Artificial Neural Networks (ANNs) are powerful machine-learning models capable of capturing intricate non-linear relationships. They are widely used nowadays across numerous scientific and engineering domains, driving advancements in both research and real-world applications. In our recent work, we focused on the statics and dynamics of a particular subclass of ANNs, which we refer to as binary ANNs. A binary ANN is a feed-forward network in which both inputs and outputs are restricted to binary values, making it particularly suitable for a variety of practical use cases. Our previous study approached binary ANNs through the lens of belief-change theory, specifically the Alchourron, Gardenfors and Makinson (AGM) framework, yielding several key insights. Most notably, we demonstrated that the knowledge embodied in a binary ANN (expressed through its input-output behaviour) can be symbolically represented using a propositional logic language. Moreover, the process of modifying a belief set (through revision or contraction) was mapped onto a gradual transition through a series of intermediate belief sets. Analogously, the training of binary ANNs was conceptualized as a sequence of such belief-set transitions, which we showed can be formalized using full-meet AGM-style belief change. In the present article, we extend this line of investigation by addressing some critical limitations of our previous study. Specifically, we show that Dalal's method for belief change naturally induces a structured, gradual evolution of states of belief. More importantly, given the known shortcomings of full-meet belief change, we demonstrate that the training dynamics of binary ANNs can be more effectively modelled using robust AGM-style change operations -- namely, lexicographic revision and moderate contraction -- that align with the Darwiche-Pearl framework for iterated belief change.

Despite the significant interest in extending the AGM paradigm of belief change beyond finitary logics, the computational aspects of AGM have remained almost untouched. We investigate the computability of AGM contraction on non-finitary logics, and show an intriguing negative result: there are infinitely many uncomputable AGM contraction functions in such logics. Drastically, even if we restrict the theories used to represent epistemic states, in all non-trivial cases, the uncomputability remains. On the positive side, we identify an infinite class of computable AGM contraction functions on Linear Temporal Logic (LTL). We use B\"uchi automata to construct such functions as well as to represent and reason about LTL knowledge.

Belief contraction is the operation of removing from the set K of initial beliefs a particular belief φ . One reason for doing so is, for example, the discovery that some previously trusted evidence supporting φ was faulty. For instance, a prosecutor might form the belief that the defendant is guilty on the basis of his confession; if the prosecutor later discovers that the confession was extorted, she might abandon the belief of guilt, that is, become open minded about whether the defendant is guilty or not. In their seminal contribution to belief change, Alchourrón, Gärdenfors and Makinson ([1]) defined the notion of "rational and minimal" contraction by means of a set of eight properties, known as the AGM axioms or postulates. They did so within a syntactic approach where the initial belief set K is a consistent and deductively closed set of propositional formulas and the result of removing φ from K is a new set of propositional formulas, denoted by K φ . We provide a new characterization of AGM belief contraction based on a so-far-unnoticed connection between the notion of belief contraction and the Stalnaker-Lewis theory of conditionals ([34, 21]).

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.

The AGM paradigm of belief change studies the dynamics of belief states in light of new information. Finding, or even approximating, those beliefs that are dependent on or relevant to a change is valuable because, for example, it can narrow the set of beliefs considered during belief change operations. A strong intuition in this area is captured by Gärdenforss preservation criterion (GPC), which suggests that formulas independent of a belief change should remain intact. GPC thus allows one to build dependence relations that are linked with belief change. Such dependence relations can in turn be used as a theoretical benchmark against which to evaluate other approximate dependence or relevance relations. Fariñas and Herzig axiomatize a dependence relation with respect to a belief set, and, based on GPC, they characterize the correspondence between AGM contraction functions and dependence relations. In this paper, we introduce base dependence as a relation between formulas with respect to a belief base, and prove a more general characterization that shows the correspondence between kernel contraction and base dependence. At this level of generalization, different types of base dependence emerge, which we show to be a result of possible redundancy in the belief base. We further show that one of these relations that emerge, strong base dependence, is parallel to saturated kernel contraction. We then prove that our latter characterization is a reversible generalization of Fariñas and Herzigs characterization. That is, in the special case when the underlying belief base is deductively closed (i.e., it is a belief set), strong base dependence reduces to dependence, and so do their respective characterizations. Finally, an intriguing feature of Fariñas and Herzigs formalism is that it meets other criteria for dependence, namely, Keyness conjunction criterion for dependence (CCD) and Gärdenforss conjunction criterion for independence (CCI). We prove that our base dependence formalism also meets these criteria. Even more interestingly, we offer a more specific criterion that implies both CCD and CCI, and show our base dependence formalism also meets this new criterion.

