
8.1 Notational conventions
8.1.1 Namespaces
Throughout this document, the xsd: prefix stands for the XML Schema namespace URI http://www.w3.org/2001/XMLSchema#, the rdf: prefix stands for http://www.w3.org/1999/02/22-rdf-syntax-ns#, and rif: stands for the URI of the RIF namespace, http://www.w3.org/2007/rif#.
Syntax such as xsd:string should be understood as a compact URI [CURIE] -- a macro that expands to a concatenation of the character sequence denoted by the prefix xsd and the string string. The 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.
8.1.2 BNF pseudo-schemas
The XML syntax of RIF-PRD is specified for each component as a pseudo-schema, as part of the description of the component. The pseudo-schemas use BNF-style conventions for attributes and elements: "?" denotes optionality (i.e. zero or one occurrences), "*" denotes zero or more occurrences, "+" one or more occurrences, "[" and "]" are used to form groups, and "|" represents choice. Attributes are conventionally assigned a value which corresponds to their type, as defined in the normative schema. Elements are conventionally assigned a value which is the name of the syntactic class of their content, as defined in the normative schema.
Three kinds of syntactic components are used to specify RIF-PRD:
Abstract classes are defined only by their subclasses: they are not visible in the XML markup and can be thought of as extension points. In this document, abstract constructs will be denoted with all-uppercase names;
Concrete classes have a concrete definition, and they are associated with specific XML markup. In this document, concrete constructs will be denoted with CamelCase names with leading capital letter;
Properties, or roles, define how two classes relate to each other. They have concrete definitions and are associated with specific XML markup. In this document, properties will be denoted with camelCase names with leading smallcase letter.
8.2 Relative IRIs and XML base
Relative IRIs are allowed in RIF-PRD XML syntax, anywhere IRIs are allowed, including constant types, symbol spaces, location, and profile. The attribute xml:base [XML-Base] is used to make them absolute.
8.3 Conditions
This section specifies the XML constructs that are used in RIF-PRD to serialize condition formulas.
8.3.1 TERM
The TERM class of constructs is used to serialize terms, be they simple terms, that is, constants and variables; lists; or positional terms, the latter 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 | List | External ]
8.3.1.1 Const
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 be a rif:iri;
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 rdf:plainLiteral. A compliant implementation MUST ignore the xml:lang attribute if the type of the Const is not rdf:plainLiteral.
The content of the Const element is the constant's literal, which can be any Unicode character string.
<Const type=rif:iri [xml:lang=xsd:language]? >
Any Unicode string
</Const>
Example 8.1.
a. A constant with built-in type xsd:integer and value 2,000:
<Const type="xsd:integer">2000</Const>
b. The Customer class, in the running example, is identified by a constant of type rif:iri, in the namespace http://example.com/2009/prd2#:
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 NCName.
<Var> xsd:NCName </Var>
8.3.1.3 List
In RIF, the List element is used to serialize a list.
The List element contains an optional items element, that contains one or more TERMs (without variables) that serialize the elements of the list. The order of the sub-elements is significant and MUST be preserved. This is emphasized by the fixed value "yes" of the mandatory attribute ordered in the items element.
As a TERM, the External element is used to serialize a positional term. In RIF-PRD, a positional term represents always a call to an externally defined function, e.g. a built-in, a user-defined function, a query to an external data source, etc.
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:
The External and the content elements 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 alone;
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 externally defined function to be applied to the args TERMs;
The optional args element contains one 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 attribute ordered.
Example 8.3. The example shows one way to serialize, in RIF-PRD, the product of a variable named ?value and the xsd:decimal value 0.9, where the operation conforms to the specification of the built-in func:numeric-multiply, as specified in [RIF-DTB].
The ATOMIC class is used to serialize atomic formulas: positional 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 ]
The Atom element contains one op element, followed by zero or one args element:
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 one 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 is emphasized by the required value "yes" of the attribute ordered.
