### Abstract

Abstract. 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-wellfounded 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. Keywords: Belief Revision, Dynamic Epistemic Logic, Formal Learning Theory, Truthtracking Introduction At the basis of the modeling of intelligent behavior lies the idea that agents integrate new information into their prior beliefs and knowledge. Intelligent agents are assumed to be endowed with some learning methods, which allow them to change their beliefs on the basis of assessing new information. But how effective is an agent's learning method in eventually finding the truth? To make this question precise and to answer it, we borrow concepts from formal learning theory and adapt them to the commonly used model of beliefs, knowledge, and belief change, namely that of possible worlds. A set S of possible worlds, let us call it a state space, together with a family O of observable properties, represents the agent's epistemic space, her knowledge. Note that the sets S and O do not have to be finite, in fact throughout the paper we will assume that both S and O are at most countable. Intuitively, the epistemic space represents the uncertainty range 2 Alexandru Baltag, Nina Gierasimczuk, and Sonja Smets of the agent. She can consider some possible worlds to be more plausible than others. This is captured by a total preorder on possible worlds, called a plausibility preorder. It captures the agent's assessments concerning which of any two worlds s, s is more plausible to be the actual one. Such an assessment can obviously be based on many different factors, in particular on the assessed level of simplicity, or on consistency with previous observations. An epistemic space, together with a plausibility preorder is called a plausibility space. The above paragraph describes a static epistemic (plausibility) space. To represent the dynamic aspects of knowledge and belief we will define methods that, triggered by an incoming information, change the epistemic (plausibility) space. The change can occur through, e.g., removal of the states incompatible with the new information, or through a revision of the plausibility relation. Many belief-revision policies proposed in the literature are formulated, or can be reconstructed, within this setting. In this paper we investigate three basic policies: conditioning, minimal revision We obtain our results by defining learning methods which are based on belief-revision policies. We show that learning from positive data via conditioning and lexicographic revision is universal, i.e., those learning methods can uniformly learn the real world, when starting in any epistemic space in which the real world is learnable (via any learning method). However, this happens only if the agent's prior plans/dispositions for belief revision are suitably chosen; and not every such prior is suitable. Furthermore, the most conservative belief-revision method, minimal revision, is not universal. Our approach brings together methods of formal learning theory [FLT, see, e.g., We are chiefly concerned with the possible-world based counterpart of one of the central notions in formal learning theory, namely identifiability in Truth-Tracking by Belief Revision 3 the limit Notation and basic definitions Let S be a possibly infinite set of possible worlds and let O ⊆ P(S) be a possibly infinite (but at most countable) set of observable properties. An observable property is henceforth identified with the set of those possible worlds which make the property true. These properties can be observed by an agent and hence can be viewed as data or evidence for learning: an agent can learn whether or not they hold. This does not mean that they are necessarily all observable at the same time. Indeed, it is natural to assume that only finitely many of them could be observed at a given moment. To simplify we here assume that at each step of the learning process only one observable property is accessed by the learner. The agent is represented by her epistemic space, i.e., a range of possible worlds that satisfy relevant observable properties. Definition 1. Let S be a set of possible worlds and O ⊆ P(S). The pair S = (S, O) is then called an epistemic space. The epistemic space represents an agent who does not favor any possibility over others. We extend epistemic spaces to capture such a case by introducing a total preorder called a plausibility relation. Definition 2. Let S = (S, O) be an epistemic space, and ⊆ S × S be a total preorder. 2 The structure B S = (S, O, ) is called a plausibility space. Since we allow for the epistemic space to be infinite, the question of well-foundedness of the plausibility space becomes very relevant. We do not restrict our considerations to well-founded spaces, but we will take them into account as a special class of plausibility spaces. Because of their popularity in the literature we call them standard plausibility spaces. 3 Definition 3. A standard plausibility space B S = (S, O, ) is one whose plausibility relation is well-founded (i.e., there is no infinite descending chain s 0 s 1 . . . s n . . ., where ≺ is the strict plausibility relation, given by: s t and t s. 1 The emergence of the stronger epistemic state of irrevocable knowledge can be linked to a more restrictive kind of identifiability, finite identifiability [see 2 In other words, the binary relation is total, reflexive, and transitive in S. 3 Note that we interpret s ≺ t as 's is more plausible than t'.