On Revising Fuzzy Belief Bases

We look at the problem of revising fuzzy belief bases, i.e., belief base revision in which both formulas in the base as well as revision-input formulas can come attached with varying degrees. Working within a very general framework for fuzzy logic which is able to capture certain types of uncertainty calculi as well as truth-functional fuzzy logics, we show how the idea of rational change from “crisp” base revision, as embodied by the idea of partial meet (base) revision, can be faithfully extended to revising fuzzy belief bases. We present and axiomatise an operation of partial meet fuzzy base revision and illustrate how the operation works in several important special instances of the framework. We also axiomatise the related operation of partial meet fuzzy base contraction.This paper is an extended version of a paper presented at the Nineteenth Conference on Uncertainty in Arti.cial Intelligence (UAI’03).     

Reversing the Levi identity

The AGM (Alchourrón-GÄrdenfors-Makinson) model of belief change is extended to cover changes on sets of beliefs that arenot closed under logical consequence (belief bases). Three major types of change operations, namely contraction, internal revision, and external revision are axiomatically characterized, and their interrelations are studied. In external revision, the Levi identity is reversed in the sense that onefirst adds the new belief to the belief base, and afterwards contracts its negation. It is argued that external revision represents an intuitively plausible way of revising one's beliefs. Since it typically involves the temporary acceptance of an inconsistent set of beliefs, it can only be used in belief representations that distinguish between different inconsistent sets of belief.     

Infinitary Belief Revision

This paper extends the AGM theory of belief revision to accommodate infinitary belief change. We generalize both axiomatization and modeling of the AGM theory. We show that most properties of the AGM belief change operations are preserved by the generalized operations whereas the infinitary belief change operations have their special properties. We prove that the extended axiomatic system for the generalized belief change operators with a Limit Postulate properly specifies infinite belief change. This framework provides a basis for first-order belief revision and the theory of revising a belief state by a belief state.     

Minimal Belief Change and the Pareto Principle

This paper analyzes the notion of a minimal belief change that incorporates new information. I apply the fundamental decision-theoretic principle of Pareto-optimality to derive a notion of minimal belief change, for two different representations of belief: First, for beliefs represented by a theory – a deductively closed set of sentences or propositions – and second for beliefs represented by an axiomatic base for a theory. Three postulates exactly characterize Pareto-minimal revisions of theories, yielding a weaker set of constraints than the standard AGM postulates. The Levi identity characterizes Pareto-minimal revisions of belief bases: a change of belief base is Pareto-minimal if and only if the change satisfies the Levi identity (for “maxichoice” contraction operators). Thus for belief bases, Pareto-minimality imposes constraints that the AGM postulates do not. The Ramsey test is a well-known way of establishing connections between belief revision postulates and axioms for conditionals (“if p, then q”). Pareto-minimal theory change corresponds exactly to three characteristic axioms of counterfactual systems: a theory revision operator that satisfies the Ramsey test validates these axioms if and only if the revision operator is Pareto-minimal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Conditional Ranking Revision

In the context of a general framework for belief dynamics which interprets revision as doxastic constraint satisfaction, we discuss a proposal for revising quasi-probabilistic belief measures with finite sets of graded conditionals. The belief states are ranking measures with divisible values (generalizing Spohn’s epistemology), and the conditionals are interpreted as ranking constraints. The approach is inspired by the minimal information paradigm and based on the principle-guided canonical construction of a ranking model of the input conditionals. This is achieved by extending techniques known from conditional default reasoning. We give an overview of how it handles different principles for conditional and parallel revision and compare it with similar accounts.     

Iterated Belief Revision

This is a discussion of the problem of extending the basic AGM belief revision theory to iterated belief revision: the problem of formulating rules, not only for revising a basic belief state in response to potential new information, but also for revising one’s revision rules in response to potential new information. The emphasis in the paper is on foundational questions about the nature of and motivation for various constraints, and about the methodology of the evaluation of putative counterexamples to proposed constraints. Some specific constraints that have been proposed are criticized. The paper emphasizes the importance of meta-information—information about one’s sources of information—and argues that little of substance can be said about constraints on iterated belief revision at a level of abstraction that lacks the resources for explicit representation of meta-information.

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A consistency-based framework for merging knowledge bases

We present a framework for expressing various merging operators for belief sets. This framework generalises our earlier work on consistency-based belief revision and contraction. Two primary merging operators are identified: in the first approach, belief sources are consistently combined so that the result of merging knowledge bases K₁,…,K_n is a maximal consistent (if possible) set of formulas comprising the joint knowledge of the knowledge bases. This approach then accords with one's intuitions as to what a “merge” operator should do. The second approach is more akin to a generalised belief revision operator. Knowledge bases K₁,…,K_n are “projected” onto another (in the simplest case the knowledge base where only tautologies are known). Properties of these operators are investigated, primarily by comparing their properties with postulates that have been identified previously in the literature. Notably, the approach is independent of syntax, in that merging knowledge bases K₁,…,K_n is independent of how each K_i is expressed. As well, we investigate the role of entailment-based and consistency-based integrity constraints, the interrelationships between these approaches and belief revision, and the expression of further merging operators.     

