AGM Revision Research Articles

A Basis for AGM Revision in Bayesian Probability Revision

In standard Bayesian probability revision, the adoption of full beliefs (propositions with probability 1) is irreversible. Once an agent has full belief in a proposition, no subsequent revision can remove that belief. This is an unrealistic feature, and it also makes probability revision incompatible with belief change theory, which focuses on how the set of full beliefs is modified through both additions and retractions. This problem in probability theory can be solved in a model that (i) lets the codomain of the probability function be a hyperreal-valued rather than the real-valued closed interval [0, 1], and (ii) identifies the full beliefs as the propositions whose probability is either 1 or infinitesimally smaller than 1. In this model, changes in the probability function will result in changes in the set of full beliefs (belief set), which constitutes a submodel that can be conceived as the “tip of the iceberg” within the larger model that also contains beliefs on lower levels of probability. The patterns of change in the set of full beliefs in this modified Bayesian model coincides with the corresponding pattern in a slightly modified version of AGM revision, which is commonly conceived as the gold standard of (dichotomous) belief change. The modification only concerns the marginal case of revision by an inconsistent input sentence. These results show that probability revision and dichotomous belief change can be unified in one and the same framework, or – if we so wish – that belief change theory can be subsumed under a modified version of probability revision that allows for iterated change and for the removal of full beliefs.

Iterated AGM Revision Based on Probability Revision

Close connections between probability theory and the theory of belief change emerge if the codomain of probability functions is extended from the real-valued interval [0, 1] to a hyperreal interval with the same limits. Full beliefs are identified as propositions with a probability at most infinitesimally smaller than 1. Full beliefs can then be given up, and changes in the set of full beliefs follow a pattern very close to that of AGM revision. In this contribution, iterated revision is investigated. The iterated changes in the set of full beliefs generated by repeated revisions of a hyperreal probability function can, semantically, be modelled with the same basic structure as the sphere models of belief change theory. The changes on the set of full beliefs induced by probability revision satisfy the Darwiche–Pearl postulates for iterated belief change.

Four Approaches to Supposition

Suppositions can be introduced in either the indicative or subjunctive mood. There are also two types of judgments that may be initiated by suppositions of either sort: qualitative (binary) judgments and quantitative (numerical) ones. The former are judgments about whether a given proposition is acceptable and the latter are about how acceptable it is. We systematically explicate the relationships between canonical representatives of each of the four available types of theories. For the qualitative accounts, our representative theories of indicative and subjunctive supposition are based on the belief change operations given by AGM revision and KM update respectively; for the quantitative ones, we consider those given by conditionalization and imaging. This choice is motivated by the familiar approach of understanding supposition as ‘provisional belief revision’ wherein one temporarily treats the supposition as true and forms judgments by making appropriate changes to their other opinions. To compare the numerical judgments recommended by the quantitative theories with the binary ones recommended by the qualitative accounts, we appeal to a suitably adapted version of the Lockean thesis. Ultimately, we establish a number of new results that we interpret as vindicating the often-repeated claim that conditionalization is a probabilistic version of revision, while imaging is a probabilistic version of update.

Filtered Belief Revision: Syntax and Semantics

In an earlier paper [Rational choice and AGM belief revision, Artificial Intelligence, 2009] a correspondence was established between the set-theoretic structures of revealed-preference theory (developed in economics) and the syntactic belief revision functions of the AGM theory (developed in philosophy and computer science). In this paper we extend the re-interpretation of those structures in terms of one-shot belief revision by relating them to the trichotomous attitude towards information studied in Garapa (Rev Symb Logic, 1–21, 2020) where information may be either (1) fully accepted or (2) rejected or (3) taken seriously but not fully accepted. We begin by introducing the syntactic notion of filtered belief revision and providing a characterization of it in terms of a mixture of both AGM revision and contraction. We then establish a correspondence between the proposed notion of filtered belief revision and the above-mentioned set-theoretic structures, interpreted as semantic partial belief revision structures. We also provide an interpretation of the trichotomous attitude towards information in terms of the degree of implausibility of the information.

