Craig interpolation and uniform interpolation have many applications in knowledge representation, including explainability, forgetting, modularization and reuse, and even learning. At the same time, many relevant knowledge representation formalisms do in general not have Craig or uniform interpolation, and computing interpolants in practice is challenging. We have a closer look at two prominent knowledge representation formalisms, description logics and logic programming, and discuss theoretical results and practical methods for computing interpolants.

Datalog$^\neg$ is a central formalism used in a variety of domains ranging from deductive databases and abstract argumentation frameworks to answer set programming. Its model theory is the finite counterpart of the logical semantics developed for normal logic programs, mainly based on the notions of Clark's completion and two-valued or three-valued canonical models including supported, stable, regular and well-founded models. In this paper we establish a formal link between Datalog$^\neg$ and Boolean network theory first introduced for gene regulatory networks. We show that in the absence of odd cycles in a Datalog$^\neg$ program, the regular models coincide with the stable models, which entails the existence of stable models, and in the absence of even cycles, we prove the uniqueness of stable partial models and regular models. This connection also gives new upper bounds on the numbers of stable partial, regular, and stable models of a Datalog$^\neg$ program using the cardinality of a feedback vertex set in its atom dependency graph. Interestingly, our connection to Boolean network theory also points us to the notion of trap spaces. In particular we show the equivalence between subset-minimal stable trap spaces and regular models.

A wide variety of nonmonotonic semantics can be expressed as approximators defined under AFT (Approximation Fixpoint Theory). Using traditional AFT theory, it is not possible to define approximators that rely on information computed in previous iterations of stable revision. However, this information is rich for semantics that incorporate classical negation into nonmonotonic reasoning. In this work, we introduce a methodology resembling AFT that can utilize priorly computed upper bounds to more precisely capture semantics. We demonstrate our framework's applicability to hybrid MKNF (minimal knowledge and negation as failure) knowledge bases by extending the state-of-the-art approximator.

We propose a novel approach to logic-based learning which generates assumption-based argumentation (ABA) frameworks from positive and negative examples, using a given background knowledge. These ABA frameworks can be mapped onto logic programs with negation as failure that may be non-stratified. Whereas existing argumentation-based methods learn exceptions to general rules by interpreting the exceptions as rebuttal attacks, our approach interprets them as undercutting attacks. Our learning technique is based on the use of transformation rules, including some adapted from logic program transformation rules (notably folding) as well as others, such as rote learning and assumption introduction. We present a general strategy that applies the transformation rules in a suitable order to learn stratified frameworks, and we also propose a variant that handles the non-stratified case. We illustrate the benefits of our approach with a number of examples, which show that, on one hand, we are able to easily reconstruct other logic-based learning approaches and, on the other hand, we can work out in a very simple and natural way problems that seem to be hard for existing techniques.

In this paper we recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. Then, we show the equivalence between bets on conditionals and conditional bets, by reviewing de Finetti's trivalent analysis of conditionals. But our approach goes beyond de Finetti's early trivalent logical analysis and is based on his later ideas, aiming to take his proposals to a higher level. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity and compare them with our approach to compound conditionals. We prove a Probabilistic Deduction Theorem for conditional events. After that, we study some probabilistic deduction theorems, by presenting several examples. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic Deduction Theorem. We use the inference from a disjunction, "$A$ or $B$", to the conditional,"if not-$A$ then $B$", as an example to show the invalidity of the Import-Export principle. We also introduce a General Import-Export principle and we illustrate it by examining some p-valid inference rules of System P. Finally, we briefly discuss some related work relevant to AI.

The use of meta-rules in logic, i.e., rules whose content includes other rules, has recently gained attention in the setting of non-monotonic reasoning: a first logical formalisation and efficient algorithms to compute the (meta)-extensions of such theories were proposed in Olivieri et al (2021) This work extends such a logical framework by considering the deontic aspect. The resulting logic will not just be able to model policies but also tackle well-known aspects that occur in numerous legal systems. The use of Defeasible Logic (DL) to model meta-rules in the application area we just alluded to has been investigated. Within this line of research, the study mentioned above was not focusing on the general computational properties of meta-rules. This study fills this gap with two major contributions. First, we introduce and formalise two variants of Defeasible Deontic Logic with Meta-Rules to represent (1) defeasible meta-theories with deontic modalities, and (2) two different types of conflicts among rules: Simple Conflict Defeasible Deontic Logic, and Cautious Conflict Defeasible Deontic Logic. Second, we advance efficient algorithms to compute the extensions for both variants.

The rich expressivity provided by the cognitive event calculus (CEC) knowledge representation framework allows for reasoning over deeply nested beliefs, desires, intentions, and so on. I put CEC to the test by attempting to model the complex reasoning and deceptive planning used in an episode of the popular television show Breaking Bad. CEC is used to represent the knowledge used by reasoners coming up with plans like the ones devised by the fictional characters I describe. However, it becomes clear that a form of nonmonotonic reasoning is necessary--specifically so that an agent can reason about the nonmonotonic beliefs of another agent. I show how CEC can be augmented to have this ability, and then provide examples detailing how my proposed augmentation enables much of the reasoning used by agents such as the Breaking Bad characters. I close by discussing what sort of reasoning tool would be necessary to implement such nonmonotonic reasoning.

Decision theory and nonmonotonic logics are formalisms that can be employed to represent and solve problems of planning under uncertainty. We analyze the usefulness of these two approaches by establishing a simple correspondence between the two formalisms. The analysis indicates that planning using nonmonotonic logic comprises two decision-theoretic concepts: probabilities (degrees of belief in planning hypotheses) and utilities (degrees of preference for planning outcomes). We present and discuss examples of the following lessons from this decision-theoretic view of nonmonotonic reasoning: (1) decision theory and nonmonotonic logics are intended to solve different components of the planning problem; (2) when considered in the context of planning under uncertainty, nonmonotonic logics do not retain the domain-independent characteristics of classical (monotonic) logic; and (3) because certain nonmonotonic programming paradigms (for example, frame-based inheritance, nonmonotonic logics) are inherently problem specific, they might be inappropriate for use in solving certain types of planning problems. We discuss how these conclusions affect several current AI research issues.

We propose an approach to the definition of microservices with an Answer Set Programming (ASP) `core', where microservices are a successful abstraction for designing distributed applications as suites of independently deployable interacting components. Such ASP-based components might be employed in distributed architectures related to Cloud Computing or to the Internet of Things (IoT).

In complex areas such as law and science, knowledge has been in centuries formulated by primarily describing prototypical instances and properties, and then by overriding the general theory to include possible exceptions. For example, many laws are formulated by adding new norms that, in case of conflicts, may partially or completely override the previous ones. Similarly, biologists have been incrementally introducing exceptions to general properties. For instance, the human heart is usually located in the left-hand half of the thorax. Still there are exceptional individuals, with so-called situs inversus, whose heart is located on the opposite side. Eukariotic cells are those with a proper nucleus, by definition. Still they comprise mammalian red blood cells, that in their mature stage have no nucleus.