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Rule Interchange Format

XChange: A rule-based language for programming reactive behavior on the Web

Reactivity on the Web is an emerging research issue covering: updating data on the Web, exchanging information about events (such as executed updates) between Web nodes, and reacting to combinations of such events. Reactivity plays an important role for upcoming Web systems such as online marketplaces, adaptive Web and Semantic Web systems, as well as Web services and Grids.

Following a declarative approach to reactivity on the Web, a novel reactive, rule-based language called XChange (see publications) has been developed. The language XChange follows clear paradigms that aim at providing a better language understanding and ease programming. XChange paradigms:

    • Event vs. Event Query. An event is a happening to which each Web site may decide to react in a particular way or not to react to at all. One might argue that defining an event in such a way is too vague. The intention here is to emphasise that one can conceive every kind of changes on the Web as events. However, each Web-based reactive system can be interested in different types of events or in different combinations of (like a given temporal order between) such events. Thus, the large spectra of possible events are always filtered relatively to one’s interests. In order to notify Web nodes about events and to process event data, events need to have a data representation. In XChange, incoming events are represented as XML documents.Event queries are queries against event data. Event query specifications differ considerably from event representations (e.g. event queries may contain variables for selecting parts of the events’ representation). Most proposals dealing with reactivity do not significantly differentiate between event and event query. Overloading the notion of event precludes a clear language semantics and thus, makes the implementation of the language and its usage much more difficult. Event queries in XChange serve a double purpose: detecting events of interest and temporal combinations of them, and selecting data items from events’ representation. This double purpose is novel in the field of reactivity and reactive rules.
    • Volatile vs. Persistent Data. The development of the XChange language – its design and its implementation – reflects the novel view over the Web data that differentiates between volatile data (event data communicated on the Web between XChange programs) and persistent data (data of Web resources, such as XML or HTML documents). The clear distinction between volatile and persistent data aims at easing programming and avoiding the emergence of a parallel Web of events.
    • Rule-Based Language. Reactivity can be specified and realised by means of reactive rules. XChange is a rule-based language that uses reactive rules for specifying the desired reactive behavior and deductive rules for constructing views over Web resources’ data.An XChange program is located at one Web site and contains reactive rules, more precisely Event-Condition-Action rules (ECA rules) of the form Event query – Web query – Action. Every incoming event is queried using the event query (query against volatile data). If an answer is found and the Web query (query to persistent data) has also an answer, then the Action is executed. The fact that the event query and the Web query have answers determines the rule to be fired; the answers influence the action to be executed, as information contained in the answers are generally used in the action part.
    • Pattern-Based Approach. XChange is a pattern-based language: event queries, Web queries, event raising specifications, and updates describe patterns for events requiring a reaction, Web data, raising event messages, and updating Web data, respectively. Patterns are templates that closely resemble the structure of the data to be queried, constructed, or modified, thus being very intuitive and also straight-forward to visualize.
    • Processing of Events
      • Local Processing and Incremental Evaluation. Event queries are processed locally at each XChange-aware Web node. For efficiency reasons, (composite) event queries are evaluated in an incremental manner.
      • Bounded Event Lifespan. An essential aspect of XChange is that each XChange-aware Web node controls its own event memory usage. In particular, the size of the event history kept in memory depends only on the event queries posed at this Web node. This is consistent with the clear distinction between events as volatile data and standard Web data as persistent data.
  • Relationship Between Reactive and Query Languages. A working hypothesis of the XChange project is that a reactive language for the Web should build upon a Web query language. XChange embeds the (Semantic) Web query language Xcerpt.

    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:

    Prefix(ex <http://example.com/2008/prd1#>)
    (* ex:rule_1 *)
    Forall ?customer ?purchasesYTD (
     If   And( ?customer#ex:Customer
               ?customer[ex:purchasesYTD->?purchasesYTD]
               External(pred:numeric-greater-than(?purchasesYTD 5000)) )
     Then Do( Modify(?customer[ex:status->"Gold"]) ) )

    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:

    1. Match: the rules are instantiated based on the definition of the rule conditions and the current state of the data source;
    2. Conflict resolution: a decision algorithm, often called the conflict resolution strategy, is applied to select which rule instance will be executed;
    3. 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:

    1. Gold rule: A “Silver” customer with a shopping cart worth at least $2,000 is awarded the “Gold” status.
    2. Discount rule: “Silver” and “Gold” customers are awarded a 5% discount on the total worth of their shopping cart.
    3. 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.
    4. 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 symbols Const,
    • a countably infinite set of variable symbols Var (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).