Relevance in Belief Update

It has been pointed out by Katsuno and Mendelzon that the so-called AGM revision operators, defined by Alchourrón, Gärdenfors and Makinson, do not behave well in dynamically-changing applications. On that premise, Katsuno and Mendelzon formally characterized a different type of belief-change operators, typically referred to as KM update operators, which, to this date, constitute a benchmark in belief update. In this article, we show that there exist KM update operators that yield the same counter-intuitive results as any AGM revision operator. Against this non-satisfactory background, we prove that a translation of Parikh’s relevance-sensitive axiom (P), in the realm of belief update, suffices to block this liberal behaviour of KM update operators. It is shown, both axiomatically and semantically, that axiom (P) for belief update, essentially, encodes a type of relevance that acts at the possible-worlds level, in the context of which each possible world is locally modified, in the light of new information. Interestingly, relevance at the possible-worlds level is shown to be equivalent to a form of relevance that acts at the sentential level, by considering the building blocks of relevance to be the sentences of the language. Furthermore, we concretely demonstrate that Parikh’s notion of relevance in belief update can be regarded as (at least a partial) solution to the frame, ramification and qualification problems, encountered in dynamically-changing worlds. Last but not least, a whole new class of well-behaved, relevance-sensitive KM update operators is introduced, which generalize Forbus’ update operator and are perfectly-suited for real-world implementations.

PROBABILISTIC STABILITY, AGM REVISION OPERATORS AND MAXIMUM ENTROPY

AbstractSeveral authors have investigated the question of whether canonical logic-based accounts of belief revision, and especially the theory of AGM revision operators, are compatible with the dynamics of Bayesian conditioning. Here we show that Leitgeb’s stability rule for acceptance, which has been offered as a possible solution to the Lottery paradox, allows to bridge AGM revision and Bayesian update: using the stability rule, we prove that AGM revision operators emerge from Bayesian conditioning by an application of the principle of maximum entropy. In situations of information loss, or whenever the agent relies on a qualitative description of her information state—such as a plausibility ranking over hypotheses, or a belief set—the dynamics of AGM belief revision are compatible with Bayesian conditioning; indeed, through the maximum entropy principle, conditioning naturally generates AGM revision operators. This mitigates an impossibility theorem of Lin and Kelly for tracking Bayesian conditioning with AGM revision, and suggests an approach to the compatibility problem that highlights the information loss incurred by acceptance rules in passing from probabilistic to qualitative representations of belief.

On uniform belief revision

Abstract Rational belief-change policies are encoded in the so-called AGM revision functions, defined in the prominent work of Alchourrón, Gärdenfors and Makinson. The present article studies an interesting class of well-behaved AGM revision functions, called herein uniform-revision operators (or UR operators, for short). Each UR operator is uniquely defined by means of a single total preorder over all possible worlds, a fact that in turn entails a significantly lower representational cost, relative to an arbitrary AGM revision function, and an embedded solution to the iterated-revision problem, at no extra representational cost. Herein, we first demonstrate how weaker, more expressive—yet, more representationally expensive—types of uniform revision can be defined. Furthermore, we prove that UR operators, essentially, generalize a significant type of belief change, namely, parametrized-difference revision. Lastly, we show that they are (to some extent) relevance-sensitive, as well as that they respect the so-called principle of kinetic consistency.

Full Characterization of Parikh's Relevance-Sensitive Axiom for Belief Revision

In this article, the epistemic-entrenchment and partial-meet characterizations of Parikh's relevance-sensitive axiom for belief revision, known as axiom (P), are provided. In short, axiom (P) states that, if a belief set $K$ can be divided into two disjoint compartments, and the new information $\varphi$ relates only to the first compartment, then the revision of $K$ by $\varphi$ should not affect the second compartment. Accordingly, we identify the subclass of epistemic-entrenchment and that of selection-function preorders, inducing AGM revision functions that satisfy axiom (P). Hence, together with the faithful-preorders characterization of (P) that has already been provided, Parikh's axiom is fully characterized in terms of all popular constructive models of Belief Revision. Since the notions of relevance and local change are inherent in almost all intellectual activity, the completion of the constructive view of (P) has a significant impact on many theoretical, as well as applied, domains of Artificial Intelligence.

A Generalisation of AGM Contraction and Revision to Fragments of First-Order Logic


 
 
 AGM contraction and revision assume an underlying logic that contains propositional logic. Consequently, this assumption excludes many useful logics such as the Horn fragment of propositional logic and most description logics. Our goal in this paper is to generalise AGM contraction and revision to (near-)arbitrary fragments of classical first-order logic. To this end, we first define a very general logic that captures these fragments. In so doing, we make the modest assumptions that a logic contains conjunction and that information is expressed by closed formulas or sentences. The resulting logic is called first-order conjunctive logic or FC logic for short. We then take as the point of departure the AGM approach of constructing contraction functions through epistemic entrenchment, that is the entrenchment-based contraction. We redefine entrenchment-based contraction in ways that apply to any FC logic, which we call FC contraction. We prove a representation theorem showing its compliance with all the AGM contraction postulates except for the controversial recovery postulate. We also give methods for constructing revision functions through epistemic entrenchment which we call FC revision; which also apply to any FC logic. We show that if the underlying FC logic contains tautologies then FC revision complies with all the AGM revision postulates. Finally, in the context of FC logic, we provide three methods for generating revision functions via a variant of the Levi Identity, which we call contraction, withdrawal and cut generated revision, and explore the notion of revision equivalence. We show that withdrawal and cut generated revision coincide with FC revision and so does contraction generated revision under a finiteness condition.
 
