Hybrid MKNF knowledge bases under the well-founded semantics integrate Description Logics with Logic Programming. However, they do not support classical negation in the rule component, limiting their ability to represent explicit negative knowledge. This limitation is particularly significant in safety-critical applications, where reasoning often requires explicit negative information rather than interpreting the absence of information as evidence of absence. To address this issue, we introduce an extension of Hybrid MKNF that supports classical negation in the rule component. We formally define the syntax and semantics of the extended language and present a general procedure for computing its well-founded model.
Combining RDF rule languages, such as N3 or SHACL Rules, with default negation is challenging. Existing methods to stratify negation often fail for RDF rules, since individual triples do not carry enough information to meaningfully restrict potential dependencies. Blank nodes in rule heads further complicate the matter, since the order of rule applications may determine whether new values are created, which in turn can change the applicability of rules with negation. To solve these open problems, we propose chain stratification as a robust new condition that guarantees a well-behaved semantics for RDF rules with negation, and existential rules in general. Our condition combines an elaborate analysis of potential multistep derivations with a mechanism for using integrity constraints to discard impossible cases. Applying rules in any order that respects chain stratification is guaranteed to derive an RDF graph that is unique, lean, and justified under the usual negation-as-failure semantics. To show the practicality, we also provide a prototype implementation.
Negation is a common and important semantic feature in natural language, yet Large Language Models (LLMs) struggle when negation is involved in natural language understanding tasks. Commonsense knowledge, on the other hand, despite being a well-studied topic, lacks investigations involving negation. In this work, we show that commonsense knowledge with negation is challenging for models to understand. We present a novel approach to automatically augment existing commonsense knowledge corpora with negation, yielding two new corpora containing over 2M triples with if-then relations. In addition, pre-training LLMs on our corpora benefits negation understanding.
In various data models, the classical triple is a typical semantic data model. However, due to the design of the triple as a simple structure for representing positive assertions, it cannot sufficiently express different forms of negation present in the triple and its elements. This paper conceptually proposes that there are three distinct forms of negation within triples and their elements: contradictory negation, opposite negation and intermediary negation. Based on the the set SCOI and the logic LCOI+PLCOI with three kinds of negation, we propose an extension of triple that can distinguish and express these three different negations in the triple and its elements, called the TCOI triple with contradictory negation, opposite negation and intermediary negation. The TCOI triple is a semantic and structural extension of the classical triple. While retaining the ability to express positive assertions, it systematically introduces the three semantic dimensions of three negations, allowing these negations to independently act on the elements of the triple and on the whole triple. This significantly enhances the triple model capability to represent and reasoning about complex negative information. This paper also explores the expressive power and reasoning of the TCOI triple, as well as the application of TCOI triple implication reasoning in counterfactuals and counterfactual reasoning. We propose a truth-value (continuous value) algorithm for TCOI triple implication reasoning and perform its calculation through an example of the counterfactuals and counterfactual reasoning.