Example 8.4. The example shows the RIF XML serialization of the positional atom ex1:gold(?customer), where the predicate symbol gold is defined in the example namespace http://example.com/2009/prd2#.
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, that serialize the terms of the equality. The order of the sub-elements is not significant.
<Equal>
<left> TERM </left>
<right> TERM </right>
</Equal>
The Member element contains two required sub-elements:
the instance elements must be a construct from the TERM abstract class that serializes the reference to the object;
the class element must be a construct from the TERM abstract class that serializes the reference to the class.
<Member>
<instance> TERM </instance>
<class> TERM </class>
</Member>
Example 8.5. The example shows the RIF XML serialization of class membership atom that tests whether a variable named ?customer belongs to a class identified by the name Customer in the namespace http://example.com/2009/prd2#
an object element, that contains an element of the TERM abstract class that serializes the reference to the frame's object;
zero to many slot elements, each serializing 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 ordered.
<Frame>
<object> TERM </object>
<slot ordered="yes"> TERMTERM </slot>*
</Frame>
Example 8.6. The example shows the RIF XML serialization of an expression that states that the object denoted by the variable ?customer has the value denoted by the string "Gold" for the property identified by the symbol status that is defined in the example namespace http://example.com/2009/prd2#
When it is an 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.
The opConst 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.
<External>
<content>
Atom
</content>
</External>
Example 8.7. The example below shows the RIF XML serialization of an externally defined atomic formula that tests whether the value denoted by the variable named ?value is greater than or equal to the integer value 2000, where the test is intended to behave like the built-in predicate pred:numeric-greater-than-or-equal as specified in [RIF-DTB]:
The Or element contains zero or more formula sub-elements, each containing an element of the FORMULA group, that serializes one of the disjuncts.
<Or>
<formula> FORMULA </formula>*
</Or>
8.3.3.4 INeg
The kind of negation that is used in RIF-PRD is serialized using the INeg element.
The INeg element contains exactly one formula sub-element. The formula element contains an element of the FORMULA group, that serializes the negated statement.
one or more declare sub-elements, each containing one 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, that serializes the formula in the scope of the quantifier.
<Exists>
<declare> Var </declare>+
<formula> FORMULA </formula>
</Exists>
Example 8.8. The example shows the RIF XML serialization of a condition on the existence of a value greater than or equal to 2.000, in the Gold rule of the running example, as represented in example 4.2.
The Retract element has one target sub-element that contains an Atom, a Frame, a TERM, or a pair of TERM constructs that represent the target of the action. The target element has an optional attribute, ordered, that MUST be present when the element contains two TERM sub-elements: the order of the sub-elements is significant and MUST be preserved. This is emphasized by the required value "yes" of the attribute.
<Retract>
<target ordered="yes"?>
[ Atom | Frame | TERM | TERM TERM ]
</target>
</Retract>
The Modify element has one target sub-element that contains one Frame that represents the target of the action.
<Modify>
<target> Frame </target>
</Modify>
Example 8.9. The example shows the RIF XML representation of the action that updates the status of a customer, in the Gold rule, in the running example, as represented in example 4.2: Modify(?customer[status->"Gold"])
The Execute element has one target sub-element that contains an Atom, that represents the externally defined action to be executed.
The opConst in the Atom element must be a symbol of type rif:iri that must uniquely identify the externally defined action to be applied to the args TERMs.
<Execute>
<target> Atom </target>
</Execute>
Example 8.10. The example shows the RIF XML serialization of the message printing action, in the Unknonw status rule, in the running example, using the act:print action built-in.
The namespace for RIF-PRD action built-ins is http://www.w3.org/2007/rif-builtin-action#.
If action variables are declared in the action part of a rule, or if some 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 SHOULD be serialized like the conclusions of logic rules using RIF-Core or RIF-BLD, that is, as a single asserted Atom or Frame, or as a conjunction of the asserted facts, using the And element.