    1. Constants and variables. If tConst or tVar then t is a simple term;
    2. 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;
    3. Positional terms. If tConst 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:

    1. Positional atomic formulas. If tConst and t1, …, tn, n≥0, are terms then t(t1 ... tn) is a positional atomic formula (or simply an atom)
    2. Equality atomic formulas. t = s is an equality atomic formula (or simply an equality), if t and s are terms
    3. 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
    4. Subclass atomic formulas. t##s is a subclass atomic formula (or simply a subclass) if t and s are terms
    5. 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
    6. 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:

    1. Atomic formula: If φ is an atomic formula then it is also a condition formula.
    2. 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.
    3. 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.
    4. Negation: If φ is a condition formula, then so is Not(φ), called a negative formula.
    5. 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:
    And( ?customer # ex1:Customer
         ?customer[ex1:status->"New"]
         Exists ?shoppingCart ?item ( And ( ?customer[ex1:shoppingCart->?shoppingCart]
                                            ?shoppingCart[ex1:containsItem->?item]
                                            ?item # ex1:Widget) )
                                    )
        )

    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 eConst, then Var(e) = ∅;
    • if eVar, 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…n Var(argi);
    • if p(arg1…argn), n ≥ 0, is an atom, then, Var(p(arg1…argn) = Var(External(p(arg1…argn)) = ∪i=1…n Var(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…n Var(ki)i=1…n Var(vi).
    • if fi, i = 0…n, n ≥ 0, are condition formulas, then Var([And|Or](f1…fn)) = ∪i=0…n Var(fi);
    • if f is a condition formula, then Var(Not(f)) = Var(f);
    • if f is a condition formula and xiVar 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 {xVar | 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): ∀ xDom(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 σ matches ψ 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 = l2iΦ,
        • 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 = t2Φ, 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 = t2Φ, 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 = c2Φ, then c2 = c1Φ;
    • for every triple of constants c1, c2 and c3, if c1 = c2Φ and c2 = c3Φ, then c1 = c3Φ;
    • 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:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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:

    1. 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.
    2. 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.
    3. 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 (Action). A action is either an atomic action or a compound 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:

    1. 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());
    2. 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:

    Do( (?oldValue ?shoppingCart[ex1:value->?oldValue])
        Modify( ?shoppingCart[ex1:value->func:numeric-multiply(?oldValue 0.90)] ) )

    3.1.3 Well-formed action blocks

    Not all action blocks are well-formed in RIF-PRD:

    • 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 action α 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 (?v p) is well-formed if and only if one of the following is true:

    • the action variable binding, p, is the declaration of a new frame object: p = New(), 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-PRDW × 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-PRDW × L × W. (w, α, w’) ∈ →RIF-PRD if and only if wW, 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’:

    1. α is Assert(φ), where φ is a ground atomic formula, and Φ’ = Φ ∪ {φ};
    2. α is Retract(φ), where φ is a ground atomic formula, and Φ’ = Φ \ {φ};
    3. α is Retract(o s), where o and s are constants, and Φ’ = (Φ \ {o[s->v] | for all the values of v});
    4. 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};
    5. 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:

    Initial state:

    • w0 = {_c1#ex1:Customer _v1#ex1:Voucher _s1#ex1:ShoppingCart _c1[ex1:voucher->_v1] _c1[ex1:shoppingCart->_s1] _v1[ex1:value->5] _s1[ex1:value->500]}
    1. 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
      • w1 = {_c1#ex1:Customer _v1#ex1:Voucher _s1#ex1:ShoppingCart _c1[ex1:voucher->_v1] _c1[ex1:shoppingCart->_s1] _v1[ex1:value->5] _s1[ex1:value->500] _c1[ex1:status->"New"]}
    2. 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:
      • w2 = {_c1#ex1:Customer _v1#ex1:Voucher _s1#ex1:ShoppingCart _c1[ex1:shoppingCart->_s1] _v1[ex1:value->5] _s1[ex1:value->500] _c1[ex1:status->"New"]}
    3. 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:
      • w3 = {_c1#ex1:Customer _s1#ex1:ShoppingCart _c1[ex1:shoppingCart->_s1] _s1[ex1:value->500] _c1[ex1:status->"New"]}
    4. 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
      • w4 = {_c1#ex1:Customer _s1#ex1:ShoppingCart _c1[ex1:shoppingCart->_s1] _c1[ex1:status->"New"]}
    5. 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:
      • w5 = {_c1#ex1:Customer _s1#ex1:ShoppingCart _c1[ex1:shoppingCart->_s1] _s1[ex1:value->450] _c1[ex1:status->"New"]}
    6. 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:
      • w6 = w5 = {_c1#ex1:Customer _s1#ex1:ShoppingCart _c1[ex1:shoppingCart->_s1] _s1[ex1:value->450] _c1[ex1:status->"New"]}

