Belief Revision (Computer Science)

In this paper we propose a geometric approach to the theory of evidence based on convex geometric interpretations of its two key notions of belief function and Dempster's sum. On one side, we analyze the geometry of belief functions as points of a polytope in the Cartesian space called belief space, and discuss the intimate relationship between basic probability assignment and convex combination. On the other side, we study the global geometry of Dempster's rule by describing its action on those convex combinations. By proving that Dempster's sum and convex closure commute we become able to depict the geometric structure of conditional subspaces, i.e. sets of belief functions conditioned by a given function b. Natural applications of these geometric methods to classical problems like probabilistic approximation and canonical decomposition are outlined.

In this paper, we develop a notion of doxastic actions, general enough to cover all examples of communication actions and most other belief-changing actions encountered in the literature, but also flexible enough to deal with the issue of (static and dynamic) revision of beliefs. This can be seen as a natural extension of the work in [3, 4] on “epistemic actions”, incorporating ideas from the semantics of belief revision and of conditional belief, along the lines pioneered in [2] and [11], but using the conditional belief approach adopted in [22, 10, 9] and adapted in [25] to the context of dynamic belief revision.

In this paper, we analyze from a geometric perspective the meaningful relations taking place between belief and probability functions in the framework of the geometric approach to the theory of evidence. Starting from the case of binary domains, we identify and study three major geometric entities relating a generic belief function (b.f.) to the set of probabilities P: 1) the dual line connecting belief and plausibility functions; 2) the orthogonal complement of P; and 3) the simplex of consistent probabilities. Each of them is in turn associated with a different probability measure that depends on the original b.f. We focus in particular on the geometry and properties of the orthogonal projection of a b.f. onto P and its intersection probability, provide their interpretations in terms of degrees of belief, and discuss their behavior with respect to affine combination.

We present a logic of conditional doxastic actions, obtained by incorporating ideas from belief revision theory into the usual dynamic logic of epistemic actions. We do this by extending to actions the setting of epistemic plausibility models, developed in Baltag and Smets (2006) for representing (static) conditional beliefs. We introduce
a natural extension of the notion of update product from Baltag and Moss (2004) to plausibility models.

We present a complete, decidable logic for reasoning about a notion of completely trustworthy (" conclusive ") evidence and its relations to justifiable (implicit) belief and knowledge, as well as to their explicit justifications. This logic makes use of a number of evidence-related notions such as availability, admissibility, and " goodness " of a piece of evidence, and is based on an innovative modification of the Fitting semantics for Artemov's Justification Logic designed to preempt Gettier-type counterexamples. We combine this with ideas from belief revision and awareness logics to provide an account for explicitly justified (defeasible) knowledge based on conclusive evidence that addresses the problem of (logical) omniscience.

In this paper we discuss the semantics and properties of the relative belief transform, a probability transformation of belief functions closely related to the classical plausibility transform. We discuss its rationale in both the probability-bound and Shafer's interpretations of belief functions. Even though the resulting probability (as it is the case for the plausibility transform) is not consistent with the original belief function, an interesting rationale in terms of optimal strategies in a non-cooperative game can be given in the probability-bound interpretation to both relative belief and plausibility of singletons. On the other hand, we prove that relative belief commutes with Dempster's orthogonal sum, meets a number of properties which are the duals of those met by the relative plausibility of singletons, and commutes with convex closure in a similar way to Dempster's rule. This supports the argument that relative plausibility and belief transform are indeed naturally associated with the D-S framework, and highlights a classification of probability transformations in two families, according to the operator they relate to. Finally, we point out that relative belief is only a member of a class of \relative mass" mappings, which can be interpreted as low-cost proxies for both plausibility and pignistic transforms.

On the basis of impossibility results on probability, belief revision, and conditionals, it is argued that conditional beliefs differ from beliefs in conditionals qua mental states. Once this is established, it will be pointed out in what sense conditional beliefs are still conditional, even though they may lack conditional contents, and why it is permissible to still regard them as beliefs, although they are not beliefs in conditionals. Along the way, the main logical, dispositional, representational, and normative properties of conditional beliefs are studied, and it is explained how the failure of not distinguishing conditional beliefs from beliefs in conditionals can lead philosophical and empirical theories astray.