As an abstract class, ACTION_BLOCK is not associated with specific XML markup in RIF-PRD instance documents.
[ Do | And | Atom | Frame ]
8.4.2.1 New
The New element is used to serialize the construct used to create a new frame identifer, in an action variable declaration.
The New element is always empty.
<New />
8.4.2.2 Do
An action block is serialized using the Do element.
A Do element contains:
zero or more actionVar sub-elements, each of them used to serialize one action variable declaration. Accordingly, an actionVar element must contain a Var sub-element, that serializes the declared variable; followed by the serialization of an action variable binding, that assigns an initial value to the declared variable, that is: either a frame or the empty element New;
one actions sub-element that serializes the sequence of actions in the action block, and that contains, accordingly, a sequence of one or more sub-elements of the ACTION class. The order of the actions is significant, and the order MUST be preserved, as emphasized by the required ordered="yes" attribute.
<Do>
<actionVar ordered="yes">
Var
[ New | Frame ]
</actionVar>*
<actions ordered="yes">
ACTION+
</actions>
</Do>
Example 8.11. The example shows the RIF XML serialization of an action block that asserts that a customer gets a new $5 voucher.
This section specifies the XML constructs that are used, in RIR-PRD, to serialize rules and groups.
8.5.1 RULE
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.
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;
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.
The Import directive is used to serialize the reference to an RDF graph, an OWL ontology or another RIF document to be combined with a RIF document. The Import directive is inherited from [RIF-Core]. The abstract syntax and semantics of RDF graph and OWL ontology imports are specified in [RIF-RDF-OWL].
The Import directive contains:
exactly one location sub-element, that contains an IRI, that serializes the location of the RDF or OWL document to be combined with the RIF document;
zero or one profile sub-element, that contains an IRI. The admitted values for that constant and their semantics are listed in the section Profiles of Imports, in [RIF-RDF-OWL].
The Document is the root element of any RIF-PRD instance document.
The Document contains zero or more directive sub-elements, each containing an Import directive, and zero or one payload sub-element, that must contain a Group element.
<Document>
<directive> Import </directive>*
<payload> Group </payload>?
</Document>
The semantics of a document that imports RDF and/or OWL documents is specified in [RIF-RDF-OWL] and [RIF-BLD]. The semantics of a document that does not import other documents is the semantics of the rule set that is serialised by the Group in the document's payload sub-element, if any.
8.7 Constructs carrying no semantics
8.7.1 Annotation
Annotations can be associated with any concrete class element in RIF-PRD: those are the elements with a CamelCase tagname starting with an upper-case character:
CLASSELT = [ TERM | ATOMIC | FORMULA | ACTION | ACTION_BLOCK | New | RULE | Group | Document | Import ]
An identifier can be associated to any instance element of the abstract CLASSELT class of constructs, as an optional id sub-element that MUST contain a Const of type rif:iri.
Annotations can be included in any instance of a concrete class element using the meta sub-element.
The Frame construct is used to serialize annotations: the content of the Frame's object sub-element identifies the object to which the annotation is associated:, and the Frame's slots represent the annotation properly said as property-value pairs.
If all the annotations are related to the same object, the meta element can contain a single Frame sub-element. If annotations related to several different objects need be serialized, the meta role element can contain an And element with zero or more formula sub-elements, each containing one Frame element, that serializes the annotations relative to one identified object.
<any concrete element in CLASSELT>
<id> Const </id>?
<meta>
[ Frame
|
<And>
<formula> Frame </formula>*
</And>
]
</meta>?
other CLASSELT content
</any concrete element in CLASSELT>
Notice that the content of the meta sub-element of an instance of a RIF-PRD class element is not necessarily associated to that same instance element: only the content of the object sub-element of the Frame that represents the annotations specifies what the annotations are about, not where it is included in the instance RIF document.