A feature-based inference model of numerical estimation: The split-seed effect

Prior research has identified two modes of quantitative estimation: numerical retrieval and ordinal conversion. In this paper we introduce a third mode, which operates by a feature-based inference process. In contrast to prior research, the results of three experiments demonstrate that people estimate automobile prices by combining metric information associated with two critical features: product class and brand status. In addition, Experiments 2 and 3 demonstrated that when participants are seeded with the actual current base price of one of the to-be-estimated vehicles, they respond by revising the general metric and splitting the information carried by the seed between the two critical features. As a result, the degree of post-seeding revision is directly related to the number of these features that the seed and the transfer items have in common. The paper concludes with a general discussion of the practical and theoretical implications of our findings.     

Stratified Belief Bases Revision with Argumentative Inference

We propose a revision operator on a stratified belief base, i.e., a belief base that stores beliefs in different strata corresponding to the value an agent assigns to these beliefs. Furthermore, the operator will be defined as to perform the revision in such a way that information is never lost upon revision but stored in a stratum or layer containing information perceived as having a lower value. In this manner, if the revision of one layer leads to the rejection of some information to maintain consistency, instead of being withdrawn it will be kept and introduced in a different layer with lower value. Throughout this development we will follow the principle of minimal change, being one of the important principles proposed in belief change theory, particularly emphasized in the AGM model. Regarding the reasoning part from the stratified belief base, the agent will obtain the inferences using an argumentative formalism. Thus, the argumentation framework will decide which information prevails when sentences of different layers are used for entailing conflicting beliefs. We will also illustrate how inferences are changed and how the status of arguments can be modified after a revision process.     

Belief Revision and Verisimilitude Based on Preference and Truth Orderings

In this rather technical paper we establish a useful combination of belief revision and verisimilitude according to which better theories provide better predictions, and revising with more verisimilar data results in theories that are closer to the truth. Moreover, this paper presents two alternative definitions of refined verisimilitude, which are more perspicuous than the algebraic version used in previous publications.     

Iterated belief change based on epistemic entrenchment

In this paper it is argued that, in order to solve the problem of iterated belief change, both the belief state and its input should be represented as epistemic entrenchment (EE) relations. A belief revision operation is constructed that updates a given EE relation to a new one in light of an evidential EE relation. It is shown that the operation in question satisfies generalized versions of the Gärdenfors revision postulates. The account offered is motivated by Spohn's ordinal conditionalization functions, and can be seen as the Jeffrization of a proposal considered by Rott.I am indebted John G. Bennett and Henry E. Kyburg. jr. for their insightful comments on an earlier draft of this paper. I thank the referees forErkenntnis who, apart from giving extended comments and suggestions, provided me with some hard-to-find relevant material. I also thank Prashanta Bandyopadhyay, Norman Foo, Maurice Pagnucco, Hans Rott and Mary-Anne Williams for their suggestions. The errors that remain are, of course, mine.     

Being proven wrong elicits learning in children – but only in those with higher executive function skills

This study investigated whether prompting children to generate predictions about an outcome facilitates activation of prior knowledge and improves belief revision. 51 children aged 9–12 were tested on two experimental tasks in which generating a prediction was compared to closely matched control conditions, as well as on a test of executive functions (EF). In Experiment 1, we showed that children exhibited a pupillary surprise response to events that they had predicted incorrectly, hypothesized to reflect the transient release of noradrenaline in response to cognitive conflict. However, children's surprise response was not associated with better belief revision, in contrast to a previous study involving adults. Experiment 2 revealed that, while generating predictions helped children activate their prior knowledge, only those with better inhibitory control skills learned from incorrectly predicted outcomes. Together, these results suggest that good inhibitory control skills are needed for learning through cognitive conflict. Thus, generating predictions benefits learning – but only among children with sufficient EF capacities to harness surprise for revising their beliefs.     

Reasoning with prioritized information by iterative aggregation of distance functions

We introduce a general framework for reasoning with prioritized propositional data by aggregation of distance functions. Our formalism is based on a possible world semantics, where conclusions are drawn according to the most ‘plausible’ worlds (interpretations), namely: the worlds that are as ‘close’ as possible to the set of premises, and, at the same time, are as ‘faithful’ as possible to the more reliable (or important) information in this set. This implies that the consequence relations that are induced by our framework are derived from a pre-defined metric on the space of interpretations, and inferences are determined by a ranking function applied to the premises. We study the basic properties of the entailment relations that are obtained by this framework, and relate our approach to other methods of maintaining incomplete and inconsistent information, most specifically in the contexts of (iterated) belief revision, consistent query answering in database systems, and integration of prioritized data sources.     