This paper argues that there is a conflict between two theses held by John McDowell, namely i) the claim that we are under a standing obligation to revise our beliefs if reflection demands it; and ii) the view that veridical experience is a mode of direct access to the world. Since (i) puts no bounds on what would constitute reasonable doubt, it invites skeptical concerns which overthrow (ii). Conversely, since (ii) says that there are some experiences which we are entitled to trust, it undermines the prescriptive scope of (i). Drawing on C. S. Peirce’s distinction between genuine and contrived doubt, I maintain that critical revisions of beliefs should be triggered only by unwanted disruptions of habits, thereby restoring unity between McDowell’s two theses.

The abilities of detecting contradictions and rearranging the cognitive space in order to cope with them are important to be embedded in the BDI architecture of an agent acting in a complex and dynamic world. However, to be accomplished in a multi-agent environment, “belief revision” must depart considerably from its original definitions. According to us, the main changes should be the following ones: replacing the “priority to the incoming information principle” with the “recoverability principle”: any previously believed piece of information must belong to the current cognitive state whenever it is possible dealing not just with pieces of information but with couples <source, information> since the reliability of the source affects the credibility of the information and vice-versa. The “belief-function” formalism is here accepted as a simple and intuitive way to transfer the sources' reliability to the information's credibility.

Is information always true? According to some authors, including Dretske, Grice, Barwise, and recently, Floridi, who has defended the Veridicality Thesis, the answer is positive. For, on Floridi’s view, there is an intimate relation between information and knowledge, which is always true. It is argued in this article that information used in inferential knowledge can, nevertheless, be false, thereby showing that the Veridicality thesis is false.

The single most important statement that can be made with regard to the logical status of human and physical geographical reasoning is that it belongs to the class of non-monotonic reasoning. In other words,  geographical reasoning is defeasible (i.e. earlier drawn conclusions can be defeated by the addition of new facts, some of which are generated by the very application of our ideas outside academe) and non-demonstrative (i.e. our beliefs about reality cannot be demonstrated once and for ever; we must do with building provisional arguments based on the current weight of the evidence, which itself changes with the times). Indeed, unlike mathematics, which is a closed world of formalisms that relies heavily on deduction, demonstration, and proof, geography studies and attempts to improve the real, messy world out there. Because that wild, unruly world is always in a process of becoming, the conclusions drawn yesterday may have to be abandoned today, and so on, without end in sight. The core virtue of geographical reasoning is not certainty, but adaptability. As the world changes, so do geography’s entertained beliefs about how to improve it. The point we are trying to make is that an alternative window into the nature of geographical reasoning, besides the one provided by the discipline’s key concepts, offers an opening toward the more subtle problematic of the mechanisms by which the updating of geographical beliefs operate.

In this paper, we investigate the problem of truth approximation via belief merging, i.e., we ask whether, and under what conditions, a group of inquirers merging together their beliefs makes progress toward the truth about the underlying domain. We answer this question by proving some formal results on how belief merging operators perform with respect to the task of truth approximation, construed as increasing verisimilitude or truthlikeness. Our results shed new light on the issue of how rational (dis)agreement affects the inquirers' quest for truth. In particular, they vindicate the intuition that scientific inquiry, and rational discussion in general, benefits from some heterogeneity in opinion and interaction among different viewpoints. The links between our approach and related analyses of truth tracking, judgment aggregation, and opinion dynamics, are also highlighted.

This paper outlines a distinction between distributed truth maintenance and distributed belief revision. The latter has a more complex conceptualization than the former and it needs the evaluation of special features as the relationship between the informant's reliability and the information's credibility. We point out some criteria to judge the qualities of a distributed belief revision strategy from a global perspective. However, the performances of such a strategy can be estimated only on a simulation basis. The general framework for assumption-based distributed belief revision that we present has been tested on our specific “CLUEDO” multi-agent simulation testbed. The model has been continuously improved during many cycles of modification of the model, simulation, and evaluation of the performances, till the results presented in this paper are achieved