It is suggested to use Dublin Core, RDFS, and OWL properties for annotations, 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 8.13. The example shows the structure of the document that contains the runnig example rule set, as represented in example 4.2, including annotations such as rule set and rule names.
To make it easier to read, a non-normative, lightweight notation was introduced to complement the mathematical English specification of the abstract syntax and the semantics of RIF-PRD. This section specifies a presentation syntax for RIF-PRD, that extends that notation. The presentation syntax is not normative. However, it may help implementers by providing a more succinct overview of RIF-PRD syntax.
The EBNF for the RIF-PRD presentation syntax is given as follows. For convenience of reading we show the entire EBNF in its four parts (rules, conditions, actions, and annotations).
A NEGATEDFORMULA can be written using either Not or INeg. INeg is short for inflationary negation and is preferred over 'Not' to avoid ambiguity about the semantics of the negation.
The RIF-PRD 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 an ANGLEBRACKIRI or CURIE that represents the identifier of the symbol space of the constant, and UNICODESTRING is a Unicode string from the lexical space of that symbol space. ANGLEBRACKIRI and CURIE are defined in the section Shortcuts for Constants in RIF's Presentation Syntax in [RIF-DTB]. Constant symbols can also have several shortcut forms, which are represented by the non-terminal CONSTSHORT. These shortcuts are also defined in the same section of [RIF-DTB]. One of them is the CURIE shortcut, which is extensively used in the examples in this document.
Names are XML NCNames. Variables are composed of NCName symbols prefixed with a ?-sign.
Example 9.1. Here is the transcription, in the RIF-PRD presentation syntax, of the complete RIF-PRD document corresponding to the running example:
Document(
Prefix( ex1 <http://example.com/2009/prd2> )
(* ex1:CheckoutRuleset *)
Group (
(* ex1:GoldRule *)
Group (
Forall ?customer such that And(?customer # ex1:Customer
?customer[ex1:status -> "Silver"])
(Forall ?shoppingCart such that ?customer[ex1:shoppingCart -> ?shoppingCart]
(If Exists ?value (And(?shoppingCart[ex1:value -> ?value]
External(pred:numeric-greater-than-or-equal(?value 2000))))
Then Do(Modify(?customer[ex1:status -> "Gold"])))))
(* ex1:DiscountRule *)
Group (
Forall ?customer such that ?customer # ex1:Customer
(If Or( ?customer[ex1:status -> "Silver"]
?customer[ex1:status -> "Gold"])
Then Do ((?s ?customer[ex1:shoppingCart -> ?s])
(?v ?s[ex1:value -> ?v])
Modify(?s [ex1:value -> External(func:numeric-multiply (?v 0.95))]))))
(* ex1:NewCustomerAndWidgetRule *)
Group (
Forall ?customer such that And(?customer # ex1:Customer
?customer[ex1:status -> "New"] )
(If Exists ?shoppingCart ?item
(And(?customer[ex1:shoppingCart -> ?shoppingCart]
?shoppingCart[ex1:containsItem -> ?item]
?item # ex1:Widget ) )
Then Do( (?s ?customer[ex1:shoppingCart -> ?s])
(?val ?s[ex1:value -> ?val])
Retract(?customer ex1:voucher)
Modify(?s[ex1:value -> External(func:numeric-multiply(?val 0.90))]))))
(* ex1:UnknownStatusRule *)
Group (
Forall ?customer such that ?customer # ex1:Customer
(If Not(Exists ?status
(And(?customer[ex1:status -> ?status]
External(pred:list-contains(List("New" "Bronze" "Silver" "Gold") ?status)) )))
Then Do( Execute(act:print(External(func:concat("New customer: " ?customer))))
Assert(?customer[ex1:status -> "New"]))))
)
)
☐
10 Acknowledgements
This document is the product of the Rules Interchange Format (RIF) Working Group (see below) whose
members deserve recognition for their time and commitment. The editors extend special thanks to Harold Boley and Changhai Ke for their thorough reviews and insightful discussions; the working group chairs, Chris Welty and Christian de Sainte Marie, for their invaluable technical help and inspirational leadership; and W3C staff contact Sandro Hawke, a constant source of ideas, help, and feedback.