 

AGM Contraction and Revision of Rules

In this paper we study AGM contraction and revision of rules using input/output logical theories. We replace propositional formulas in the AGM framework of theory change by pairs of propositional formulas, representing the rule based character of theories, and we replace the classical consequence operator Cn by an input/output logic. The results in this paper suggest that, in general, results from belief base dynamics can be transferred to rule base dynamics, but that a similar transfer of AGM theory change to rule change is much more problematic. First, we generalise belief base contraction to rule base contraction, and show that two representation results of Hansson still hold for rule base contraction. Second, we show that the six so-called basic postulates of AGM contraction are consistent only for some input/output logics, but not for others. In particular, we show that the notorious recovery postulate can be satisfied only by basic output, but not by simple-minded output. Third, we show how AGM rule revision can be defined in terms of AGM rule contraction using the Levi identity. We highlight various topics for further research.

Believability Relations for Select-Direct Sentential Revision

A set of sentential revision operations can be generated in a select-direct way within a new framework for belief change named descriptor revision firstly introduced in Hansson [8]. In this paper, we adopt another constructive approach to these operations, based on a relation {preceq} on sentences named believability relation. Intuitively, {varphi preceq psi} means that the subject is at least as prone to believe or accept {varphi} as to believe or accept {psi}. We demonstrate that so called H-believability relations and basic believability relations, the second of which the is axiomatically characterized with a set of weak postulates, are faithful alternative models for two typical select-direct sentential revision operations. Then we investigate additional postulates on believability relations that correlate with properties of the generated revision operations. Finally, we show that traditional AGM revision operations can be reconstructed from a strengthened variant of the basic believability relation and there is a close connection between this relation and the standard epistemic entrenchment relation.

Inter-Definability of Horn Contraction and Horn Revision

There have been a number of publications in recent years on generalising the AGM paradigm to the Horn fragment of propositional logic. Most of them focused on adapting AGM contraction and revision to the Horn setting. It remains an open question whether the adapted Horn contraction and Horn revision are inter-definable as in the AGM case through the Levi and Harper identities. In this paper, we give a positive answer by providing methods for generating contraction and revision from their dual operations. Noticeably, we cannot apply the Levi and Harper identities directly in such methods as the Horn fragment does not fully support negation. To overcome this difficulty, a Horn approximation technique called Horn strengthening is used. We show that Horn contraction generated from Horn revision is always plausible whereas Horn revision generated from Horn contraction is, in general, implausible and, to regain plausibility, the generating contraction has to be properly restricted.

Relevance in belief revision

Possible-world semantics are provided for Parikh's relevance-sensitive axiom for belief revision, known as axiom (P). Loosely speaking, axiom (P) states that if a belief set K can be divided into two disjoint compartments, and the new information ϕ relates only to the first compartment, then the second compartment should not be effected by the revision of K by ϕ. Using the well-known connection between AGM revision functions and preorders on possible worlds as our starting point, we formulate additional constraints on such preorders that characterise precisely Parikh's axiom (P). Interestingly, the additional constraints essentially generalise a criterion of plausibility between possible worlds that predates axiom (P). A by-product of our study is the identification of two possible readings of Parikh's axiom (P), which we call the strong and the weak versions of the axiom. Regarding specific operators, we show that Dalal's belief revision operator satisfies both weak and strong (P), and it is therefore relevance-sensitive.

A Monoselective Presentation of AGM Revision

A new equivalent presentation of AGM revision is introduced, in which a preference-based choice function directly selects one among the potential outcomes of the operation. This model differs from the usual presentations of AGM revision in which the choice function instead delivers a collection of sets whose intersection is the outcome. The new presentation confirms the versatility of AGM revision, but it also lends credibility to the more general model of direct choice among outcomes (descriptor revision) of which AGM revision is shown here to be a special case.

