Artificial Neural Networks (ANNs) are powerful computational models that are able to reproduce complex non-linear processes, and are being widely used in a plethora of contemporary disciplines. In this article, we study the statics and dynamics of a certain class of ANNs, called binary ANNs, from the perspective of belief-change theory. A binary ANN is a feed-forward ANN whose inputs and outputs take binary values, and as such, it is suitable for a wide range of practical applications. For this type of ANNs, we point out that their knowledge (expressed via their input-output relationship) can symbolically be represented in terms of a propositional logic language. Furthermore, in the realm of belief change, we identify the process of changing (revising/contracting) an initial belief set to a modified belief set, as a process of a gradual transition of intermediate belief sets — such a gradualist approach to belief change is more congruent with the behaviors of real-world agents. Along these lines, we provide natural metrics for measuring the distance between these intermediate belief sets, effectively quantifying the disparity in their encoded knowledge. Thereafter, we demonstrate that, similar to belief change, the training process of binary ANNs, through backpropagation, can be emulated via a sequence of successive transitions of belief sets, the distance between which is intuitively related through one of the aforementioned metrics. We also prove that the alluded successive transitions of belief sets can be modeled by means of rational revision and contraction operators, defined within the fundamental belief-change framework of Alchourrón, Gärdenfors and Makinson (AGM). Thus, the process of machine learning (specifically, training binary ANNs) is framed as an operation of AGM-style belief change, offering a modular and logically structured perspective on neural learning. • This article explores the statics and dynamics of binary Artificial Neural Networks (ANNs) through belief-change theory. • A binary ANN is a feed-forward ANN with binary inputs and outputs, suitable for various applications. • The study aligns machine-learning operations like backpropagation with formalized logic theories. • This alignment advances neural-computation understanding and sets the stage for future AI-logical framework integration. • The work paves the way for more interpretable, robust, neuro-symbolic intelligent systems.

Belief revision and belief update are two fundamental and well-studied processes of belief change. In the present article, we introduce a consistency principle which dictates that the revision and update policies employed by a rational agent are not independent, but ought to be related in a certain coherent way. We formalize our consistency principle both axiomatically and semantically, and we establish a representation result explicitly connecting the two formalizations. Furthermore, we show that two important concrete types of belief change, namely uniform belief change and parametrized-difference belief change, serve as proof-of-concept examples for the introduced consistency principle, as they fully comply with it. Additionally, we identify an intriguing property of uniform belief change in which revision and update become indistinguishable when an epistemic input contradicts the initial state of belief, as both processes produce identical outcomes. Lastly, it is shown that, unlike parametrized-difference belief change, uniform belief change is incompatible with Parikh’s notion of relevance. Consequently, building on previous results, it is demonstrated that parametrized-difference belief change is relevance-sensitive — indicating that the proposed principle of consistency is compatible with relevance — while uniform belief change is not.

Abstract The belief-structures model of Chopra and Parikh is perfectly suitable for representing the statics and dynamics of the beliefs of resource-bounded agents. Nevertheless, despite its favourable properties, it lies outside the AGM framework, the well-established paradigm for codifying rational belief revision of belief sets (i.e. logical theories), proposed by Alchourrón, Gärdenfors and Makinson. This feature can be attributed to two primary reasons: firstly, a multivalued logic is utilized for evaluating queries relative to a belief structure, and secondly, the objects under revision in the context of the belief-structures model are belief structures, rather than belief sets. Against this background, we present, in this article, a natural approach for adapting the revision of belief structures to an AGM-style revision while retaining the favourable properties of the belief-structures model. Under certain realistic assumptions, including the mutual disjointness of sublanguages of belief structures, a collection of interesting properties of the proposed method are identified, by investigating the satisfiability and violation of principal postulates of the belief-revision theory. Overall, it is demonstrated that the introduced approach constitutes a meaningful relevance-sensitive tool for integrating disjoint belief structures within the AGM framework, with potential applications in emerging fields such as digital twins, where precise and dynamic belief changes are essential for maintaining accurate simulations of real-world systems.

In this article, we study the dynamics of collective beliefs. As a first step, we formulate David Westlund’s Principle of Collective Change (PCC) —a criterion that characterizes the evolution of collective knowledge— in the realm of belief revision. Thereafter, we establish a number of unsatisfiability results pointing out that the widely-accepted revision operators of Alchourrón, Gärdenfors and Makinson, combined with fundamental types of merging operations —including the ones proposed by Konieczny and Pino Pérez as well as Baral et al.— collide with the PCC. These impossibility results essentially extend in the context of belief revision the negative results established by Westlund for the operations of contraction and expansion. At the opposite of the impossibility results, we also establish a number of satisfiability results, proving that, under certain (rather strict) requirements, the PCC is indeed respected for specific merging operators. Overall, it is argued that the PCC is a rather unsuitable property for characterizing the process of collective change. Last but not least, mainly in response to the unsatisfactory situation related to the PCC, we explore some alternative criteria of collective change, and evaluate their compliance with belief revision and belief merging.