We study the learning power of iterated belief-revision methods. Successful learning is understood as convergence to correct, i.e., true, beliefs. We focus on the issue of universality: whether or not a particular belief-revision method is able to learn everything that in principle is learnable. We provide a general framework for interpreting belief-revision policies as learning methods. We focus on three popular cases: conditioning , lexicographic revision, and minimal revision. Our main result is that conditioning and lexicographic revision can drive a universal learning mechanism, provided that the observations include only and all true data, and provided that a non-standard, i.e., non-well-founded prior plausibility relation is allowed. We show that a standard, i.e., well-founded belief-revision setting is in general too narrow to guarantee universality of any learning method based on belief revision. We also show that minimal revision is not universal. Finally , we consider situations in which observational errors (false observations) may occur. Given a fairness condition, which says that only finitely many errors occur, and that every error is eventually corrected, we show that lexicographic revision is still universal in this setting, while the other two methods are not.

Most approaches to iterated belief revision are accompanied by
some motivation for the use of the proposed revision operator (or family of operators), and typically encode enough information in the epistemic state of an agent for uniquely determining one step revision. But in those approaches describing a family of operators there is usually little indication of how to proceed uniquely after the first revision step. In this paper we contribute towards addressing that deficiency by providing a formal framework which goes beyond the first revision step in two ways. First, the framework is obtained by enriching the epistemic state of an agent starting from the following intuitive idea: we associate to each world x two abstract objects x+ and x−, and we assume that, in addition to preferences over the set of worlds, we are given preferences over this set of objects as well. The latter can be considered as meta-information encoded in the epistemic state which enables us to go beyond the first revision step of the revision operator being applied, and to obtain a unique set of preferences over worlds. We then extend this framework to consider, not only the revision of preferences over worlds, but also the revision of this extended structure itself. We look at some desirable properties for revising the structure and prove the consistency of these properties by giving a concrete operator satisfying all of them. Perhaps more importantly, we show that this framework has strong connections with two other types of constructions in related areas. Firstly, it can be seen as a special case of preference aggregation which opens up the possibility of extending the framework presented here into a full-fledged framework for preference aggregation and social choice theory. Secondly, it is related to existing work on the use of interval orderings in a number of different contexts.

In modeling the knowledge processing structure of an Agent in a Multi-Agent world it becomes necessary to enlarge the traditional concept of Belief Revision. For detecting contradictions and identifying their sources it is sufficient to maintain informations about what has been told; but to "solve" a contradiction it is necessary to keep informations about who said it or, in general, about the source where that knowledge came from. We can take as certain the fact that an agent gave an information, but we can take the given information only as a revisable assumption. The Belief Revision system can't leave the sources of the informations out of consideration because of their relevance in giving the additional notion of "strength of belief" [Galliers 89]. In fact, the reliability of the source affects the credibility of the information and vice-versa. It is necessary to develop systems that deal with couples <assumption, source of the assumption>. In [Dragoni 91] we've proposed a system that moves in this direction. Here we give a short description of that system. In the first two parts we describe the agent's knowledge processing structure using a particular characterization of the "Assumption Based Belief Revision" concept; in part three we outline the project of an embedded device that enables the overall system to deal with couples <assumption, source of the assumption> in an rather anthropomorphic manner.

An argumentation framework can be seen as expressing, in an abstract way, the conflicting information of an under- lying logical knowledge base. This conflicting information often allows for the presence of more than one possible rea- sonable position (extension/labelling) which one can take. A relevant question, therefore, is how much these positions differ from each other. In the current paper, we will examine the issue of how to define meaningful measures of distance between the (complete) labellings of a given argumentation framework. We provide concrete distance measures based on argument-wise label difference, as well as based on the notion of critical sets, and examine their properties.