The regular attendees at meetings of the Rule Interchange Format (RIF) Working Group at the time of the publication were:
Adrian Paschke (Freie Universitaet Berlin),
Axel Polleres (DERI),
Chris Welty (IBM),
Christian de Sainte Marie (IBM),
Dave Reynolds (HP),
Gary Hallmark (ORACLE),
Harold Boley (NRC),
Jos de Bruijn (FUB),
Leora Morgenstern (IBM),
Michael Kifer (Stony Brook),
Mike Dean (BBN),
Sandro Hawke (W3C/MIT), and
Stella Mitchell (IBM).
This alternative specification is provided for the convenience of the reader, for compatibility with other RIF specifications, such as [RIF-DTB] and [RIF-RDF-OWL], and to make explicit the interoperability with RIF logic dialects, in particular [RIF-Core] and [RIF-BLD].
12.1 Semantic structures
The key concept in a model-theoretic semantics of a logic language is the notion of a semantic structure [Enderton01, Mendelson97].
Definition (Semantic structure).
A semantic structure, I, is a tuple of the form
<TV, DTS, D, Dind,
Dfunc, IC, IV, Ilist,
IP, Iframe,
Isub, Iisa, I=,
Iexternal, Itruth>. Here D is a non-empty set of elements called the Herbrand domain of I, that is, the set of all ground terms which can be formed by using the elements of Const. Dind, Dfunc are nonempty subsets of D. Dind is used to interpret the elements of Const that are individuals and Dfunc 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 in the RIF data types and builtins specification [RIF-DTB] for the semantics of datatypes).
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:
IC maps Const to D.
This mapping interprets constant symbols. In addition:
If a constant, c ∈ Const,
is an individual then it is required that
IC(c) ∈ Dind.
If c ∈ Const, is a function symbol then it is required that IC(c) ∈ Dfunc.
IV maps Var to Dind.
This mapping interprets variable symbols.
Ilist : Dind* → Dind is used to interpret lists.
In addition, this mapping is required to satisfy the following conditions:
Ilist is an injective one-to-one function.
Ilist(Dind) is disjoint from the value spaces of all data types in DTS.
IP maps D to functions D*ind → D (here D*ind is a set of all sequences of any finite length over the domain Dind).
This mapping interprets positional terms atoms.
Iframe maps Dind to total functions of the form SetOfFiniteBags(Dind × Dind) → D.
This mapping interprets frame terms. An argument, d ? Dind, to Iframe represents an object and the finite bag {<a1,v1>, ..., <ak,vk>} represents a bag of attribute-value pairs for d. We will see shortly how Iframe 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. 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 while ?B and ?D are instantiated with b. (We shall see later that o[a->b a->b] is equivalent to o[a->b].)
Isub gives meaning to the subclass relationship. It is a mapping of the form Dind × Dind → D.
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 condition formulas;
Iisa gives meaning to class membership. It is a mapping of the form Dind × Dind → 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;
I= is a mapping of the form Dind × Dind → D.
It gives meaning to the equality operator.
Itruth is a mapping of the form D → TV.
It is used to define truth valuation for formulas.
Iexternal is a mapping from the coherent set of schemas for externally defined functions to total functions D* → D. For each external schema σ = (?X1 ... ?Xn; τ) in the coherent set of external schemas associated with the language, Iexternal(σ) is a function of the form Dn → D.
