In this chapter, we are interested in formalizing the reasoning of a single agent who can make observations on a dynamic system and considers actions to perform on it. Reasoning about action and change is among the first issues addressed within Artificial Intelligence (AI); especially, it was the subject of the seminal article by McCarthy and Hayes [1969]. Research in this area has been very productive until the late 1990s. Among other things, solutions to the various problems to be faced when dealing with action representation were put forward and a classification of action languages according to their expressive power was undertaken. Moreover, much progress towards the automatization of reasoning about action and change was made, for example through the design and the evaluation of algorithms implementing the reasoning processes of the main action languages and the investigation of the computational complexity of such processes. The reasons why an agent may wish to act in order to modify the current state of a dynamic system or to learn more about it are numerous.

A recent method for solving zero-sum partially observable stochastic games (zs-POSGs) embeds the original game into a new one called the occupancy Markov game. This reformulation allows applying Bellman's principle of optimality to solve zs-POSGs. However, improving a current solution requires solving a linear program with exponentially many potential constraints, which significantly restricts the scalability of this approach. This paper exploits the optimal value function's novel uniform continuity properties to overcome this limitation. We first construct a new operator that is computationally more efficient than the state-of-the-art update rules without compromising optimality. In particular, improving a current solution now involves a linear program with an exponential drop in constraints. We then also show that point-based value iteration algorithms utilizing our findings improve the scalability of existing methods while maintaining guarantees in various domains.

Belief revision and update, two significant types of belief change, both focus on how an agent modify her beliefs in presence of new information. The most striking difference between them is that the former studies the change of beliefs in a static world while the latter concentrates on a dynamically-changing world. The famous AGM and KM postulates were proposed to capture rational belief revision and update, respectively. However, both of them are too permissive to exclude some unreasonable changes in the iteration. In response to this weakness, the DP postulates and its extensions for iterated belief revision were presented. Furthermore, Rodrigues integrated these postulates in belief update. Unfortunately, his approach does not meet the basic requirement of iterated belief update. This paper is intended to solve this problem of Rodrigues's approach. Firstly, we present a modification of the original KM postulates based on belief states. Subsequently, we migrate several well-known postulates for iterated belief revision to iterated belief update. Moreover, we provide the exact semantic characterizations based on partial preorders for each of the proposed postulates. Finally, we analyze the compatibility between the above iterated postulates and the KM postulates for belief update.

Existing methods for dealing with knowledge updates differ greatly depending on the underlying knowledge representation formalism. When Classical Logic is used, update operators are typically based on manipulating the knowledge base on the model-theoretic level. On the opposite side of the spectrum stand the semantics for updating Answer-Set Programs where most approaches need to rely on rule syntax. Yet, a unifying perspective that could embrace all these approaches is of great importance as it enables a deeper understanding of all involved methods and principles and creates room for their cross-fertilisation, ripening and further development.

Over the last couple of decades, there has been a considerable effort devoted to the problem of updating logic programs under the stable model semantics (a.k.a. answer-set programs) or, in other words, the problem of characterising the result of bringing up-to-date a logic program when the world it describes changes. Whereas the state-of-the-art approaches are guided by the same basic intuitions and aspirations as belief updates in the context of classical logic, they build upon fundamentally different principles and methods, which have prevented a unifying framework that could embrace both belief and rule updates. In this paper, we will overview some of the main approaches and results related to answer-set programming updates, while pointing out some of the main challenges that research in this topic has faced.

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.