Classic entrenchment-based contraction is not applicable to many useful logics, such as description logics. This is because the semantic construction refers to arbitrary disjunctions of formulas, while many logics do not fully support disjunction. In this paper, we present a new entrenchment-based contraction which does not rely on any logical connectives except conjunction. This contraction is applicable to all fragments of first-order logic that support conjunction. We provide a representation theorem for the contraction which shows that it satisfies all the AGM postulates except for the controversial Recovery Postulate, and is a natural generalisation of entrenchment-based contraction.

The AGM framework is the benchmark approach in belief change. Since the framework assumes an underlying logic containing classical Propositional Logic, it can not be applied to systems with a logic weaker than Propositional Logic. To remedy this limitation, several researchers have studied AGM-style contraction and revision under the Horn fragment of Propositional Logic (i.e., Horn logic). In this paper, we contribute to this line of research by investigating the Horn version of the AGM entrenchment-based contraction. The study is challenging as the construction of entrenchment-based contraction refers to arbitrary disjunctions which are not expressible under Horn logic. In order to adapt the construction to Horn logic, we make use of a Horn approximation technique called Horn strengthening. We provide a representation theorem for the newly constructed contraction which we refer to as entrenchment-based Horn contraction. Ideally, contractions defined under Horn logic (i.e., Horn contractions) should be as rational as AGM contraction. We propose the notion of Horn equivalence which intuitively captures the equivalence between Horn contraction and AGM contraction. We show that, under this notion, entrenchment-based Horn contraction is equivalent to a restricted form of entrenchment-based contraction.

Two essential tasks in managing Description Logic (DL) ontologies are eliminating problematic axioms and incorporating newly formed axioms. Such elimination and incorporation are formalised as the operations of contraction and revision in belief change.In this paper, we deal with contraction and revision for the DL-Lite family through a model-theoretic approach.Standard DL semantics yields infinite numbers of models for DL-Lite TBoxes, thus it is not practical to develop algorithms for contraction and revision that involve DL models. The key to our approach is the introduction of an alternative semantics called type semantics which is more succinct than DL semantics. More importantly, with a finite signature, type semantics always yields finite humber of models.We then define model-based contraction and revision for DL-Lite TBoxesunder type semantics and provide representation theorems for them.Finally, the succinctness of type semantics allows us to develop tractable algorithms for both operations.

In classical, AGM-style belief change, it is assumed that the underlying logic contains classical propositional logic. This is clearly a limiting assumption, particularly in Artificial Intelligence. Consequently there has been recent interest in studying belief change in approaches where the full expressivity of classical propositional logic is not obtained. In this paper we investigate belief contraction in Horn knowledge bases. We point out that the obvious extension to the Horn case, involving Horn remainder sets as a starting point, is problematic. Not only do Horn remainder sets have undesirable properties, but also some desirable Horn contraction functions are not captured by this approach. For Horn belief set contraction, we develop an account in terms of a model-theoretic characterisation involving weak remainder sets. Maxichoice and partial meet Horn contraction is specified, and we show that the problems arising with earlier work are resolved by these approaches. As well, constructions of the specific operators and sets of postulates are provided, and representation results are obtained. We also examine Horn package contraction, or contraction by a set of formulas. Again, we give a construction and postulate set, linking them via a representation result. Last, we investigate the closely-related notion of forgetting in Horn clauses. This work is arguably interesting since Horn clauses have found widespread use in AI; as well, the results given here may potentially be extended to other areas which make use of Horn-like reasoning, such as logic programming, rule-based systems, and description logics. Finally, since Horn reasoning is weaker than classical reasoning, this work sheds light on the foundations of belief change