SILK is an expressive Semantic Web rule language and system equipped with scalable reactive higher-order defaults. We present two of its latest novel features. Omni rules are prioritized defeasible clauses that provide a higher-level expressive abstraction for users by making rules omni-directional. They handle multi-way conflicts. And they are soundly interoperable with FOL and W3C OWL as well as with RIF-BLD, SPARQL, and RDF. RIF-SILK is a new RIF dialect that extends RIF-BLD via RIF-FLD.
several key advanced expressive features: defaults , with strong negation and
priorities, cf. courteous LP [
The SILK language includes an ASCII presentation syntax, an abstract syntax, and a RIF dialect (RIF-SILK) which includes markup syntax. The hyper LP semantics includes a model theory and proof theory. Essentially, its implementation can be viewed as transforming knowledge from higher to lower abstraction levels. Higher is good for knowledge acquisition (human cognition); lower is good for reasoning (code reuse and optimization). Despite hyper LP’s expressive power, inferencing in it is highly scalable computationally. It is tractable under non-onerous restrictions, having practically the same computational complexity as Horn LP (similar to Horn first-order logic (FOL)). Hyper LP retains the pragmatic quality of LP: it is intuitionistic, i.e., conclusions drawn are definite, and lacks general “reasoning by cases”.
SILK knowledge interchange can be via load (import), or query (external,
during inferencing), or event (update received). KR languages supported for
interchange include: SPARQL and RDF(S); SQL and ODBC (e.g., Excel
spreadsheets); SILK, RIF (-BLD and -SILK), and OWL (-RL); Cyc; and AURA [
We have developed a novel major expressive extension in hyper LP, called omni rules, and implemented support for it in SILK. Omni rules are prioritized defeasible axioms that provide a yet higher-level expressive abstraction for users. Omni rules handle multi-way conflicts. And omni rules are soundly interoperable with FOL and W3C OWL as well as with RIF-BLD, SPARQL, and RDF.
SILK syntax: A SILK rule has the form “@[tag->G] H :- B ;”. Here, “:-” is the usual LP rule implication connective, i.e., informally “if”. H and B are called the rule head and body, respectively; and “;” is the statement ending delimiter. “:- B” may be absent, i.e., the body may be empty (true). a # c means that a is an instance of class c. Skolems are prefixed by the underscore character (“ ”). A courteous rule label term, used for prioritization, is called a tag. SILK has recently been extended to incorporate rule annotations . An explicit tag is specified in a rule via an annotation which optionally begins the rule: @[tag->G] H :- B ; specifies a rule with tag G. The tag (G) may be strict (silk:strict), i.e., non-defeasible; if so, the “tag->silk:” may be omitted. An annotation can also be used to specify an explicit unique ID for the rule and/or other processing directives.
Clausal case: Omni rules generalize the permitted form of rule head formula. The basic case is a clausal omni rule; this has the form @[tag->G] F ; . Here, the body is empty. The head F is a disjunction of literals. Each such head literal is an atom or the neg of an atom; F is NAF-free (as usual in LP). E.g.: @[tag->hi] wet(?Loc,today) or neg occur(rain,?Loc,yesterday) ; The tag is optional. Outer universal quantification can be left implicit.
The semantics of omni rules is defined (and implemented currently) via a transformation. A clausal omni rule:
@[tag->G] L1 or L2 or ... or Lk ; — where each Li is an atom or the neg of an atom, and k > 0 — is transformed into a set of k directional rules, one for each choice of head literal — i.e., one for each “contrapositive variant”: @[tag->G] L1 :- neg L2 and ... and neg Lk ; ... @[tag->G] Li :- neg L1 and ... and neg Li−1 and
neg Li+1 and ... and neg Lk ; ...