Standard belief change assumes an underlying logic containing full classical propositional logic. However, there are good reasons for considering belief change in less expressive logics as well. In this paper we build on recent investigations by Delgrande on contraction for Horn logic. We show that the standard basic form of contraction, partial meet, is too strong in the Horn case. This result stands in contrast to Delgrande’s conjecture that orderly maxichoice is the appropriate form of contraction for Horn logic. We then define a more appropriate notion of basic contraction for the Horn case, influenced by the convexity property holding for full propositional logic and which we refer to as infra contraction. The main contribution of this work is a result which shows that the construction method for Horn contraction for belief sets based on our infra remainder sets corresponds exactly to Hansson’s classical kernel contraction for belief sets, when restricted to Horn logic. This result is obtained via a detour through contraction for belief bases. We prove that kernel contraction for belief bases produces precisely the same results as the belief base version of infra contraction. The use of belief bases to obtain this result provides evidence for the conjecture that Horn belief change is best viewed as a ‘hybrid’ version of belief set change and belief base change. One of the consequences of the link with base contraction is the provision of a representation result for Horn contraction for belief sets in which a version of the Core-retainment postulate features.

Belief revision is the process of rearranging a knowledge base to preserve global consistency while accomodating incoming information. Early approaches to belief revision used symbolic model-theoretic, considering the problem as one of changing a logical theory. More recent approaches have adopted qualitative syntactic methods, taking them into the area of "truth maintenance systems", and numerical mathematical methods, thus moving into the mainstream literature of uncertainty management. Multi-agent systems, in which information may come from a variety of human or artificial sources with different degrees of reliability, seem to be a natural domain for belief revision. The aim of this paper is to give a synoptic perspective of this composite subject from the clear air of the high theoretical peaks down to the muddy plain of practical algorithms

The paper builds a belief hierarchy as a common framework to all uncertainty measures for which an actor is ambiguous about his uncertain beliefs. The belief hierarchy is moreover interpreted by distinguishing physical and psychical worlds, associated to objective and subjective probabilities. Various principles are introduced in order to transform a belief hierarchy, which are applied to establish some links between different contexts of belief change. The numerous belief change rules proposed in the literature can then be associated to specific belief hierarchies and to specific contexts of belief change and epistemic justification of these rules given. The papers brings new and original analysis of the relation between subjective and objective beliefs and on the possibility of learning some generic probabilistic belief despite its resiliency.

We introduce Propositional Typicality Logic (PTL), a logic for reasoning about typicality. We do so by enriching classical propositional logic with a typicality operator of which the intuition is to capture the most typical (or normal) situations in which a formula holds. The semantics is in terms of ranked models as studied in KLM-style preferential reasoning. This allows us to show that rational consequence relations can be embedded in our logic. Moreover we show that we can define consequence relations on the language of PTL itself, thereby moving beyond the propositional setting. Building on the existing link between propositional rational consequence and belief revision, we show that the same correspondence holds for rational consequence and belief revision on PTL. We investigate entailment for PTL, and propose two appropriate notions thereof.

In this paper we propose a credal representation of the interval probability associated with a belief function (b.f.), and show how it relates to several classical Bayesian transformations of b.f.s through the notion of “focus” of a pair of simplices.
While a belief function corresponds to a polytope of probabilities consistent with it, the related interval probability is geometrically represented by a pair of upper and lower simplices. Starting from the interpretation of the pignistic function as the center of mass of the credal set of consistent probabilities, we prove that relative belief of singletons, relative plausibility of singletons and intersection probability can all be described as foci of different pairs of simplices in the region of all probability measures. The formulation of frameworks similar to the Transferable Belief Model for such Bayesian transformations appears then at hand.

In a recent pair of publications, Richard Bradley has offered two novel no-go theorems involving the principle of 'Preservation' for conditionals, which guarantees that one's prior conditional beliefs will exhibit a certain degree of inertia in the face of a change in one's non-conditional beliefs. We first note that Bradley's original discussions of these results—in which he finds motivation for rejecting Preservation, first in a principle of 'Commutativity' , then in a doxastic analogue of the rule of Modus Ponens—are problematic in a significant number of respects. We then turn to a recent u-turn on his part, in which he winds up rescinding his commitment to Modus Ponens, on the grounds of a tension with the rule of Import-Export for conditionals. Here we offer an important positive contribution to the literature, settling the following crucial question that Bradley leaves unanswered: Assuming that one gives up on full-blown Modus Ponens on the grounds of its incompatibility with Import-Export, what weakened version of the principle should one be settling for instead? Our discussion of the issue turns out to unearth an interesting connection between epistemic undermining and the apparent failures of Modus Ponens in McGee's famous counterexamples.

