A Variable Occurrence-Centric Framework for Inconsistency Handling (Extended Version)
Yakoub Salhi
TL;DR
This paper tackles inconsistency in propositional bases by introducing occurrence-based constructs, notably Minimal Inconsistency Relations ($MIR$) and Maximal Consistency Relations ($MCR$), as syntactic duals on $Occ(K)$ that capture conflicts missed by MIS/MCS. It develops non-explosive inference by repairing the base through $MCR$-driven renaming via a fixed C-renaming $R$, yielding renaming-based bases $ ho_{ hicksim}(K)$ and associated inference relations that operate under classical entailment, while preserving all formulas and variables. An unusual occurrence-based semantics assigns truth to variable occurrences (via o-interpretations and a-minimal/b-minimal o-models) and links these models to $MCR$s, providing a robust foundation for occurrence-aware reasoning and connecting to existing frameworks like forgetting and LP_m. The work establishes MIR/MCR dualities, introduces O-MIS and BMCR concepts, and demonstrates how these notions give rise to multiple, non-explosive inference schemes with clear theoretical relationships to traditional approaches and potential practical benefits in reasoning under inconsistency. Overall, it lays a syntactic, occurrence-centric framework for diagnosing and repairing inconsistencies with potential extensions to non-classical logics and computational causes.
Abstract
In this paper, we introduce a syntactic framework for analyzing and handling inconsistencies in propositional bases. Our approach focuses on examining the relationships between variable occurrences within conflicts. We propose two dual concepts: Minimal Inconsistency Relation (MIR) and Maximal Consistency Relation (MCR). Each MIR is a minimal equivalence relation on variable occurrences that results in inconsistency, while each MCR is a maximal equivalence relation designed to prevent inconsistency. Notably, MIRs capture conflicts overlooked by minimal inconsistent subsets. Using MCRs, we develop a series of non-explosive inference relations. The main strategy involves restoring consistency by modifying the propositional base according to each MCR, followed by employing the classical inference relation to derive conclusions. Additionally, we propose an unusual semantics that assigns truth values to variable occurrences instead of the variables themselves. The associated inference relations are established through Boolean interpretations compatible with the occurrence-based models.