Parikh proposed his relevance-sensitive axiom to remedy the weakness of the classical AGM paradigm in addressing relevant change. An insufficiency of Parikh’s criterion, however, is its dependency on the contingent beliefs of a belief set to be revised, since the former only constrains the revision process of splittable theories (i.e., theories that can be divided in mutually disjoint compartments). The case of arbitrary non-splittable belief sets remains out of the scope of Parikh’s approach. On that premise, we generalize Parikh’s criterion, introducing (both axiomatically and semantically) a new notion of relevance, which we call relevance at the sentential level . We show that the proposed notion of relevance is universal (as it is applicable to arbitrary belief sets) and acts in a more refined way as compared to Parikh’s proposal; as we illustrate, this latter feature of relevance at the sentential level potentially leads to a significant drop in the computational resources required for implementing belief revision. Furthermore, we prove that Dalal’s popular revision operator respects, to a certain extent, relevance at the sentential level. Last but not least, the tight relation between local and relevance-sensitive revision is pointed out.

Abstract In the present article, a solver for the well-behaved concrete revision operators, named parametrized-difference (PD) revision operators, is described. The solver is developed by means of the powerful framework of answer set programming, which constitutes a contemporary modelling tool, oriented towards difficult search problems. Several useful functionalities are supported by the system, namely, dynamic PD revision, query-answering capabilities, integrity-constraints handling and revision under the closed-world assumption. The solver exhibits high performance in a plethora of revision instances, including hard ones of the SATLIB library; the fact that PD revision respects an intuitive relevance-sensitive principle, identified herein, contributes to this high performance. A graphical user interface provides easy interaction with the implemented system, making it a high-end standalone revision-tool for potential artificial intelligence applications.

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.

The policy of rational belief revision is encoded in the so-called AGM revision functions. Such functions are characterized (both axiomatically and constructively) within the well-known AGM paradigm, proposed by Alchourrón, Gärdenfors and Makinson. In this article, we show that - although not in a straightforward way - a sufficient extension of the underlying language allows for the modelling of any AGM revision function (defined at the initial language), by means of a Hamming-based rule for belief revision introduced by Dalal (defined at the extended language). The established results enrich the applicability of Dalal's proposal, leading to a conceptual and ontological reduction, as well as open new doors for the construction of any type of revision function in a practical context, given the intuitive appeal and simplicity of Dalal's construction.

The AGM paradigm for belief change, as originally introduced by Alchourron, Gärdenfors and Makinson, lacks any guidelines for the process of iterated revision. One of the most influential work addressing this problem is Darwiche and Pearl's approach (DP approach, for short), which, despite its well-documented shortcomings, remains to this date the most dominant. In this article, we make further observations on the DP approach. In particular, we prove that the DP postulates are, in a strong sense, inconsistent with Parikh's relevance-sensitive axiom (P), extending previous initial conflicts. Immediate consequences of this result are that an entire class of intuitive revision operators, which includes Dalal's operator, violates the DP postulates, as well as that the Independence postulate and Spohn's conditionalization are inconsistent with axiom (P). The whole study, essentially, indicates that two fundamental aspects of the revision process, namely, iteration and relevance, are in deep conflict, and opens the discussion for a potential reconciliation towards a comprehensive formal framework for knowledge dynamics.

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.

Abstract Parikh’s relevance-sensitive axiom (P) for belief revision is open to two different interpretations, i.e. the weak and the strong version of (P), both of which are plausible depending on the context. Given that strong (P) has not received the attention it deserves, in this article, an extended examination of it is conducted. In particular, we point out interesting properties of the semantic characterization of the strong version of (P), as well as a vital feature of it that, potentially, results in a significant drop on the resources required for an implementation of a belief-revision system. Lastly, we shed light on the natural connection between global and local revision functions, via their corresponding semantic characterization, hence, a means for constructing global revision functions from local ones, and vice versa, is provided.

Notwithstanding the extensive work on iterated belief revision, there is, still, no fully satisfactory solution within the classical AGM paradigm. The seminal work of Darwiche and Pearl (DP approach, for short) remains the most dominant, despite its well-documented shortcomings. In this article, we make further observations on the DP approach. Firstly, we prove that the DP postulates are, in a strong sense, inconsistent with Parikh's relevance-sensitive axiom (P), extending previous initial conflicts. Immediate consequences of this result are that an entire class of intuitive revision operators, which includes Dalal's operator, violates the DP postulates, as well as that the Independence postulate and Spohn's conditionalization are inconsistent with (P). Lastly, we show that the DP postulates allow for more revision polices than the ones that can be captured by identifying belief states with total preorders over possible worlds, a fact implying that a preference ordering (over possible worlds) is an insufficient representation for a belief state.

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.

Answer Set Programming is a declarative problem solving approach, initially tailored to modelling problems in the area of Knowledge Representation and Reasoning. In this article, we provide a knowledge-based system, capable of representing and reasoning about legal knowledge in the context of Answer Set Programming - thus, modelling non-monotonicity that is inherent in legal arguments. The work, although limited to a specific indicative domain, namely, university regulations, has a variety of extensions. The overall approach constitutes a representative implementation of the Answer Set Programming's modelling methodology, as well as an enhancing of the bond between Artificial Intelligence and Legal Science, bringing us a step closer to a successful development of an automated legal reasoning system for real-world applications.

In this article, we provide the epistemic-entrenchment characterization of the weak version of Parikh’s relevance-sensitive axiom for belief revision — known as axiom (P) — for the general case of incomplete theories. 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 affected by the revision of K by φ. The above-mentioned characterization, essentially, constitutes additional constraints on epistemic-entrenchment preorders, that induce AGM revision functions, satisfying the weak version of Parikh’s axiom (P).