[Paper Review] Distributing Knowledge into Simple Bases

Adrian Haret, Jean-Guy Mailly+1 more|arXiv (Cornell University)|Mar 31, 2016

Logic, Reasoning, and Knowledge16 references3 citations

TL;DR

This paper introduces the concept of knowledge distributability, where an arbitrary knowledge base is decomposed into simpler fragments (e.g., 1CNF, 2CNF, Horn) such that merging via specific operators reconstructs the original knowledge. The key finding is that drastic-distance-based merging enables full distributability across fragments, while Hamming-distance-based operators show varying power depending on the fragment and aggregation function, with 2CNF and Horn allowing non-trivial distribution under certain operators.

ABSTRACT

Understanding the behavior of belief change operators for fragments of classical logic has received increasing interest over the last years. Results in this direction are mainly concerned with adapting representation theorems. However, fragment-driven belief change also leads to novel research questions. In this paper we propose the concept of belief distribution, which can be understood as the reverse task of merging. More specifically, we are interested in the following question: given an arbitrary knowledge base $K$ and some merging operator $Δ$, can we find a profile $E$ and a constraint $μ$, both from a given fragment of classical logic, such that $Δ_μ(E)$ yields a result equivalent to $K$? In other words, we are interested in seeing if $K$ can be distributed into knowledge bases of simpler structure, such that the task of merging allows for a reconstruction of the original knowledge. Our initial results show that merging based on drastic distance allows for an easy distribution of knowledge, while the power of distribution for operators based on Hamming distance relies heavily on the fragment of choice.

Motivation & Objective

• To investigate whether arbitrary knowledge bases can be decomposed into simpler logical fragments such that merging reconstructs the original knowledge.
• To analyze the power of different belief merging operators (based on drastic and Hamming distances) in reconstructing arbitrary knowledge from profiles in restricted fragments.
• To explore the theoretical and practical implications of knowledge distribution, including information hiding and storage constraints in limited-logic systems.
• To characterize the limits of distributability and simplifiability across fragments like 1CNF, 2CNF, and Horn, under various merging operators.

Proposed method

• Proposes a formal framework for knowledge distributability, where a knowledge base K is reconstructed from a profile E and integrity constraint μ via a merging operator Δμ.
• Uses IC-merging with integrity constraints and evaluates reconstruction using model equivalence (Δμ(E) ≡ K).
• Applies merging operators based on drastic distance (ΔD) and Hamming distance (ΔH) with three aggregation functions: Σ, GMax, GMin.
• Employs model-theoretic analysis to compare distances between interpretations and knowledge base models, ensuring minimal changes under the chosen distance metric.
• Constructs explicit profiles of knowledge bases in target fragments (e.g., Horn-expressible KBs) to demonstrate reconstruction for specific cases.
• Analyzes cases based on model structure (e.g., pairwise incomparability, subset relations) to prove distributability or limitations.

Experimental results

Research questions

• RQ1 Can any arbitrary knowledge base be distributed into a profile of simpler logical fragments such that merging reconstructs the original knowledge?
• RQ2 How does the choice of merging operator—specifically based on drastic or Hamming distance—affect the distributability of knowledge across fragments?
• RQ3 Which fragments (1CNF, 2CNF, Horn) allow full distributability, and which only partial or trivial cases, under different merging operators?
• RQ4 Can knowledge be simplified (i.e., distributed into a single simpler knowledge base) using specific merging operators?
• RQ5 What is the role of aggregation functions (Σ, GMax, GMin) in determining the power of merging operators for knowledge distribution?

Key findings

• Knowledge distributability using drastic-distance-based merging (ΔD) is possible for any knowledge base across all fragments, including 1CNF, 2CNF, and Horn.
• For Hamming-distance-based merging with Σ aggregation (ΔH,Σ), full distributability is only possible for 2CNF and Horn fragments, not for 1CNF.
• Distributability with ΔH,GMax and ΔH,GMin is possible for 2CNF, but not for 1CNF, and limited for Horn, with no full coverage.
• Simplifiability (distribution into a single knowledge base) is only possible for trivial cases in 1CNF and 2CNF under ΔD, but not in general.
• For 2CNF, arbitrary knowledge bases can be both distributed and simplified using appropriate merging operators, indicating high expressive power.
• The results show that merging operators behave differently depending on the distance metric and aggregation function, with no uniform categorization possible.