    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.

    4.1 Abstract syntax

    The alphabet of the RIF-PRD rule language includes the alphabets of the RIF-PRD condition language and the RIF-PRD action language and adds symbols for:

    • 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"] ) ) )

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

    • if f is an action block that declares action variables ?v1 ... ?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:

    1. 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;
    2. 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

    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:

    w0 = {_john#ex1:Customer _john[ex1:status->"Silver"] _s1#ex1:ShoppingCart _john[ex1:shoppingCart->_s1] _s1[ex1:value->2000]}

    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:

    w1 = {_john#ex1:Customer _john[ex1:status->"Gold"] _s1#ex1:ShoppingCart _john[ex1:shoppingCart->_s1] _s1[ex1:value->2000]}

    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:

    w2 = {_john#ex1:Customer _john[ex1:status->"Gold"] _s1#ex1:ShoppingCart _john[ex1:shoppingCart->_s1] _s1[ex1:value->1900]}

    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:

    1. All the rules are rewritten in disjunctive normal form as described in section Safeness;
    2. Each rule is replaced by a group of rules
      • with the same priority as the rule it replaces,
      • 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 match w if and only if one of the following is true:

    • rule(ri) is an unconditional action block;
    • rule(ri) is a conditional action block: If condition, Then action, and substitution(ri) matches the condition formula condition to the 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: (vi New()), 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: (vi o[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

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

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

    • S is a set of system states, called the system cycle states;
    • A is a set of transition labels, where each transition label is a sequence of ground RIF-PRD atomic actions;
    • The transition relation →PRSS × A × S, is defined as follows:
      ∀ (s, a, s’ ) ∈ S × A × S, (s, a, s’ ) ∈ →PRS if and only if all of the following hold:

      1. (facts(s), a, facts(s’)) ∈ →*RIF-PRD, where →*RIF-PRD denotes the transitive closure of the transition relation →RIF-PRD that is determined by the specification of the semantics of the atomic actions supported by RIF-PRD;
      2. a = actions(picked(s));
    • T ⊆ S, a set of final system states.   ☐

    Intuitively, the first condition in the definition of the transition relation →PRS states that a production rule system can transition from one system 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.

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

    1. the conflict resolution strategy, that is, how rule instances are precisely selected for execution from the rule instances that match a given state of the fact base, and
    2. 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

    1. Refraction is applied to the conflict set, that is, all the refracted rule instances are removed from further consideration;
    2. The remaining rule instances are ordered by decreasing priority, and only the rule instances with the highest priority are kept for further consideration;
    3. The remaining rule instances are ordered by decreasing recency, and only the most recent rule instances are kept for further consideration;
    4. 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;
    • else, lastPicked(ri, s) = 1 + lastPicked(ri, previous(s)).

    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:

    Document (
      Prefix( ex2 <http://example.com/2009/prd3#> )
      (* ex2:ExampleRuleSet *)
      Group (
        (* ex2:Rule_1 *) Forall ...
        (* ex2:HighPriorityRules *)
        Group 10 (
          (* ex2:Rule_2 *) Forall ...
          (* ex2:Rule_3 *) 
          Group 9 (Forall ... ) )
        (* ex2:NoPriorityRules *)
        Group (
          (* ex2:Rule_4 *) Forall ...
          (* ex2:Rule_5 *) Forall ... )
     )

    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:

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

    1. initialize picked(s) with the conflict set, that a rule set RS determines in a system state s: picked(s) = conflictSet(RS, s);
    2. apply the refraction rule to all the rule instances in picked(s);
    3. then apply the priority rule to all the remaining instances in picked(s);
    4. then apply the recency rule to all the remaining instances in picked(s);
    5. then apply the tie-break rule to the remaing instance in picked(s);
    6. 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):

    conflictSet(ex1:CheckoutRuleset, s2) = { ex1:DiscountRule/{(_john/?customer)} } = 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.

