Iterated Revision as Prioritized Merging

Standard accounts of iterated belief revision assume a static world, about which an agent receives a sequence of observations. More recent items are assumed to have priority over less recent items. We argue that there is no reason, given a static world, for giving priority to more recent items. Instead we suggest that a sequence of observations should be merged with the agent’s beliefs. Since observations may have differing reliability, arguably the appropriate belief change operator is prioritized merging. We develop this view here, suggesting postulates for prioritized merging, and examining existing merging operators with respect to these postulates. As well, we examine other suggested postulates for iterated revision, to determine how well they fit with the prioritized merging interpretation. All postulates for iterated revision that we examine, except for Darwiche and Pearl’s controversial C2, are consequences of our suggested postulates for prioritized merging. Introduction In knowledge representation, the area of belief change addresses the specification and construction of systems for reasoning about a possibly uncertain and possibly evolving world. A fundamental belief change operation is belief revision (along with its dual operation of belief contraction). Belief revision concerns the situation in which new information may be inconsistent with the reasoner’s beliefs, and needs to be incorporated in a consistent manner where possible. The common assumption is that in revision an agent receives information about a purely inertial (or static) world1. information about a purely inertial (or static) world. That is, the agent performs no actions that can cause the world to evolve, nor do any exogenous actions occur. A belief revision operator is not arbitrary, but rather is usually guided by various rationality criteria. One of the Copyright c © 2006, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. 1(Friedman & Halpern 1999) argue that more generally, what counts in belief revision is not that the world itself is static but that the propositions used to describe the world are static, therefore, time-stamping variables also allows for dealing with an evolving world in a pure belief revision setting. Thus, belief revision is also typical of the situation where an agent investigates a past event and tries to reason about what was the real state of the world when this event took place. most widely accepted of the rationality criteria is the success postulate: that a new item of information (which we’ll refer to as an observation) is always accepted. Thus if we use K to denote the agent’s initial belief state, and the agent receives the observation α, then in the revised state K ∗ α, α is believed. Much attention has been paid to the problem of iterated belief revision, in which an agent receives a stream or sequence of (possibly conflicting) observations. An assumption common to all approaches to iterated revision is that revision takes place whenever an observation is received. Hence for a sequence of observations α1, . . . , αn, the result of revising by this information is (. . . (K ∗α1) ∗ · · · ∗αn−1) ∗ αn. This assumption, together with the success postulate, implies that more recent observations are assigned a higher priority than less recently received observations. For example, p will be believed in the state resulting from (K ∗ ¬p) ∗ p. This ordering of observations for revision is reasonable in a dynamic framework, where events can occur and induce unpredicted changes in the world. However, this does not carry over to a purely inertial framework, where the state of the world does not change. In this case, the order in which the observations are made is not really significant in itself, and we might have received the very same observations in a different order. Thus, for example, imagine coming home to three messages on your telephone answering machine, each from a friend independently reporting on a party that you failed to attend; clearly the order of messages is irrelevant. In fact there are examples wherein priority is given to the older items of information. Thus in history, all other things being equal, older reports concerning some ancient event may be given more weight than more recent reports, since they are closer to the event itself and so presumably more accurate. Even if in some contexts it makes sense that reliability coincides with recency, these contexts are specific and should not be considered as a general case. In this paper we address iterated belief revision based upon these intuitions. Thus, given an inertial framework, the order of observations is irrelevant. However, it is quite possible that some observations may be more reliable than others (since sources of information may have varying degrees of reliability). Hence, for us the problem of iterated revision is, in fact, a problem of prioritized merging of information: The agent has (perhaps as part of an epistemic state) a set of beliefs, and must modify its epistemic state so as to account for the new observation. Each observation is attached an evaluation of its reliability (or priority); these evaluations induce a total preorder over the observations (and, in fact, the agent’s beliefs), and the problem becomes one of merging this information into a single set of beliefs while taking the priorities into account. Thus, “standard” iterated belief revision corresponds to the situation in which observations are linearly ordered, and an observation’s priority corresponds with its temporal position. We give a number of postulates governing prioritized merging, and examine existing approaches with respect to these postulates. It turns out that these postulates also imply the AGM revision postulates. As well, we examine postulates that have been proposed for iterated revision. It proves to be the case that all postulates for iterated revision that we examine, except for Darwiche and Pearl’s controversial C2, are consequences