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The Realizability of Revision and Contraction Operators in Epistemic Spaces

Kai Sauerwald, Matthias Thimm

TL;DR

The paper investigates the realizability of AGM-style belief revision and contraction within the framework of epistemic spaces $\mathbb{E}=\langle \mathcal{E}, \mathrm{Bel}\rangle$, showing that such operators exist only in precisely characterized spaces. It introduces linear change operators as canonical realizations when realizable and provides necessary and sufficient conditions for the realizability of revision and contraction, including equivalences across contraction types and between full-meet and maxichoice variants. The results demonstrate that revision and contraction need not be interdefinable in epistemic spaces, and that full-meet revision requires strong unbiased conditions. Practically, these findings enable pre-checks for the applicability of belief-change in a domain and guide the design of domain-specific revision/contraction mechanisms, with avenues for iterated and more expressive belief-change formalisms.

Abstract

This paper studies the realizability of belief revision and belief contraction operators in epistemic spaces. We observe that AGM revision and AGM contraction operators for epistemic spaces are only realizable in precisely determined epistemic spaces. We define the class of linear change operators, a special kind of maxichoice operator. When AGM revision, respectively, AGM contraction, is realizable, linear change operators are a canonical realization.

The Realizability of Revision and Contraction Operators in Epistemic Spaces

TL;DR

The paper investigates the realizability of AGM-style belief revision and contraction within the framework of epistemic spaces , showing that such operators exist only in precisely characterized spaces. It introduces linear change operators as canonical realizations when realizable and provides necessary and sufficient conditions for the realizability of revision and contraction, including equivalences across contraction types and between full-meet and maxichoice variants. The results demonstrate that revision and contraction need not be interdefinable in epistemic spaces, and that full-meet revision requires strong unbiased conditions. Practically, these findings enable pre-checks for the applicability of belief-change in a domain and guide the design of domain-specific revision/contraction mechanisms, with avenues for iterated and more expressive belief-change formalisms.

Abstract

This paper studies the realizability of belief revision and belief contraction operators in epistemic spaces. We observe that AGM revision and AGM contraction operators for epistemic spaces are only realizable in precisely determined epistemic spaces. We define the class of linear change operators, a special kind of maxichoice operator. When AGM revision, respectively, AGM contraction, is realizable, linear change operators are a canonical realization.
Paper Structure (10 sections, 10 theorems, 14 equations, 2 figures)

This paper contains 10 sections, 10 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Propositional Logic, Minima and Orders
  3. Epistemic Spaces, Revision and Contraction
  4. Realizability of Contraction Operators
  5. Realizability of Revision Operators
  6. Consequences for the Interdefinability of Revision and Contraction
  7. Conclusion and Discussion
  8. Supplementary Material
  9. Background
  10. Proofs

Key Result

Theorem 6

Let $\mathbb{E} = \langle \mathcal{E},\mathop{\mathrm{Bel}}\nolimits \rangle$ be an epistemic space. AGM contraction is realizable in $\mathbb{E}$ if and only if $\mathbb{E}$ satisfies:

Figures (2)

  • Figure 1: Graphical representation of the contraction operator $\div$ for $\mathbb{E}$ given in Example \ref{['ex:es_contraction_operator']}. Nodes are the epistemic states of $\mathbb{E}$. Edges represent the behaviour of $\div$; there is an edge from $\Psi_1$ to $\Psi_2$ with label $x$ if $x\equiv\alpha$ implies $\Psi_2 = \Psi_1 \div \alpha$ hold. We use $*$ as placeholder label that stand for all $x$ that are not explicitly mentioned.
  • Figure 2: Graphical representation of the revision operator $\star$ for $\mathbb{E}$ given in Example \ref{['ex:es_revision_operator']}. Due to space reasons we omitted most labels.

Theorems & Definitions (31)

  • Definition 1: ? ?; adapted
  • Definition 2
  • Definition 3
  • Definition 4: Adapted, ?, ?
  • Definition 5: Contraction Realizability
  • Theorem 6
  • Definition 7
  • Definition 8
  • Example 9
  • Proposition 10
  • ...and 21 more