For every external schema, σ, associated with the language, Iexternal(σ) is assumed to be specified externally in some document (hence the name external schema). In particular, if σ is a schema of a RIF built-in predicate, function or action, Iexternal(σ) is specified so that:
If σ is a schema of a built-in function then Iexternal(σ) must be the function defined in [RIF-DTB];
If σ is a schema of a built-in predicate then
Itruth ο (Iexternal(σ)) (the composition of Itruth and Iexternal(σ), a truth-valued function) must be as specified in [RIF-DTB];
If σ is a schema of a built-in action then
Itruth ο (Iexternal(σ)) (the composition of Itruth and Iexternal(σ), a truth-valued function) must be as specified in the section Built-in actions in this document.
For convenience, we also define the following mapping I from terms to D:
I(k) = IC(k), if k is a symbol in Const;
I(?v) = IV(?v), if ?v is a variable in Var;
For list terms, the mapping is defined as follows:
I(List( )) = Ilist(<>). Here <> denotes an empty list of elements of Dind. (Note that the domain of Ilist is Dind*, so Dind0 is an empty list of elements of Dind.)
I(List(t1 ... tn)) = Ilist(I(t1), ..., I(tn)), if n>0.
I(p(t1 ... tn)) = IP(I(p))(I(t1),...,I(tn));
I(o[a1->v1 ... ak->vk]) = Iframe(I(o))({<I(a1),I(v1)>, ..., <I(an),I(vn)>}) Here {...} denotes a bag of attribute/value pairs.
I(c1##c2) = Isub(I(c1), I(c2));
I(o#c) = Iisa(I(o), I(c));
I(x=y) = I=(I(x), I(y));
I(External(t)) = Iexternal(σ)(I(s1), ..., I(sn)), if t is an instance of the external schema σ = (?X1 ... ?Xn; τ) by substitution ?X1/s1 ... ?Xn/s1. Note that, by definition, External(t) is well-formed only if t is an instance of an external schema. Furthermore, by the definition of coherent sets of external schemas, t can be an instance of at most one such schema, so I(External(t)) is well-defined.
The effect of datatypes. The set DTS must include the datatypes described in Section Primitive Datatypes of the RIF data types and builtins specification [RIF-DTB].
The datatype identifiers in DTS impose the following restrictions. Given dt ∈ DTS, let LSdt denote the lexical space of dt, VSdt denote its value space, and Ldt: LSdt → VSdt the lexical-to-value-space mapping (for the definitions of these concepts, see Section Primitive Datatypes of the RIF data types and builtins specification [RIF-DTB]. Then the following must hold:
VSdt ⊆ Dind; and
For each constant "lit"^^dt such that lit ∈ LSdt, IC("lit"^^dt) = Ldt(lit).
That is, IC must map the constants of a datatype dt in accordance with Ldt.
RIF-PRD does not impose restrictions on IC for constants in symbol spaces that are not datatypes included in DTS. ☐
12.2 Interpretation of condition formulas
This section defines how a semantic structure, I, determines
the truth value TValI(φ) of a condition formula, φ.
We define a mapping, TValI, from the set of all condition formulas to TV. Note that the definition implies that TValI(φ) is defined only if the set DTS of the datatypes of I includes all the datatypes mentioned in φ and Iexternal is defined on all externally defined functions and predicates in φ.
Definition (Truth valuation).