Belief revision in Horn theories

This paper investigates belief revision where the underlying logic is that governing Horn clauses. We show that classical (AGM) belief revision doesn't immediately generalise to the Horn case. In particular, a standard construction based on a total preorder over possible worlds may violate the accepted (AGM) postulates. Conversely, in the obvious extension to the AGM approach, Horn revision functions are not captured by total preorders over possible worlds. We address these difficulties by introducing two modifications to the AGM approach. First, the semantic construction is restricted to “well behaved” orderings, what we call Horn compliant orderings. Second, the revision postulates are augmented by an additional postulate. Both restrictions are redundant in the AGM approach, but not in the Horn case. In a representation result we show that the class of revision functions captured by Horn compliant total preorders over possible worlds is precisely that given by the (extended) set of Horn revision postulates. Further, we show that Horn revision is compatible with work in iterated revision and work concerning relevance in revision. We also consider specific revision operators. Arguably this work is interesting for several reasons. It extends AGM revision to inferentially-weaker Horn theories; hence it sheds light on the theoretical underpinnings of belief change, as well as generalising the AGM paradigm. Thus, this work is relevant to revision in areas that employ Horn clauses, such as deductive databases and logic programming, as well as areas in which inference is weaker than classical logic, such as in description logic.

BRIDGING JEFFREY'S RULE, AGM REVISION AND DEMPSTER CONDITIONING IN THE THEORY OF EVIDENCE

Belief revision characterizes the process of revising an agent's beliefs when receiving new evidence. In the field of artificial intelligence, revision strategies have been extensively studied in the context of logic-based formalisms and probability kinematics. However, so far there is not much literature on this topic in evidence theory. In contrast, combination rules proposed so far in the theory of evidence, especially Dempster rule, are symmetric. They rely on a basic assumption, that is, pieces of evidence being combined are considered to be on a par, i.e. play the same role. When one source of evidence is less reliable than another, it is possible to discount it and then a symmetric combination operation is still used. In the case of revision, the idea is to let prior knowledge of an agent be altered by some input information. The change problem is thus intrinsically asymmetric. Assuming the input information is reliable, it should be retained whilst the prior information should be changed minimally to that effect. To deal with this issue, this paper defines the notion of revision for the theory of evidence in such a way as to bring together probabilistic and logical views. Several revision rules previously proposed are reviewed and we advocate one of them as better corresponding to the idea of revision. It is extended to cope with inconsistency between prior and input information. It reduces to Dempster rule of combination, just like revision in the sense of Alchourrón, Gärdenfors, and Makinson (AGM) reduces to expansion, when the input is strongly consistent with the prior belief function. Properties of this revision rule are also investigated and it is shown to generalize Jeffrey's rule of updating, Dempster rule of conditioning and a form of AGM revision.

Double preference relations for generalised belief change

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 ordering, 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.

A program-level approach to revising logic programs under the answer set semantics

AbstractAn approach to the revision of logic programs under the answer set semantics is presented. For programs P and Q, the goal is to determine the answer sets that correspond to the revision of P by Q, denoted P * Q. A fundamental principle of classical (AGM) revision, and the one that guides the approach here, is the success postulate. In AGM revision, this stipulates that α ∈ K * α. By analogy with the success postulate, for programs P and Q, this means that the answer sets of Q will in some sense be contained in those of P * Q. The essential idea is that for P * Q, a three-valued answer set for Q, consisting of positive and negative literals, is first determined. The positive literals constitute a regular answer set, while the negated literals make up a minimal set of naf literals required to produce the answer set from Q. These literals are propagated to the program P, along with those rules of Q that are not decided by these literals. The approach differs from work in update logic programs in two main respects. First, we ensure that the revising logic program has higher priority, and so we satisfy the success postulate; second, for the preference implicit in a revision P * Q, the program Q as a whole takes precedence over P, unlike update logic programs, since answer sets of Q are propagated to P. We show that a core group of the AGM postulates are satisfied, as are the postulates that have been proposed for update logic programs.

A sketch of a dynamic epistemic semiring

This paper proposes a semiring formulation for reasoning about an agent’s changing beliefs: a dynamic epistemic semiring ( DES). A DES is a modal semiring extended with epistemic-action operators. The paper concentrates on the revision operator by proposing an axiomatisation, developing a basic calculus and deriving the classical AGM revision axioms in the algebra. Iterated action is also considered.

BELIEF REVISION IN NON-CLASSICAL LOGICS

In this article, we propose a belief revision approach for families of (non-classical) logics whose semantics are first-order axiomatisable. Given any such (non-classical) logic $L$, the approach enables the definition of belief revision operators for $L$, in terms of a belief revision operation satisfying the postulates for revision theory proposed by Alchourrón, Gärdenfors and Makinson (AGM revision, Alchourrón et al. (1985)). The approach is illustrated by considering the modal logic K, Belnap's four-valued logic, and Łukasiewicz's many-valued logic. In addition, we present a general methodology to translate algebraic logics into classical logic. For the examples provided, we analyse in what circumstances the properties of the AGM revision are preserved and discuss the advantages of the approach from both theoretical and practical viewpoints.