Native Closed-World Reasoner

• Native closed-world reasoner is a logic-based system that assumes missing information is false, enabling non-monotonic inferences and integrating closed-world and open-world paradigms.
• It employs hybrid MKNF semantics with alternating fixpoint iterations and SLG(O) resolution to efficiently combine rule-based and ontological reasoning.
• The architecture features integrated DL tableau, bridge predicates, and robust optimizations like tabling and early termination to achieve scalable and sound query evaluation.

A native closed-world reasoner is a reasoning engine or algorithmic framework designed to natively support the closed-world assumption (CWA) within a logic-based knowledge representation formalism. In this paradigm, missing information is assumed to be false, enabling non-monotonic inferences and negative conclusions from the absence of data. Native closed-world reasoners are characterized by their tight coupling of classical closed-world, non-monotonic rule processing with open-world ontological knowledge, embedding closed-world inference into their semantic and operational core. The following sections discuss foundational semantics, implementation principles, architecture, integration with ontologies, optimizations, and practical illustrations, with primary exemplification through the CDF-Rules system and related approaches (Gomes et al., 2011).

1. Hybrid MKNF Semantics and the Well-Founded Model

Native closed-world reasoning in hybrid knowledge bases is grounded in the Minimal Knowledge and Negation as Failure (MKNF) semantic framework. An MKNF knowledge base consists of a tuple $K=(\mathcal{O},\mathcal{P})$, where:

• $\mathcal{O}$ is a monotonic open-world Description Logic ontology.
• $\mathcal{P}$ is a finite set of MKNF rules using non-monotonic, closed-world well-founded semantics.

An MKNF rule is of the modal form:

$$K H \gets\, K A_1, \ldots, K A_n,\, \text{not}\; B_1, \ldots, \text{not}\; B_m$$

where the modal operator $K$ denotes "known" in the hybrid KB, and "not" is default negation.

To realize a three-valued well-founded semantics analogous to that of logic programs, a series of operators are defined:

• The Coherence Transform augments the KB by introducing fresh predicates for closed-world checking of DL-predicates in rule heads.
• $R_K$ evaluates rule-derived knowledge, while $D_K$ incorporates DL entailments given the current hypothesis set.
• The alternating fixpoint construction iterates between expansions based on default negation and positive reasoning, until convergence to a least fixpoint $(T_\omega, TU_\omega)$, partitioning modal atoms as true/false/undefined.
• This fixpoint directly generalizes classical well-founded semantics to the hybrid open/closed-world MKNF setting, ensuring soundness and embedding both non-monotonic and ontological reasoning.

2. Goal-Directed Query Answering via SLG(O) Resolution

To avoid materializing the full well-founded model, native closed-world engines such as CDF-Rules and related SLG(O) systems employ a query-oriented, top-down evaluation strategy:

• Queries initiate a controlled fixpoint computation: an outer loop for default negation (alternating $\Gamma$-operator cycles), and inner loops for interleaved rule and ontology entailment ($R_K$, $D_K$ operators).
• Recursive rule evaluation and DL entailment are tabled, parameterized by fixpoint iteration indices $(\text{Outer},\text{Inner})$.
• Tabling guarantees both termination and re-use across iterations.
• Default negation is computed incrementally by inspecting tables from prior fixpoint iterations, ensuring non-monotonic inference while preserving computational tractability.
• This approach tightly integrates external DL oracles, supporting arbitrary (decidable) description logics via pluggable entailment checks.

The architecture supports answering of monotonic and non-monotonic queries directly, determining the status of modal atoms as true, false, or undefined depending on their presence in the terminal fixpoint sets.

3. Ontology-Rule Interfacing and CDF Type-1 Integration

A key property of native closed-world reasoners is the algorithmic synchronization between ontology entailment and closed-world rule inference:

• In CDF-Rules, the ontology is formalized as a CDF Type-1 (ALCQ) theory, interfaced via a dedicated tableau prover.
• Intensional facts derived by rules (e.g., known/3) are provided to the tableau for “mixed” ontological inferences, enabling bidirectional flow of information.
• At each fixpoint step, new individuals introduced by DL role assertions are dynamically added to the relevance set and become subject to subsequent rule application.
• Closed-world status of DL-predicates in rule heads is enforced via augmented negative checks using supplemental predicates and default negation in the coherence transform.
• The data structures consist of indexed tables for modal atoms, default negations, and mappings from individuals to iteration indices, all necessary for correct incremental operation.

This tightly-coupled design realizes hybrid reasoning without sacrificing the expressivity of the underlying ontology language.

4. Architectural Modules and Algorithmic Optimizations

Native closed-world reasoners adopt a modularized architecture targeting scalability and efficient closed-world computation:

• XSB-Prolog with SLG tabling: Implements well-founded rule semantics, default negation, and ensures termination.
• CDF Type-0 and Type-1 layers: Provide fast, closed-world inheritance queries and open-world ALQ tableau reasoning, respectively.
• Bridge predicates: Mediate intensional updates between the rule and ontology components.
• Tabling and relevance restriction: Ensure that only query-relevant individuals are considered and that redundant computation is eliminated.

Key optimizations include:

• Early termination: Skipping outer fixpoint computation if no new modal atoms are produced in an inner iteration.
• Type-0 fast-path: Bypasses the DL tableau oracle for atoms handled purely by the closed-world extensional layer.
• Tabled, incremental evaluation: Minimizes redundant entailment checks and supports incremental updates.

These optimizations ensure tractable reasoning even in large KBs, maintaining polynomial data complexity under suitable DLs.

5. Illustrative Example: Iterative Closed-World Reasoning Trace

A customs inspection scenario illustrates the interplay of open- and closed-world inference in the CDF-Rules framework:

• An ontology axiomatizes region and shipment safety, while rules define inspect(X) based on hasShipment and negated safeCountry status.
• The answer to a query such as known(inspect(vessel42),0,0) unfolds over outer/inner fixpoint iterations:
  • The initial phase absorbs ontology assertions.
  • Further rounds propagate new knowledge from rules, conditionally fire inspect, and re-assess negation under updated knowledge.
• The system converges to a fixpoint assigning truth values according to both ontological entailment and closed-world conditions imposed by the rules.
• The evaluation is relevance-driven: only individuals reachable by role predicates are explored.

This stepwise fixpoint evolution exemplifies the operational semantics realized by a native closed-world reasoning system.

6. Comparison and Theoretical Guarantees

The native closed-world approach guarantees:

• Sound implementation of hybrid MKNF semantics: Every derived truth assignment for modal atoms agrees with Motik & Rosati’s intended semantics for hybrid knowledge bases.
• Termination under well-founded semantics: For bounded or acyclic rules, tabling ensures that the evaluation terminates for any fixed query.
• Polynomial data complexity: Under tractable DLs (e.g., EL+, DL-Lite), the entire pipeline scales linearly or quadratically in data size, suitable for use in large, industrial KBs.
• Generalization of classical well-founded inference: Both OWA (ontology) and CWA (rules) are supported in a single engine, with the SLD-residual architecture allowing for further extension (e.g., to disjunctive, probabilistic, or temporal logics with closed-world components) (Alferes et al., 2010).

Native closed-world reasoners thus provide a rigorous, efficient, and semantically integrated platform for closed-world reasoning in knowledge bases with ontological and non-monotonic rule content (Gomes et al., 2011).