of our suggested postulates for prioritized merging. First though, we need to address some crucial questions about the meaning of belief revision, especially when it comes to revising epistemic states and iteration. We fully agree with Friedman and Halpern (1996) claim that before stating postulates for iterated revision, the concerned problem addressed by the theory must be laid bare, namely what they call the “underlying ontology or scenario”. This is the topic of the next section, where we will argue that there are (at least) three different scenarii for the revision of epistemic states. In particular, we note that the view of iterated belief revision as a merging problem adopted in this paper is at odds with the view of belief revision as resulting from non-monotonic inference based on background knowledge, originally suggested by Gärdenfors and Makinson as a reinterpretation of the belief revision axioms. In the following section, we develop a general approach for prioritized merging and investigate properties of prioritized merging as well as its relation to unprioritized merging. After this, in the next section we come back to iterated revision, and show that it is essentially a specific case of prioritized merging. In the next to last section, we briefly question the assumption that the result of merging is a plain formula, and consider the case of merging complex observations into complex observations. Last, we conclude with a summary and some remarks concerning future research. Revision of epistemic states: three views Background Interest in belief revision as a foundational topic in artificial intelligence arguably goes back to the AGM approach (Alchourrón, Gärdenfors, & Makinson 1985), which provides a well-known set of rationality postulates for belief revision. This approach assumes that belief states are modelled by sets of sentences, called belief sets, closed under the logical consequence operator of a logic that includes classical propositional logic. An important assumption is that belief revision takes place in an inertial (or static) world, so that the input information is always with respect to the same, static world. The axiomatic framework given by the rationality postulates has a corresponding semantic model, given by a so-called epistemic entrenchment relation between propositions of the language. Properties of an epistemic entrenchment make it representable by means of a complete plausibility ordering over possible worlds, and the resulting belief set, after receiving input α, is viewed as the set of propositions that are true in the most plausible worlds where α holds. An epistemic entrenchment relation (or plausibility ordering on worlds) can be viewed as an epistemic state, containing not just an agent’s set of beliefs, but also sufficient information for carrying out revision, given any input. The AGM approach is silent on what happens to an epistemic state following a revision. Subsequent work on iterated revision has addressed this issue. However, we suggest that the study of iterated revision has led to a number of misunderstandings. Some researchers have claimed that, once the belief set has been revised by some input information, the epistemic entrenchment relation is simply lost, thus precluding the possibility of any further iteration. Others have claimed that the epistemic entrenchment relation changes along with the belief state. This has led to axioms being proposed which are intended to govern the change of the plausibility ordering of worlds; these additional axioms are viewed as extending the AGM axioms. This approach has led to a view of belief revision as a form of prioritized merging, where the priority assignment to pieces of input is reflected by their recency. I Belief revision as defeasible inference However, this view of iterated revision seems to be at odds with (Gärdenfors & Makinson 1994), which argues that belief revision is the other side of non-monotonic reasoning. This view can be characterised as belief revision as defeasible inference (BRDI). The BRDI problem can be stated as follows: given a plausibility ordering on worlds describing background knowledge and an input information α representing sure observations about the case at hand, find the most plausible worlds where α is true. It is thus assumed that the agent possesses three kinds of information: an epistemic entrenchment, induced by a plausibility ordering of worlds, a set of contingent observations about the current world, under the form of sentences, and a set of beliefs about the current world induced by observations and the epistemic entrenchment (Dubois, Fargier, & Prade 2004). The role of the latter is to guide the agent in tasks of inquiry and deliberation, sorting what is credible from what is less credible in view of the contingent observations, considered as sure facts. Under this view, the AGM approach to belief revision (as related to non-monotonic reasoning) has little to do with iterated revision as studied by subsequent researchers. According to Gärdenfors and Makinson, a revised belief set is the result of an inference step involving (nonmonotonic or defeasible) conditionals, from which propositional conclusions are tentatively drawn. These conclusions are altered by the arrival of new pieces of evidence, supposed to be sure, hence consistent (Friedman & Halpern 1996). In this framework, there is no clear reason why the conditional information, hence the plausibility ordering, should be revised upon making new contingent observations, and “iteration” in fact corresponds to the nonmonotonic inference of new conclusions – that is, different inputs simply yield different nonmonotonic conclusions. The Darwiche and Pearl (1997) axioms that were intended to extend the AGM axioms so as to allow for iterated revision by modifying the plausibility ordering seem to have little relevance here. Moreover, in the AGM theory you never need the original belief set when deriving the revised belief set (a point also made by Friedman and