Remarks on Inheritance Systems

We try a conceptual analysis of inheritance diagrams, first in abstract terms, and then compare to "normality" and the "small/big sets" of preferential and related reasoning. The main ideas are about nodes as truth values and information sources, truth comparison by paths, accessibility or relevance of information by paths, relative normality, and prototypical reasoning

A Lightweight Defeasible Description Logic in Depth: Quantification in Rational Reasoning and Beyond

Description Logics (DLs) are increasingly successful knowledge representation formalisms, useful for any application requiring implicit derivation of knowledge from explicitly known facts. A prominent example domain benefiting from these formalisms since the 1990s is the biomedical field. This area contributes an intangible amount of facts and relations between low- and high-level concepts such as the constitution of cells or interactions between studied illnesses, their symptoms and remedies. DLs are well-suited for handling large formal knowledge repositories and computing inferable coherences throughout such data, relying on their well-founded first-order semantics. In particular, DLs of reduced expressivity have proven a tremendous worth for handling large ontologies due to their computational tractability. In spite of these assets and prevailing influence, classical DLs are not well-suited to adequately model some of the most intuitive forms of reasoning. The capability for abductive reasoning is imperative for any field subjected to incomplete knowledge and the motivation to complete it with typical expectations. When such default expectations receive contradicting evidence, an abductive formalism is able to retract previously drawn, conflicting conclusions. Common examples often include human reasoning or a default characterisation of properties in biology, such as the normal arrangement of organs in the human body. Treatment of such defeasible knowledge must be aware of exceptional cases - such as a human suffering from the congenital condition situs inversus - and therefore accommodate for the ability to retract defeasible conclusions in a non-monotonic fashion. Specifically tailored non-monotonic semantics have been continuously investigated for DLs in the past 30 years. A particularly promising approach, is rooted in the research by Kraus, Lehmann and Magidor for preferential (propositional) logics and Rational Closure (RC). The biggest advantages of RC are its well-behaviour in terms of formal inference postulates and the efficient computation of defeasible entailments, by relying on a tractable reduction to classical reasoning in the underlying formalism. A major contribution of this work is a reorganisation of the core of this reasoning method, into an abstract framework formalisation. This framework is then easily instantiated to provide the reduction method for RC in DLs as well as more advanced closure operators, such as Relevant or Lexicographic Closure. In spite of their practical aptitude, we discovered that all reduction approaches fail to provide any defeasible conclusions for elements that only occur in the relational neighbourhood of the inspected elements. More explicitly, a distinguishing advantage of DLs over propositional logic is the capability to model binary relations and describe aspects of a related concept in terms of existential and universal quantification. Previous approaches to RC (and more advanced closures) are not able to derive typical behaviour for the concepts that occur within such quantification. The main contribution of this work is to introduce stronger semantics for the lightweight DL EL_bot with the capability to infer the expected entailments, while maintaining a close relation to the reduction method. We achieve this by introducing a new kind of first-order interpretation that allocates defeasible information on its elements directly. This allows to compare the level of typicality of such interpretations in terms of defeasible information satisfied at elements in the relational neighbourhood. A typicality preference relation then provides the means to single out those sets of models with maximal typicality. Based on this notion, we introduce two types of nested rational semantics, a sceptical and a selective variant, each capable of deriving the missing entailments under RC for arbitrarily nested quantified concepts. As a proof of versatility for our new semantics, we also show that the stronger Relevant Closure, can be imbued with typical information in the successors of binary relations. An extensive investigation into the computational complexity of our new semantics shows that the sceptical nested variant comes at considerable additional effort, while the selective semantics reside in the complexity of classical reasoning in the underlying DL, which remains tractable in our case

Defeasible inheritance systems and reactive diagrams

We give an analysis of defeasible inheritance diagrams, also from the perspective of reactive diagrams