@[tag->G] Lk :- neg L1 and ... and neg Lk−1 ; (Here, neg neg A is treated as equivalent to A, for any atom A.) This is called the set of directional variant rules, a.k.a. the variant set , for that clausal omni. The transformation of a clause into this set of directional rules is called its directionalization. E.g., the variant set for the omni above about wet and rain is: @[tag->hi] wet(?Loc,today) :- occur(rain,?Loc,yesterday) ; @[tag->hi] neg occur(rain,?Loc,yesterday) :- wet(?Loc,today) ; General case: More generally, a omni rule may have a body and it may have a universal FOL-formula head, i.e., take the form
@[tag->G] HU :- B ; Here, B is any hyper LP body (same as before the omni rules feature was introduced). The head HU is any syntactically FOL-looking formula that is outermost universally quantified. The quantifier-free subformula of HU may be formed syntactically from atoms using neg, and, and or freely composed. The strong implication connective <== is also permitted as a syntactic and conceptual convenience. D <== E is defined as equivalent to D or neg E. Likewise, ==> and <==> are permitted as well. E.g., the rule above about wet and rain is equivalently reformulated as:
@[tag->hi] wet(?Loc,today) <== occur(rain,?Loc,yesterday) ;
Note that the head formula may use skolemization so as to approximately
represent existentials, and thus be “nearly” full FOL in form. Moreover, the head
formula may use the higher-order (HiLog) and frame syntax/features of hyper
LP. Any universal formula HU is straightforwardly transformed (reduced) to a
conjunctive set of clauses as in FOL [
@[tag->G] HCn :- B ; In the second step, each of these clausal-head omni rules is transformed into its variant set, as described earlier except that B is also added to the body of each directional variant rule in each variant set. I.e., the i’th rule of the variant set shown above becomes: @[tag->G] Li :- neg L1 and ... and neg Li−1 and
neg Li+1 and ... and neg Lk and B ; When the omni rule’s head is a single literal, both steps of the transformation reduce to identity. Thus the previous directional kind of LP rules are simply a special case of omni rules, in hyper LP. One can thus view every rule in hyper LP as an omni rule.
We have further developed a sophisticated version of Courteous, in which
the second/clausal step of the directionalization transformation also outputs
an exclusion statement used by the argumentation theory [
Advantages for Authoring of rules: The courteous defaults feature of hyper LP enables users to author rules that mainly employ strong negation (neg) and priorities rather than negation-as-failure (naf). The omni-directional feature of hyper LP further enables users to author rules that mainly employ an “omni-directional” form of implication (<==) rather than the directional form of implication (:-) of LP rules. In this sense, omni rules abstract away the usual (i.e., previous in the LP literature) directionality of LP rules. They enable users to author FOL-style formulas, which behave omni-directionally but are nevertheless defeasible. Omni rules also allow logically-inclined users to use complex FOL formulas to represent rules in a compact, easily readable form. In this respect, hyper rules confer benefits similar to Lloyd-Topor transformations.
Figure 1 shows, in a SILK GUI [
No “Reasoning by Cases”: Despite permitting disjunction within the head, omni rules do NOT permit or perform “reasoning by cases” as FOL and Answer Set Programs (ASP) do. Omni rules thus do not incur the far greater computational complexity FOL and ASP have. Rather, the variant set in this regard behaves similarly to the unit/linear resolution strategy in FOL.
Sound Interoperability with FOL via Hypermonotonic Mapping:
We have developed a novel approach for sound interchange with FOL,
generalizing the method used in Description Logic Programs [
We can define this relationship via a hypermonotonic mapping T that consists of a pair of mappings (T 1, T 2), one for each interchange direction. T 1 maps a hyper LP rule into a universal FOL axiom. It replaces :- by <==, and ignores the tag. I.e., the hyper LP rule @[tag->G] H :- B ; is mapped into the FOL axiom H <== B ;. Certain non-onerous syntactic restrictions apply, since some expressive constructs in hyper LP have no correspondent in FOL. Thus the hyper LP rule must not contain naf nor the other body-only constructs that expressively essentially rely upon naf, including: aggregates, Lloyd-Topor forall, and Lloyd-Topor implies. And it must not contain external actions (effector literals). T 1 maps a (true) hyper LP conclusion into a FOL axiom with the same formula.
