Rational Inference in Formal Concept Analysis

TL;DR

The paper addresses the challenge of modeling defeasible, context-sensitive dependencies between attributes in Formal Concept Analysis (FCA) by importing the KLM framework for non-monotonic reasoning. It develops Contextual Attribute Logic with compound attributes and defines a defeasible conditional $\phi \mathrel|\joinrel\sim \psi$ evaluated over a preferential or ranked object order, enabling rational inference within FCA. A key contribution is ObjectRank, a method to derive a domain-consistent ranking from a defeasible theory $\Delta$ under $\Delta$-validity, leading to contextual rational closure (CRC) as the non-monotonic entailment notion in FCA; the authors prove soundness and completeness with respect to preferential and ranked semantics. The work emphasizes that contextual reasoning in FCA yields more relevant conclusions than propositional reasoning and discusses implications, examples, and future research on typical concepts and complexity of CRC.

Abstract

Defeasible conditionals are a form of non-monotonic inference which enable the expression of statements like "if $φ$ then normally $ψ$". The KLM framework defines a semantics for the propositional case of defeasible conditionals by construction of a preference ordering over possible worlds. The pattern of reasoning induced by these semantics is characterised by consequence relations satisfying certain desirable properties of non-monotonic reasoning. In FCA, implications are used to describe dependencies between attributes. However, these implications are unsuitable to reason with erroneous data or data prone to exceptions. Until recently, the topic of non-monotonic inference in FCA has remained largely uninvestigated. In this paper, we provide a construction of the KLM framework for defeasible reasoning in FCA and show that this construction remains faithful to the principle of non-monotonic inference described in the original framework. We present an additional argument that, while remaining consistent with the original ideas around non-monotonic reasoning, the defeasible reasoning we propose in FCA offers a more contextual view on inference, providing the ability for more relevant conclusions to be drawn when compared to the propositional case.

Figures (5)

• Figure 1: BaseRank
• Figure 2: RCProp
• Figure 3: A context of elements
• Figure 4: Preference over objects
• Figure 5: Illustration of contextual rational closure.

Theorems & Definitions (32)

• definition 1: Preferential Interpretation
• definition 2: Ranked Interpretation
• theorem 1: Soundness
• theorem 2: Completeness
• definition 3: Compound Attributes
• definition 4: Minimisation
• definition 5: Preferential Context
• lemma 1
• theorem 3: Soundness of Preferential Contexts
• definition 6: Derived Context