Semantics of logic programs with explicit negation

After a historical introduction, the bulk of the thesis concerns the study of a declarative semantics for logic programs. The main original contributions are: ² WFSX (Well–Founded Semantics with eXplicit negation), a new semantics for logic programs with explicit negation (i.e. extended logic programs), which compares favourably in its properties with other extant semantics. ² A generic characterization schema that facilitates comparisons among a diversity of semantics of extended logic programs, including WFSX. ² An autoepistemic and a default logic corresponding to WFSX, which solve existing problems of the classical approaches to autoepistemic and default logics, and clarify the meaning of explicit negation in logic programs. ² A framework for defining a spectrum of semantics of extended logic programs based on the abduction of negative hypotheses. This framework allows for the characterization of different levels of scepticism/credulity, consensuality, and argumentation. One of the semantics of abduction coincides with WFSX. ² O–semantics, a semantics that uniquely adds more CWA hypotheses to WFSX. The techniques used for doing so are applicable as well to the well–founded semantics of normal logic programs. ² By introducing explicit negation into logic programs contradiction may appear. I present two approaches for dealing with contradiction, and show their equivalence. One of the approaches consists in avoiding contradiction, and is based on restrictions in the adoption of abductive hypotheses. The other approach consists in removing contradiction, and is based in a transformation of contradictory programs into noncontradictory ones, guided by the reasons for contradiction

Logical tools for handling change in agent-based systems

We give a unified approach to various results and problems of nonclassical logic

A Purely Defeasible Argumentation Framework

Argumentation theory is concerned with the way that intelligent agents discuss whether some statement holds. It is a claim-based theory that is widely used in many areas, such as law, linguistics and computer science. In the past few years, formal argumentation frameworks have been heavily studied and applications have been proposed in fields such as natural language processing, the semantic web and multi-agent systems. Studying argumentation provides results which help in developing tools and applications in these areas. Argumentation is interesting as a logic-based approach to deal with inconsistent information. Arguments are constructed using a process like logical inference, with inconsistencies giving rise to conflicts between arguments. These conflicts can then be handled by well-founded means, giving a consistent set of well-justified arguments and conclusions. Dung\u27s seminal work tells us how to handle the conflicts between arguments. However, it says nothing about the structure of arguments, or how to construct arguments and attack relationships from a knowledge base. ASPIC+ is one of the most widely used systems for structured arguments. However, there are some limitations on ASPIC+ if it is to satisfy widely accepted standards of rationality. Since most of these limitations are due to the use of strict rules, it is worth considering using a purely defeasible subset of ASPIC+. The main contribution of this dissertation is the purely defeasible argumentation framework ASPIC+D. There are three research questions related to this topic which are investigated here: (1) Do we lose anything in removing the strict elements? (2) Do purely defeasible version of theories generate the same results as the original theories? (3) What do we gain by removing the strict elements? I show that using ASPIC+D, it is possible, in a well-defined sense, to capture the same information as using ASPIC+ with strict rules. In particular, I prove that under some reasonable assumptions, it is possible to take a well-defined theory in ASPIC+, that is one with a consistent set of conclusions, and translate it into ASPIC+D such that, under the grounded semantics, we obtain the same set of justified conclusions. I also show that, under some additional assumptions, the same is true under any complete-based semantics. Furthermore, I formally characterize the situations in which translating an ASPIC+ theory that is ill-defined into ASPIC+D will lead to the same sets of justified conclusions. In doing this I deal both with ASPIC+ theories that are not closed under transposition and theories that are axiom inconsistent. At last, I analyze the two systems in the context of the non-monotonic axioms. I show that ASPIC+ and ASPIC+D satisfy exactly same axioms under what I call the “argument construction” interpretation and the “justified conclusions” interpretation under the grounded semantics. Furthermore, because of the lack of strict elements, ASPIC+ satisfies more of the non-monotonic axioms than ASPIC+ in the ``justified conclusions\u27\u27 interpretation under the preferred semantic. This means that ASPIC+ and ASPIC+D may not have the same justified conclusions under the preferred semantics

Conditionals and modularity in general logics

In this work in progress, we discuss independence and interpolation and related topics for classical, modal, and non-monotonic logics

Maurinian Truths : Essays in Honour of Anna-Sofia Maurin on her 50th Birthday

This book is in honour of Professor Anna-Sofia Maurin on her 50th birthday. It consists of eighteen essays on metaphysical issues written by Swedish and international scholars