    5.2 Operational semantics of RIF-PRD documents

    The semantics of a well-formed RIF-PRD document that contains no import directive is the semantics of the rule set that is represented by the top-level group in the document, evaluated with the conflict resolution strategy that is associated to the document, and the default halting test, as specified above, in section Halting test.

    The semantics of a well-formed RIF-PRD document, D, that imports the well-formed RIF-PRD documents D1, …, Dn, n ≥ 1, is the semantics of the rule set that is the union of the rule sets represented by the top-level groups in D and the imported documents, with the rif:local constants renamed to ensure that the same symbol does not occur in two different component rule sets, and evaluated with the conflict resolution strategy that is associated to the document, and the default halting test.

    6 Built-in functions, predicates and actions

    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

    • r is a well-formed RIF-PRD rule,
    • 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 conformant RIF-PRDΤ,Ε,C,H consumer iff it implements a semantics-preserving mapping from the set of all safe RIF-PRDΤ,Ε,C,H rule sets to the language L of the processor;
    • A RIF processor is a conformant RIF-PRDΤ,Ε,C,H producer iff it implements a semantics-preserving mapping from a subset of the language L of the processor to a set of safe RIF-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:

    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:

    1. Remove redundant information. The behavior role element and all its sub-elements should be omitted in the RIF-PRD XML document;
    2. 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.;
    3. Remove patterns. If a pattern occurs in a rule with variable declaration, r1:
      • if the rule inside r1 is a unconditional action block, r2, r2 should be transformed into a conditional action block, where the condition is the pattern, and the pattern should be removed from r1,
      • 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;
    4. 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):

    <Forall>
       <declare><Var>?customer</Var></declare>
       <declare><Var>?shoppingCart</Var></declare>
       <formula>
          <Implies>
             <if>
                <And>
                   <formula>   <!-- first pattern -->
                      <And>
                         <formula><Member> ... </Member></formula>
                         <formula><Frame> ... </Frame></formula>
                      </And>
                   </formula>
                   <formula>   <!-- second pattern -->
                      <Member> ... </Member>
                   </formula>
                   <formula>   <!-- original existential condition -->
                      ...
                   </formula>
               </And>
            </if>
            <then>
               <And>
                  <formula>   <!-- serialization of ex1:Foo(?customer) -->
                     ...
                  </formula>
                  <formula>   <!-- serialization of ex1:Bar(?shoppingCart) -->
                     ...
                  </formula>
            </then>
         </Implies>
      </formula>
    </Forall>

    Source: http://www.w3.org/TR/2013/REC-rif-prd-20130205/

XChange: an ongoing research project

The design, the core language constructs, and the semantics of XChange are completed. The proof-of-concept implementation follows a modular approach that mirrors the operational semantics. Issues of efficiency of the implementation, esp. for event detection and update execution, are subject to future work.
Source: http://reactiveweb.org/xchange/

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Rule Interchange Format

RIF – Rule Interchange Format

The RIF Working Group has focused on two kinds of dialects: logic-based dialects and dialects for rules with actions. Generally, logic-based dialects include languages that employ some kind of logic, such as first-order logic (often restricted to Horn logic) or non-first-order logics underlying the various logic programming languages (e.g., logic programming under the well-founded or stable semantics). The rules-with-actions dialects include production rule systems, such as Jess, Drools and JRules, as well as reactive (or event-condition-action) rules, such as Reaction RuleML and XChange. Due to the limited resources of the RIF Working Group, it defined only two logic dialects, the Basic Logic Dialect (RIF-BLD) and a subset, the RIF Core Dialect, shared with RIF-PRD; the Production Rule Dialect (RIF-PRD) is the only rules-with-actions dialect defined by the group. Other dialects are expected to be defined by the various user communities.

Present and future RIF dialects are expected to share datatypes, built-in functions, and built-in predicates as defined by RIF Datatypes and Built-Ins (RIF-DTB). In particular, the current dialects RIF-BLD, RIF-Core, and RIF-PRD all share the foundations of RIF-DTB 1.0.

It provides a high-level explanation of RIF concepts and architecture as well as a general survey of RIF documents.

 
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