Halpern (1996)). You only need the epistemic entrenchment and the input information to construct the latter, while the original belief set is based on the most plausible worlds induced by the epistemic entrenchment. II Belief revision as incorporation of evidence A quite different view is to assume that an epistemic state represents uncertain evidence about a particular (static) world of interest. An agent now gathers and “compiles” possibly uncertain or inaccurate observations about a particular world. So the underlying plausibility ordering on worlds represents the agent’s subjective ranking as to which world is most likely the actual world, and the degree of entrenchment of a proposition (evaluated on the basis of the most plausible world that violates it (Gärdenfors 1988; Dubois & Prade 1991)) is an agent’s degree of belief that a proposition is true or not. The instigating philosophical work here arguably is (Spohn 1988), on ordinal conditional functions, and most work on iterated belief revision is placed within this framework. However the mathematical theory of evidence by Shafer (1976) discusses a very similar problem (the one of merging uncertain evidence), as well as possibility theory (Dubois & Prade 1988). This overall approach can be characterised as belief revision as incorporation of evidence (BRIE). Under this view, belief revision means changing the plausibility ordering in response to new information, and it makes sense to talk about iterating the process of revision. The success postulate then expresses the fact that the most recent information is the most reliable. However, given that observations concern a static world, it is by no means clear why the most recent should be taken as the most reliable. If all observations are equally reliable, then it seems most natural to somehow merge these observations with the agent’s beliefs. If observations come with varying degrees of reliability, then it seems most natural to exploit this reliability ordering while merging the observations with the agent’s beliefs. Comparison of the first two views To sum up the main differences between these views: under the BRDI view, the belief revision step leaves the epistemic entrenchment relation (i.e., the plausibility ordering on states) unchanged. This is because inputs and the plausibility ordering deal with different matters, resp. the particular world of interest, and the class of worlds the plausibility ordering refers to. Under this view, AGM revision is a matter of “querying” the epistemic entrenchment relation; so axioms for revising the epistemic state (e.g.(Darwiche & Pearl 1997)), cannot be seen as additional axioms completing the AGM axioms. In contrast, the BRIE view understands the AGM axioms as relevant for the revision of epistemic states including the plausibility ordering. Since the AGM axioms neither explicitly dealt with ranked belief sets, they could not address iterated revision, and additional axioms are needed to this end. Under this view, the prior epistemic state and the inputs can be handled in a homogeneous way, since they both consist of uncertain evidence about the world of interest; thus, it makes sense to have a new epistemic state be a function of the prior epistemic state and the input information; and it is then natural to iterate the process. But then, the plausibility ordering does not contain any information about how it should be revised, while, in the BRDI view, its role is precisely to provide a revision procedure. III Belief revision of background knowledge There is a third form of belief revision which we will just briefly mention: revision of background knowledge by generic information. This problem is the one, not addressed in the AGM theory, of revising the epistemic entrenchment relation within the BRDI view, not upon receiving a contingent input, but upon receiving a new piece of generic knowledge. Here, in some fashion, the generic knowledge of an agent’s epistemic state is revised. This could also be characterised as theory change (in the same sense as changing a scientific theory). Since in the BRDI view the epistemic entrenchment is equivalently represented as a set of conditionals, this problem is also the one of revising a set of conditionals by a new conditional (Boutilier & Goldszmidt 1993). Since the epistemic entrenchment is induced by a plausibility ordering on worlds, and since an input conditional can be modeled as another plausibility ordering, this third revision problem is also akin to the revision of a preference relation by another one as studied in (Freund 2004). Rules for revising a plausibility ordering can be found in (Williams 1995) and (Darwiche & Pearl 1997) in terms of Spohn’s ordinal conditional functions or (Dubois & Prade 1997) in terms of possibility theory. Several authors, as for instance Friedman and Halpern (1996), Dubois, Moral and Prade (1998) note that the results of (Darwiche & Pearl 1997) do not make it clear whether the considered issue is that of revising an epistemic entrenchment ordering, or that of allowing for iteration in the revision of belief sets. Alternatively, one might adopt a principle of minimal change of the background knowledge under the constraint of accepting the new conditional. Then this is no longer a symmetric merging process. This seems to be the rationale behind Boutilier’s natural revision (Boutilier 1993). This asymmetric approach has been mainly studied in the probabilistic literature (Domotor 1980). For instance, Jeffrey’s rule (Jeffrey 1965) revises probability distributions by enforcing a constraint P (A) = x. But there have been extensions of such probability kinematics to conditional inputs of the form P (A | B) = x (van Fraassen 1981). Devising basic principles and rational change operations for qualitative non-probabilistic representations of generic knowledge is left for further research. In the rest of this paper the focus is on prioritized merging of qualitative uncertain evidence. A principled approach to prioritized merging