T 2 maps a universal FOL axiom F; into the hyper LP omni rule F; that has the same formula as head and an empty body.
Interestingly, from a FOL viewpoint, entailment in hyper LP is sound — even though hyper LP is nonmonotonic! — and is incomplete, e.g., it lacks “reasoning by cases”. Thus we say (restricted) hyper LP is FOL-sound with respect to the interchange mapping T . When there is conflict, the incompleteness is desirable! Hyper LP entailment/reasoning is then usefully selective, unlike FOL which entails all sentences.
One usage for interchange is to import universal FOL (e.g., clauses) into hyper LP. One can assign prioritization to the imported rules, e.g., based on source authority, recency, or reliability. A second usage for interchange is to import hyper LP conclusions into FOL, e.g., in the conflict-free case.
A well-known special case of such interchange is between definite Horn LP and definite Horn FOL, especially when function-free. In this case, in the associated variant set exactly one directional clause has a positive head; the other rules in the variant set can-be/are essentially ignored. And in this case, there is completeness, not just soundness, of the LP with respect to the set of entailed ground atoms. This special case relationship is the KR foundation used in most implementations of Horn FOL reasoning — the typically used techniques essentially do LP reasoning — e.g., currently for OWL-RL, RDFS, RIF-BLD whose semantics are all based essentially on Horn FOL.
The hypermonotonic mapping T greatly generalizes that approach. It handles
strong negation (neg), and any clause not just Horn. When skolemization is
employed, it can in effect expressively cover much of FOL — thus much of
OWL, Common Logic [
Overcoming FOL’s Brittleness: Hyper LP can handle conflict robustly, always producing a consistent set of conclusions. By contrast, FOL is a “bubble” — it is perfectly brittle semantically in the face of contradictions from quality problems or merging conflicts. Any contradiction is totally contagious — the conclusions all become garbage. FOL’s extreme sensitivity to conflict limits its scalability in the number of axioms and the number of merges. The expressive opportunity arises for conflict essentially as soon as strong negation (neg) is introduced into a focal subset of FOL. For example, OWL beyond -RL and Common Logic both suffer from this problem.
A Very Large KB (VLKB) with a million or billion axioms formed by merging from multiple web sources is unlikely to have zero KB/KA conflicts from: human knowledge entry/editing; implicit context and cross-source ontology interpretation; updating cross-source; and source trustworthiness. Hyper LP’s approach provides a critical advantage for KB scalability — semantically as well as computationally. 3
Using RIF’s Framework for Logic Dialects (FLD), [
Like RIF Production Rule Dialect (PRD), RIF-SILK provides procedural attachments for external actions, etc. Unlike RIF-PRD, RIF-SILK provides a declarative semantics — and further adds higher-order, defaults, and a number of other features.
The RIF Core Answer Set Programming Dialect (RIF-CASPD) [
Contributions of this work were summarized in the Abstract, the end of section 2, and the beginning of section 3.
Next, we discuss the broad potential applicability of SILK’s core KR technology approach, and give future directions for the core technology itself.
SILK inherits a variety of use cases from previous work on courteous LP defaults, situated/production LP actions and events, and webized information integration — including in RuleML and RIF. There are many potential applications in business and government for SILK’s reactive higher-order defaults expressiveness. Horizontally, these include: policies and policy-based workflows (monitoring, trust); and ontology mapping/mediation and knowledge integration. Vertically, these include: e-commerce (shopping & advertising, contracts, customer care); defense (intelligence, operations); finance (reporting, regulatory compliance); biomed (pharma, e-science, clinical records and guidance, insurance); and mobile (personalization).
Directions for future KR work include more interchange languages, probabilistic knowledge, and parallelization of inferencing.
Acknowledgements: Thanks to all of the SILK team, including Paul Haley for suggesting the example newspaper ad policy rule and Brett Benyo for help implementing it.