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Belief Change based on Knowledge Measures

Umberto Straccia, Giovanni Casini

TL;DR

The paper develops a quantitative Belief Change framework grounded in Knowledge Measures (KMs) by introducing the principle of minimal surprise to drive contraction, expansion, and revision. It generalizes KMs via information-theoretic surp risal under a world-probability distribution, defines KM-based BC operators that satisfy the AGM postulates, and proves a representation result: any AGM BC operator can be encoded as a KM-based operator. It further provides quantitative metrics for information loss (contraction), information gain (expansion), and information change (revision), and discusses iterated revision and a severe withdrawal variant that does not satisfy the recovery postulate. The approach links probabilistic semantics with KM-based updates, offering a principled, information-theoretic foundation for rational belief management with potential applications across AI systems that reason under uncertainty.

Abstract

Knowledge Measures (KMs) aim at quantifying the amount of knowledge/information that a knowledge base carries. On the other hand, Belief Change (BC) is the process of changing beliefs (in our case, in terms of contraction, expansion and revision) taking into account a new piece of knowledge, which possibly may be in contradiction with the current belief. We propose a new quantitative BC framework that is based on KMs by defining belief change operators that try to minimise, from an information-theoretic point of view, the surprise that the changed belief carries. To this end, we introduce the principle of minimal surprise. In particular, our contributions are (i) a general information-theoretic approach to KMs for which [1] is a special case; (ii) KM-based BC operators that satisfy the so-called AGM postulates; and (iii) a characterisation of any BC operator that satisfies the AGM postulates as a KM-based BC operator, i.e., any BC operator satisfying the AGM postulates can be encoded within our quantitative BC framework. We also introduce quantitative measures that account for the information loss of contraction, information gain of expansion and information change of revision. We also give a succinct look into the problem of iterated revision, which deals with the application of a sequence of revision operations in our framework, and also illustrate how one may build from our KM-based contraction operator also one not satisfying the (in)famous recovery postulate, by focusing on the so-called severe withdrawal model as an illustrative example.

Belief Change based on Knowledge Measures

TL;DR

The paper develops a quantitative Belief Change framework grounded in Knowledge Measures (KMs) by introducing the principle of minimal surprise to drive contraction, expansion, and revision. It generalizes KMs via information-theoretic surp risal under a world-probability distribution, defines KM-based BC operators that satisfy the AGM postulates, and proves a representation result: any AGM BC operator can be encoded as a KM-based operator. It further provides quantitative metrics for information loss (contraction), information gain (expansion), and information change (revision), and discusses iterated revision and a severe withdrawal variant that does not satisfy the recovery postulate. The approach links probabilistic semantics with KM-based updates, offering a principled, information-theoretic foundation for rational belief management with potential applications across AI systems that reason under uncertainty.

Abstract

Knowledge Measures (KMs) aim at quantifying the amount of knowledge/information that a knowledge base carries. On the other hand, Belief Change (BC) is the process of changing beliefs (in our case, in terms of contraction, expansion and revision) taking into account a new piece of knowledge, which possibly may be in contradiction with the current belief. We propose a new quantitative BC framework that is based on KMs by defining belief change operators that try to minimise, from an information-theoretic point of view, the surprise that the changed belief carries. To this end, we introduce the principle of minimal surprise. In particular, our contributions are (i) a general information-theoretic approach to KMs for which [1] is a special case; (ii) KM-based BC operators that satisfy the so-called AGM postulates; and (iii) a characterisation of any BC operator that satisfies the AGM postulates as a KM-based BC operator, i.e., any BC operator satisfying the AGM postulates can be encoded within our quantitative BC framework. We also introduce quantitative measures that account for the information loss of contraction, information gain of expansion and information change of revision. We also give a succinct look into the problem of iterated revision, which deals with the application of a sequence of revision operations in our framework, and also illustrate how one may build from our KM-based contraction operator also one not satisfying the (in)famous recovery postulate, by focusing on the so-called severe withdrawal model as an illustrative example.
Paper Structure (16 sections, 42 theorems, 74 equations, 3 figures, 2 tables)

This paper contains 16 sections, 42 theorems, 74 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contribution
  3. Background
  4. An information-theoretic approach to knowledge measures
  5. Knowledge measures recap
  6. An information-theoretic approach to knowledge measures
  7. KMs-based belief change
  8. Contraction
  9. Sphere-based view of contraction
  10. A contraction without recovery
  11. Belief revision
  12. Iterated belief revision in brief
  13. Conclusions, related and future work
  14. Closely related work
  15. Future Work
  16. ...and 1 more sections

Key Result

Proposition 1

For a formula $\phi$, $\Sigma_\phi \subseteq \bar{\Sigma} \subseteq \Sigma$ and a probability distribution $\mathbb{P}_{\bar{\Sigma}}$, we have that $\mathbb{P}_\Sigma(\phi) = \mathbb{P}_{\bar{\Sigma}}(\phi)$, where $\mathbb{P}_\Sigma$ is the extension of $\mathbb{P}_{\bar{\Sigma}}$.

Figures (3)

  • Figure 1: (a) System of spheres; (b) System of spheres with contraction $\phi^\div_\alpha$ shaded.
  • Figure 2: System of spheres for severe withdrawal showing $\phi^{\mathbin{{\div}\mkern-10mu{\div}}}_\alpha$ shaded.
  • Figure 3: System of spheres where $\phi^{\div_{\kappa}}_{\alpha \land \beta} \leq_{\mathbb{P}} \beta$ holds.

Theorems & Definitions (86)

  • Remark 1
  • Remark 2
  • Remark 3
  • Example 1: Running example
  • Example 2: Straccia20
  • Example 3: Running example cont.
  • Remark 4
  • Proposition 1
  • Example 4: Running example cont.
  • Remark 5
  • ...and 76 more