We investigate the long-term behavior of iterated belief revision with higher-level doxastic information. While the classical literature on iterated belief revision [13, 11] deals only with propositional information, we are interested in learning (by an introspective agent, of some partial information about the) answers to various questions Q 1, Q 2, ..., Q n , ... that may refer to the agent’s own beliefs (or even to her belief-revision plans). Here, “learning” can be taken either in the “hard” sense (of becoming absolutely certain of the answer) or in the “soft” sense (accepting some answers as more plausible than others). If the questions are binary (“is φ true or not?”), the agent “learns” a sequence of true doxastic sentences φ 1, ..., φ n , .... “Investigating the long-term behavior” of this process means that we are interested in whether or not the agent’s beliefs, her “knowledge” and her conditional beliefs stabilize eventually or keep changing forever.

This article provides a discussion of the principle of transmission of evidential support across entailment from the perspective of belief revision theory in the AGM tradition. After outlining and briefly defending a small number of basic principles of belief change, which include a number of belief contraction analogues of the Darwiche–Pearl postulates for iterated revision, a proposal is made concerning the connection between evidential beliefs and belief change policies in rational agents. This proposal is found to be sufficient to establish the truth of a much-discussed intuition regarding transmission failure.

Abduction was first introduced in the epistemological context of scientific discovery. It was more recently analyzed in artificial intelligence, especially with respect to diagnosis analysis or ordinary reasoning. These two fields share a common view of abduction as a general process of hypotheses formation. More precisely, abduction is conceived as a kind of reverse explanation where a hypothesis H can be abduced from events E if H is a “good explanation” of E. The paper surveys four known schemes for abduction that can be used in both fields. Its first contribution is a taxonomy of these schemes according to a common semantic framework based on belief revision. Its second contribution is to produce, for each non-trivial scheme, a representation theorem linking its semantic framework to a set of postulates. Its third contribution is to present semantic and axiomatic arguments in favor of one of these schemes, “ordered abduction,” which has never been vindicated in the literature.

In this paper, we analyze Shafer’s belief functions (BFs) as geometric entities, focusing in particular on the geometric behavior of Dempster’s rule of combination in the belief space, i.e., the set of all the admissible BFs defined over a given finite domain Theta. The study of the orthogonal sums of affine subspaces allows us to unveil a convex decomposition of Dempster’s rule of combination in terms of Bayes’ rule of conditioning and prove that under specific conditions orthogonal sum and affine closure commute. A direct consequence of these results is the simplicial shape of the conditional subspaces , i.e., the sets of all the possible combinations of a given BF s. We show how Dempster’s rule exhibits a rather elegant behavior when applied to BFs assigning the same mass to a fixed subset (constant mass loci). The resulting affine spaces have a common intersection that is characteristic of the conditional subspace, called focus. The affine geometry of these foci eventually suggests an interesting geometric construction of the orthogonal sum of two BFs.

Most belief change operators in the AGM tradition assume an underlying plausibility ordering over the possible worlds which is transitive and complete. A unifying structure for these operators, based on supplementing the plausibility ordering with a second, guiding, relation over the worlds was presented in Booth et al, AIJ 2010. However it is not always reasonable to assume completeness of the underlying ordering. In this paper we generalise the structure of Booth et al to allow incomparabilities between worlds. We axiomatise the resulting class of belief removal functions, and show that it includes an important family of removal functions based on finite prioritised belief bases.

We investigate the logical and conceptual connections between abductive reasoning construed as a process of belief change, on the one hand, and truth approximation, construed as increasing (estimated) verisimilitude, on the other. We introduce the notion of '(verisimilitude-guided) abductive belief change' and discuss under what conditions abductively changing our theories or beliefs does lead them closer to the truth, and hence tracks truth approximation conceived as the main aim of inquiry. The consequences of our analysis for some recent discussions concerning belief revision aiming at truth approximation and inference to the best explanation are also highlighted.

