Belief Revision: Fundamentals and Applications

By Fouad Sabry

What Is Belief Revision

The process of altering one's views in order to take into account a new piece of knowledge is referred to as belief revision. Philosophy, database design, and artificial intelligence are all areas of study that are contributing to research on the logical formalization of belief revision for the construction of rational beings.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Belief revision

Chapter 2: Axiom

Chapter 3: Deductive reasoning

Chapter 4: Abductive reasoning

Chapter 5: Inductive logic programming

Chapter 6: Non-monotonic logic

Chapter 7: Description logic

Chapter 8: Dempster-Shafer theory

Chapter 9: Default logic

Chapter 10: Epistemic modal logic

(II) Answering the public top questions about belief revision.

(III) Real world examples for the usage of belief revision in many fields.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of belief revision.

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• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 30, 2023

Book preview

Chapter 1: Belief revision

The process of revising one's beliefs in response to new information is known as belief revision. Philosophy, databases, and artificial intelligence all conduct study on the logical formalization of belief revision for the creation of rational agents.

The fact that there may be multiple distinct ways to carry out this procedure makes belief revision non-trivial.

For example, if the current knowledge includes the three facts A is true, B is true and if A and B are true then C is true, the introduction of the new information C is false can be done preserving consistency only by removing at least one of the three facts.

Given this,, There are a minimum of three main approaches to revision.

In general, There could be a variety of methods for transforming knowledge.

Typically, two types of modifications are distinguished:

update

The old beliefs refer to the past, but the new information is about the current situation; updating is the process of revising the previous beliefs to account for the change; revision

Inconsistencies between the new and old knowledge can be explained by the likelihood that the old information is less accurate than the new information; both the old beliefs and the new information belong to the same circumstance; Revision is the process of incorporating new knowledge into an established body of beliefs without creating contradictions.

The primary premise of belief revision is that there should be little to no change; knowledge prior to and following the modification should be as comparable as possible. This principle formalizes the notion of inertia in the context of updating. This rule mandates that as much information as feasible be kept by the change in the case of revision.

The following classic example demonstrates that the actions to be taken in the two update and revision settings are different.

The example is based on two different interpretations of the set of beliefs \{a\vee b\} and the new piece of information \neg a :

update

In this instance, two satellites, The units A and B, revolve around Mars; The satellites are set up to land while sending updates to Earth; and one of the satellites sent a communication to Earth, telling people that it's still in orbit.

However, owing to obstruction, Which satellite sent the signal is unknown; subsequently, Communication that Unit A has landed reaches Earth.

This scenario can be modeled in the following way: two propositional variables a and b indicate that Unit A and Unit B, respectively, remain in orbit; the initial set of beliefs is \{a\vee b\} (either one of the two satellites is still in orbit) and the new piece of information is \neg a (Unit A has landed, and is not in orbit as a result).

The only rational result of the update is \neg a ; because the Unit A may have been the source of the initial indication that one of the two satellites had not yet landed, Unknown is the location of Unit B.

revision

In one of the two nearby theaters, the play Six Characters in Search of an Author will be presented.

This information can be denoted by \{a\vee b\} , where a and b indicates that the play will be performed at the first or at the second theatre, respectively; a further information that Jesus Christ Superstar will be performed at the first theatre indicates that \neg a holds.

Given this,, It follows naturally that Six Characters in Search of an Author will be presented at the second theater but not the first, which is represented in logic by \neg a\wedge b .

This example shows that revising the belief a\vee b with the new information \neg a produces two different results \neg a and \neg a\wedge b depending on whether the setting is that of update or revision.

A distinction between different operations that can be carried out is made in the context where all beliefs refer to the same circumstance:

contraction

elimination of a belief; expansion

adding a belief without first ensuring its consistency; revision

adding a conviction while retaining its consistency; extraction

obtaining a coherent set of beliefs and/or an order of the epistemic entrenchments; consolidation

restoring a set of beliefs' coherence; merging

the consistent integration of two or more sets of beliefs.

