Stratified Negation in RDF Rules: A Correct Approach (Extended Version)

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.

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.

Retrieval-augmented language models can fail to respect negative constraints when the retriever supplies evidence about concepts the user explicitly excluded. Beyond explicit negation, queries may ask for answers that include one concept while excluding another, or for entities that belong to a category but differ from a closely related instance. Because the excluded concept still appears in the query text, dense retrievers may assign high similarity to documents about that concept even when the user asks to avoid it. We introduce E-SENS, a training-free reranking method for negation-sensitive retrieval. E-SENS extracts a compact trap query for the excluded side and subtracts trap-query similarity from the original-query retrieval score. On ExcluIR, E-SENS shows a clear recall-violation trade-off across four embedding models and reduces trap retrieval at recall-preserving settings.