Representation Theorems for Cumulative Propositional Dependence Logics

2026-02-24

AI Summary

This study addresses cumulative propositional dependence logic and its variant with team semantics, aiming to establish precise representation theorems that characterize their entailment relations. By constructing cumulative models and asymmetric cumulative models, the work provides semantic characterizations of cumulative propositional dependence logic under System C and cumulative propositional logic with team semantics, respectively, and demonstrates their equivalence within the classical framework of cumulative logic. The research offers the first exact semantic characterization of System C in propositional dependence logic based on Kraus–Lehmann–Magidor models and extends this characterization to team semantics. Consequently, it successfully establishes representation theorems for both logical systems, delivering a general proof paradigm for cumulative logics lacking negation and material implication.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper establishes and proves representation theorems for cumulative propositional dependence logic and for cumulative propositional logic with team semantics. Cumulative logics are famously given by System C. For propositional dependence logic, we show that System C entailments are exactly captured by cumulative models from Kraus, Lehmann and Magidor. On the other hand, we show that entailment in cumulative propositional logics with team semantics is exactly captured by cumulative and asymmetric models. For the latter, we also obtain equivalence with cumulative logics based on propositional logic with classical semantics. The proofs will be useful for proving representation theorems for other cumulative logics without negation and material implication.

Problem

• cumulative logic
• propositional dependence logic
• team semantics
• representation theorem
• entailment

Innovation

• cumulative logic
• propositional dependence logic
• team semantics
• representation theorem
• System C