The first operation is carried out when the new belief to incorporate is thought to be more dependable than the old ones; hence, consistency is maintained by deleting some of the previous beliefs. This is where revision and merging vary. Since the priority among the belief sets may or may not be the same, merging is a more broad operation.

It is possible to revise something by first absorbing the new information and then consolidating everything to make it consistent. Given that the new information is not necessarily regarded as being more credible than the previous knowledge, this is actually more of a merging than a revision.

The AGM postulates (named after their proponents Alchourrón, Gärdenfors, An operator performing revision must satisfy certain conditions (e.g., and Makinson) in order to be deemed rational.

The environment being thought about is one of revision, that is, various sources of information referencing the same circumstance.

Considered are three operations: expansion (addition of a belief without a consistency check), revision (addition of a belief while maintaining consistency), both contraction (removal of a belief).

The basic AGM postulates are the first six postulates.

In the settings considered by Alchourrón, Gärdenfors, and Makinson, the current set of beliefs is represented by a deductively closed set of logical formulae K called belief set, the new piece of information is a logical formula P , and revision is performed by a binary operator * that takes as its operands the current beliefs and the new information and produces as a result a belief set representing the result of the revision.

The + operator denoted expansion: K+P is the deductive closure of K\cup \{P\} .

The AGM's proposed revisions are:

Closure: K*P is a belief set (i.e, a collection of closed deductive formulas); Success: P\in K*P

Inclusion: K*P\subseteq K+P

Vacuity:

{\text{If }}(\neg P)\not \in K,{\text{ then }}K*P=K+P

K*P is inconsistent only if P is inconsistent

Extensionality:

{\text{If }}P{\text{ and }}Q{\text{ are logically equivalent, then }}K*P=K*Q

(see logical equivalence)

K*(P\wedge Q)\subseteq (K*P)+Q{\text{If }}(\neg Q)\not \in K*P{\text{ then }}(K*P)+Q\subseteq K*(P\wedge Q)

The full meet revision is a revision operator that satisfies each of the eight postulates, in which K*P is equal to K+P if consistent, and to the deductive closure of P otherwise.

while meeting all AGM stipulations, This revision operator is viewed as being too cautious, in that if the rewriting formula is incompatible with the old knowledge base, no information from it is kept.

In particular, they are analogous to the revision operator being definable in terms of structures known as selection functions, epistemic entrenchments, systems of spheres, and preference relations. The AGM postulates are equal to a number of requirements on the revision operator. These latter relationships span a collection of models and are reflexive, transitive, and total.

Each revision operator * satisfying the AGM postulates is associated to a set of preference relations \leq _{K} , one for each possible belief set K , such that the models of K are exactly the minimal of all models according to \leq _{K} .

The revision operator and its associated family of orderings are related by the fact that K*P is the set of formulae whose set of models contains all the minimal models of P according to \leq _{K} .

This condition is equivalent to the set of models of K*P being exactly the set of the minimal models of P according to the ordering \leq _{K} .

A preference ordering \leq _{K} represents an order of implausibility among all situations, includes those that are plausible but nevertheless viewed as false.

The knowledge base models are the minimal models in accordance with such an ordering, which models are now seen as being the most likely.

In fact, all other models are deemed to be less credible and are greater than these ones.

In general, I<_{K}J indicates that the situation represented by the model I is believed to be more plausible than the situation represented by J .

The result is, revising by a formula having I and J as models should select only I to be a model of the revised knowledge base, as this model represent the most likely scenario among those supported by P .

Contraction is the operation of removing a belief P from a knowledge base K ; the result of this operation is denoted by K-P .

The identities of Levi and Harper connect the operators of revision and contractions:

K*P=(K-\neg P)+PK-P=K\cap (K*\neg P)

For contraction, eight postulates have been established. When one of the eight postulates for revision is met, the corresponding postulates for contraction are met by the associated contraction operator, and vice versa. The original contraction operator is obtained by converting a contraction operator into a revision operator, then