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Nonmonotonic Logic


Relative Expressiveness of Defeasible Logics II

arXiv.org Artificial Intelligence

(Maher 2012) introduced an approach for relative expressiveness of defeasible logics, and two notions of relative expressiveness were investigated. Using the first of these definitions of relative expressiveness, we show that all the defeasible logics in the DL framework are equally expressive under this formulation of relative expressiveness. The second formulation of relative expressiveness is stronger than the first. However, we show that logics incorporating individual defeat are equally expressive as the corresponding logics with team defeat. Thus the only differences in expressiveness of logics in DL arise from differences in how ambiguity is handled. This completes the study of relative expressiveness in DL begun in \cite{Maher12}.


Lexicographic Logic: a Many-valued Logic for Preference Representation

arXiv.org Artificial Intelligence

Logical formalisms provide a natural and concise means for specifying and reasoning about preferences. In this paper, we propose lexicographic logic, an extension of classical propositional logic that can express a variety of preferences, most notably lexicographic ones. The proposed logic supports a simple new connective whose semantics can be defined in terms of finite lists of truth values. We demonstrate that, despite the well-known theoretical limitations that pose barriers to the quantitative representation of lexicographic preferences, there exists a subset of the rational numbers over which the proposed new connective can be naturally defined. Lexicographic logic can be used to define in a simple way some well-known preferential operators, like "$A$ and if possible $B$", and "$A$ or failing that $B$". Moreover, many other hierarchical preferential operators can be defined using a systematic approach. We argue that the new logic is an effective formalism for ranking query results according to the satisfaction level of user preferences.


An application of Answer Set Programming in Distributed Architectures: ASP Microservices

arXiv.org Artificial Intelligence

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).


Defeasible reasoning in Description Logics: an overview on DL^N

arXiv.org Artificial Intelligence

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.


Towards Ranking-based Semantics for Abstract Argumentation using Conditional Logic Semantics

arXiv.org Artificial Intelligence

We propose a novel ranking-based semantics for Dung-style argumentation frameworks with the help of conditional logics. Using an intuitive translation for an argumentation framework to generate conditionals, we can apply nonmonotonic inference systems to generate a ranking on possible worlds. With this ranking we construct a ranking for our arguments. With a small extension to this ranking-based semantics we already satisfy some desirable properties for a ranking over arguments.


Exploiting Game Theory for Analysing Justifications

arXiv.org Artificial Intelligence

Justification theory is a unifying semantic framework. While it has its roots in non-monotonic logics, it can be applied to various areas in computer science, especially in explainable reasoning; its most central concept is a justification: an explanation why a property holds (or does not hold) in a model. In this paper, we continue the study of justification theory by means of three major contributions. The first is studying the relation between justification theory and game theory. We show that justification frameworks can be seen as a special type of games. The established connection provides the theoretical foundations for our next two contributions. The second contribution is studying under which condition two different dialects of justification theory (graphs as explanations vs trees as explanations) coincide. The third contribution is establishing a precise criterion of when a semantics induced by justification theory yields consistent results. In the past proving that such semantics were consistent took cumbersome and elaborate proofs. We show that these criteria are indeed satisfied for all common semantics of logic programming. This paper is under consideration for acceptance in Theory and Practice of Logic Programming (TPLP).


Nonmonotonic Inferences with Qualitative Conditionals based on Preferred Structures on Worlds

arXiv.org Artificial Intelligence

A conditional knowledge base R is a set of conditionals of the form "If A, the usually B". Using structural information derived from the conditionals in R, we introduce the preferred structure relation on worlds. The preferred structure relation is the core ingredient of a new inference relation called system W inference that inductively completes the knowledge given explicitly in R. We show that system W exhibits desirable inference properties like satisfying system P and avoiding, in contrast to e.g. system Z, the drowning problem. It fully captures and strictly extends both system Z and skeptical c-inference. In contrast to skeptical c-inference, it does not require to solve a complex constraint satisfaction problem, but is as tractable as system Z.


Nonmonotonic Reasoning

Journal of Artificial Intelligence Research

Nonmonotonic reasoning concerns situations when information is incomplete or uncertain. Thus, conclusions drawn lack iron-clad certainty that comes with classical logic reasoning. New information, even if the original one is retained, may change conclusions. Formal ways to capture mechanisms involved in nonmonotonic reasoning, and to exploit them for computation as in the answer set programming paradigm are at the heart of this research area. The six papers accepted for the special track contain significant contributions to the foundations of logic programming under the answer set semantics, to nonmonotonic extensions of description logics, to belief change in restricted settings, and to argumentation.


Applications of Linear Defeasible Logic: combining resource consumption and exceptions to energy management and business processes

arXiv.org Artificial Intelligence

Linear Logic and Defeasible Logic have been adopted to formalise different features of knowledge representation: consumption of resources, and non monotonic reasoning in particular to represent exceptions. Recently, a framework to combine sub-structural features, corresponding to the consumption of resources, with defeasibility aspects to handle potentially conflicting information, has been discussed in literature, by some of the authors. Two applications emerged that are very relevant: energy management and business process management. We illustrate a set of guide lines to determine how to apply linear defeasible logic to those contexts.


On Rational Monotony and Weak Rational Monotony for Inference Relations Induced by Sets of Minimal C-Representations

AAAI Conferences

Reasoning in the context of a conditional knowledge base containing rules of the form ’If A then usually B’ can be defined in terms of preference relations on possible worlds. These preference relations can be modeled by ranking functions that assign a degree of disbelief to each possible world. In general, there are multiple ranking functions that accept a given knowledge base. Several nonmonotonic inference relations have been proposed using c-representations, a subset of all ranking functions. These inference relations take subsets of all c-representations based on various notions of minimality into account, and they operate in different inference modes, i.e., skeptical, weakly skeptical, or credulous. For nonmonotonic inference relations, weaker versions of monotonicity like rational monotony (RM) and weak rational monotony (WRM) have been developed. In this paper, we investigate which of the inference relations induced by sets of minimal c-representations satisfy rational monotony or weak rational monotony.