Postulates for revising BDI structures

The process of rationally revising beliefs in the light of new information is a topic of great importance and long-standing interest in artificial intelligence. Moreover, significant progress has been made in understanding the philosophical, logical, and computational foundations of belief revision. However, very little research has been reported with respect to the revision of other mental states, most notably propositional attitudes such as desires and intentions. In this paper, we present a first attempt to formulate a general framework for understanding the revision of mental states. We develop an abstract belief-desire-intention model of agents, and introduce a notion of rationality for this model. We then present a series of formal postulates characterizing the processes of adding beliefs, desires, and intentions, updating costs and values, and removing beliefs, desires, and intentions. We also investigate the computational complexity of several problems involving the abstract model and comment on algorithms for revision.     

Revocable Belief Revision

Krister Segerberg proposed irrevocable belief revision, to be contrasted with standard belief revision, in a setting wherein belief of propositional formulas is modelled explicitly. This suggests that in standard belief revision is revocable: one should be able to unmake (‘revoke’) the fresh belief in the revision formula, given yet further information that contradicts it. In a dynamic epistemic logical setting for belief revision, for multiple agents, we investigate what the requirements are for revocable belief revision. By this we not merely mean recovering belief in non-modal propositions, as in the recovery principle for belief contraction, but recovering belief in modal propositions: beliefs about beliefs. These requirements are almost never met, a surprising result.     

Logic Based Merging

Belief merging aims at combining several pieces of information coming from different sources. In this paper we review the works on belief merging of propositional bases. We discuss the relationship between merging, revision, update and confluence, and some links between belief merging and social choice theory. Finally we mention the main generalizations of these works in other logical frameworks.     

Belief revision,epistemic conditionals and the Ramsey test

Epistemic conditionals have often been thought to satisfy the Ramsey test (RT): If A, then B is acceptable in a belief state G if and only if B should be accepted upon revising G with A. But as Peter Gärdenfors has shown, RT conflicts with the intuitively plausible condition of Preservation on belief revision. We investigate what happens if (a) RT is retained while Preservation is weakened, or (b) vice versa. We also generalize Gärdenfors' approach by treating belief revision as a relation rather than as a function.In our semantic approach, the same relation is used to model belief revision and to give truth-conditions for conditionals. The approach validates a weak version of the Ramsey Test (WRR) — essentially, a restriction of RT to maximally consistent belief states.We prove that alternatives (a) and (b) are both consistent, but argue that (b) is philosophically more promising. Gärdenfors' belief-revision axioms are compatible with WRR together with RT from left to right: the only direction of the test that is defensible on intuitive grounds.An earlier version of this paper was presented at the conference on the dynamics of knowledge and belief at Lund University, 24–26 August 1989. We wish to thank Sven Danielsson, Peter Gärdenfors, Sören Halldén, David Makinson, Hugh Mellor, Michael Morreau, Nils-Eric Sahlin and Brian Skyrms for their very helpful suggestions and remarks. We are also grateful for thought-provoking criticism and comments from two anonymous referees.     

Global and Iterated Contraction and Revision: An Exploration of Uniform and Semi-Uniform Approaches

In order to clarify the problems of iterated (global) belief change it is useful to study simple cases, in particular consecutive contractions by sentences that are both logically and epistemically independent. Models in which the selection mechanism is kept constant are much more plausible in this case than what they are in general. One such model, namely uniform specified meet contraction, has the advantage of being closely connected with the AGM model. Its properties seem fairly adequate for the intended type of contraction. However, the revision operator based on it via the Levi identity collapses into an implausible operation that loses all old information when revising by new information. A weaker version, semi-uniform specified meet contraction, avoids the collapse but has the disadvantage of a remarkably weak logic. It is left as an open issue whether there is an intermediate class of contraction operators that yields a more satisfactory logic.     

A Single Counterexample Leads to Moral Belief Revision

What kind of evidence will lead people to revise their moral beliefs? Moral beliefs are often strongly held convictions, and existing research has shown that morality is rooted in emotion and socialization rather than deliberative reasoning. In addition, more general issues—such as confirmation bias—further impede coherent belief revision. Here, we explored a unique means for inducing belief revision. In two experiments, participants considered a moral dilemma in which an overwhelming majority of people judged that it was inappropriate to take action to maximize utility. Their judgments contradicted a utilitarian principle they otherwise strongly endorsed. Exposure to this scenario led participants to revise their belief in the utilitarian principle, and this revision persisted over several hours. This method provides a new avenue for inducing belief revision.     

Replacement—A Sheffer Stroke for Belief Change

By replacement is meant an operation that replaces one sentence by another in a belief set. Replacement can be used as a kind of Sheffer stroke for belief change, since contraction, revision, and expansion can all be defined in terms of it. Replacement can also be defined either in terms of contraction or in terms of revision. Close connections are shown to hold between axioms for replacement and axioms for contraction and revision. Partial meet replacement is axiomatically characterized. It is shown that this operation can have outcomes that are not obtainable through either partial meet contraction or partial meet revision.