Truth valuation for well-formed condition formulas in RIF-PRD is determined using the following function, denoted TValI:
Equality: TValI(x = y) = Itruth(I(x = y)). To ensure that equality has precisely the expected properties, it is required that:
Itruth(I(x = y)) = t if I(x) = I(y) and that Itruth(I(x = y)) = f otherwise. This is tantamount to saying that TValI(x = y) = t iff I(x) = I(y);
Subclass: TValI(sc ## cl) = Itruth(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 TValI(c1 ## c2) = TValI(c2 ## c3) = t then TValI(c1 ## c3) = t;
Membership: TValI(o # cl) = Itruth(I(o # cl)). To ensure that all members of a subclass are also members of the superclass, i.e., o # cl and cl ## scl implies o # scl, the following is required:
For all o, cl, scl ∈ D, if TValI(o # cl) = TValI(cl ## scl) = t then TValI(o # scl) = t;
Frame: TValI(o[a1->v1 ... ak->vk]) = Itruth(I(o[a1->v1 ... ak->vk])). Since the bag of attribute/value pairs represents the conjunctions of all the pairs, the following is required, if k > 0:
TValI(o[a1->v1 ... ak->vk]) = t if and only if TValI(o[a1->v1]) = ... = TValI(o[ak->vk]) = t;
Externally defined atomic formula: TValI(External(t)) = Itruth(Iexternal(σ)(I(s1), ..., I(sn))), if t is an atomic formula that is an instance of the external schema σ = (?X1 ... ?Xn; τ) by substitution ?X1/s1 ... ?Xn/s1. Note that, by definition, External(t) is well-formed only if t is an instance of an external schema. Furthermore, by the definition of coherent sets of external schemas, t can be an instance of at most one such schema, so I(External(t)) is well-defined;
Conjunction: TValI(And(c1 ... cn)) = t if and only if TValI(c1) = ... = TValI(cn) = t. Otherwise, TValI(And(c1 ... cn)) = f. The empty conjunction is treated as a tautology, so TValI(And()) = t;
Disjunction: TValI(Or(c1 ... cn)) = f if and only if TValI(c1) = ... = TValI(cn) = f. Otherwise, TValI(Or(c1 ... cn)) = t. The empty disjunction is treated as a contradiction, so TValI(Or()) = f;
Negation: TValI(Not(c)) = f if and only if TValI(c) = t. Otherwise, TValI(Not(c)) = t;
Existence: TValI(Exists ?v1 ... ?vn (φ)) = t if and only if for some I*, described below, TValI*(φ) = t. Here I* is a semantic structure of the form <TV, DTS, D, Dind, Dfunc, IC, I*V, Ilist, IP, Iframe, Isub, Iisa, I=, Iexternal, Itruth>, which is exactly like I, except that the mapping I*V, is used instead of IV. I*V is defined to coincide with IV on all variables except, possibly, on ?v1,...,?vn. ☐
12.3 Condition satisfaction
We define, now, what it means for a state of the fact base to satisfy a condition formula. The satisfaction of condition formulas in a state of the fact base provides formal underpinning to the operational semantics of rule sets interchanged using RIF-PRD.
Definition (Models).
A semantic structure I is a model of a condition formula, φ, written as I |= φ, iff TValI(φ) = t. ☐
Definition (Herbrand interpretation). Given a non-empty set of constants, Const, the Herbrand domain is the set of all the ground terms that can be formed using the elements of Const, and the Herbrand base is the set of all the well-formed ground atomic formulas that can be formed with the elements in the Herbrand domain.
A semantic structure, I, is a Herbrand interpretation, if the set of all the ground formulas which are true with respect to I (that is, of which I is a model), is a subset of the corresponding Herbrand base, BI. ☐
In RIF-PRD, the semantics of condition formulas is defined with respect to semantic structures where the domain, D is the Herbrand domain that is determined by the set of all the constants, Const; that is, with respect to Herbrand interpretations.
Definition (State of the fact base). To every Herbrand interpretation I, we associate a state of the fact base, wI, that is represented by the subset of the Herbrand base that contains exactly the ground atomic formulas of which I is a model; or, equivalently, by the conjunction of all these ground atomic formulas. ☐
Definition (Condition satisfaction). A RIF-PRD condition formula φ is satisfied in a state of the fact base, wI, if and only if I is a model of φ. ☐
At the syntactic level, the interpretation of the variables by a valuation function IV is realized by a substitution. As a consequence, a ground substitution σmatches a condition formula ψ to a set of ground atomic formulas Φ if and only if σ realizes the valuation function IV of a semantics structure I that is a model of ψ and Φ is a representation of a state of the fact base, wI (as defined above), that is associated to I; that is, if and only if ψ is satisfied in wI (as defined above).