We present a general framework for representing belief-revision rules and use it to characterize Bayes' rule as a classical example and Jeffrey's rule as a non-classical one. In Jeffrey's rule, the input to a belief revision is not simply the information that some event has occurred, as in Bayes' rule, but a new assignment of probabilities to some events. Despite their differences, Bayes' and Jeffrey's rules can be characterized in terms of the same axioms: responsiveness, which requires that revised beliefs incorporate what has been learnt, and conservativeness, which requires that beliefs on which the learnt input is 'silent' do not change. To illustrate the use of non-Bayesian belief revision in economic theory, we sketch a simple decision-theoretic application.

Since the seminal, philosophical and influential works of Alchourr'on, Gardenfors and Makinson, ideas on "belief revision" have been progressively refined toward normative, effective and computable paradigms. Side by side to this "symbolic" line of research, there has been also a "numerical" approach to belief revision whose main contributes were the probabilistic and the evidence-based approaches. The opinion expressed in this paper is that, to be applied in a multi-sources environment, belief revision has to depart considerably from the original framework. In particular, it has to abandon the fundamental principle of "Priority to the Incoming Information" in preference to what we called the principle of "Recoverability". Furthermore the semantic approach should be blended with a syntactic treatment of consistency inspired by the Truth Maintenance Systems

We investigate the issue of reaching doxastic agreement among the agents of a group by " sharing " information via successive acts of sincere, persuasive and public communication within the group. The topic relates to " preference aggregation " in Social Choice theory, where the problem is to find a natural and fair merge operation for aggregating the agents' preferences into a single group preference. In this paper we study this topic within the setting of Dynamic Epistemic Logic (DEL) and its recent extensions to belief revision theory. First we interpret the agents' preference relations as " doxastic preferences " or " doxastic plausibility orders ". Using these plausibility structures, we can express several types of relevant doxastic attitudes: ranging from an agent's simple beliefs and conditional beliefs to her defeasible knowledge and irrevocable knowledge. Next, we model communication via a series of doxastic/epistemic actions that affect the agent's plausibility structures. Because these structures can change in different ways, a group of agents has to adopt a specific communication protocol in order to aggregate their doxastic attitudes. We study the type of merges that are realizable via specific communication protocols and we highlight important factors which may influence the result, such as " the order of the speakers " , or the listeners' doxastic attitudes towards the speakers (i.e their opinions concerning the reliability of the incoming information).

The merging/fusion of belief/data collections in propositional logic form is a topic that has received due attention within the domains of database and AI research. A distinction can be made between two types of scenarios to which the process of merging can be applied. In the first type, the collections represent preferences, such as the voting choices of a group of people, that need to be aggregated so as to give a consistent result that in some way best represents the collective judgement of the group. In the second type, the collections represent factual data that is to be aggregated with an aim of obtaining a result that maximises factual correctness. After introducing a general framework for belief merging via some prominent literature on the topic, this paper then introduces and considers a method for belief merging with the second type of scenario in mind. Its suitability is corroborated by demonstrating how it can be seen as a special case of a merging procedure that combines aggregation of probabilities and maximisation of expected truthlikeness.

Many belief change formalisms employ plausibility orderings over the set of possible worlds to determine how the beliefs of an agent ought to be modified after the receipt of a new epistemic input. While most such possible world semantics rely on a single order-
ing, we investigate the use of an additional preference ordering—representing, for instance, the epistemic context the agent finds itself in—to guide the process of belief change. We show that the resultant formalism provides a unifying semantics for a wide variety of belief
change operators. By varying the conditions placed on the second ordering, different families of known belief change operators can be captured, including AGM belief contraction and revision, Rott and Pagnucco’s severe withdrawal, the systematic withdrawal of Meyer et al, as well as the linear liberation and σ-liberation operators of Booth et al. Our approach also identifies novel classes of belief change operators worthy of further investigation.

The AGM theory of belief revision is based on propositional belief sets. In this paper we develop a logic for revision of temporal belief bases, containing expressions about temporal propositions (tomorrow it will rain), possibility (it may rain tomorrow), actions (the robot enters the room) and pre- and post-conditions of these actions. We prove the Katsuno-Mendelzon and the Darwiche-Pearl representation theorems by restricting the logic to formulas representing beliefs up to certain time. We illustrate our belief change model through several examples.