Dov Gabbay

In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. We then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91].

In the first part of this paper we introduced products of modal logics and proved basic results on their axiomatisability and the f.m.p. In this continuation paper we prove a stronger result - the product f.m.p. holds for products of modal logics in which some of the modalities are reflexive or serial. This theorem is applied in classical first-order logic, we identify a new Square Fragment of the classical logic, where the basic predicates are binary and all quantifiers are relativised, and for which we show the f.m.p. in the classical sense. Also we prove that SF not included in Guarded Fragment and that it can be embedded into the equational theory of relational algebras

In this paper we propoee a logic programming language which supports hypothetical updates together with integrity constraints. The language makes use of a revision mechanism, which is needed to restore consistency when an update violates some integrity constraint. The revision policy adopted is based on the simple idea that more recent information is preferred to earlier one. We show how this language can be used to represent and perform several types of defeasible reasoning. We develop a logical characterization of the language in a three-valued conditional logic, by introducing a completion construction. We show that the operational semantics is sound and complete with respect to the completion construction

Inheritance diagrams are directed acyclic graphs with two types of connections between nodes: x → y and x ↛ y . Given a diagram D, one can ask the formal question of “is there a valid path between node x and node y?” Depending on the existence of a valid path we can answer the question “x is a y” or “x is not a y”. The answer to the above question is determined through a complex inductive algorithm on paths between arbitrary pairs of points in the graph. This paper aims to simplify and interpret such diagrams and their algorithms. We approach the area on two fronts. Suggest reactive arrows to simplify the algorithms for the winning paths. We give a conceptual analysis of inheritance diagrams, and compare our analysis to the “small” and “big sets” of preferential and related reasoning. In our analysis, we consider nodes as information sources and truth values, direct links as information, and valid paths as information channels and comparisons of truth values. This results in an upward chaining, split validity, off-path preclusion inheritance formalism of a particularly simple type. We show that the small and big sets of preferential reasoning have to be relativized if we want them to conform to inheritance theory, resulting in a more cautious approach, perhaps closer to actual human reasoning. We will also interpret inheritance diagrams as theories of prototypical reasoning, based on two distances: set difference, and information difference. We will see that some of the major distinctions between inheritance formalisms are consequences of deeper and more general problems of treating conflicting information. It is easily seen that inheritance diagrams can also be analysed in terms of reactive diagrams - as can all argumentation systems. AMS Classification: 68T27, 68T30

Formal models of argumentation have been investigated in several areas, from multi-agent systems and artificial intelligence (AI) to decision making, philosophy and law. In artificial intelligence, logic-based models have been the standard for the representation of argumentative reasoning. More recently, the standard logic-based models have been shown equivalent to standard connectionist models. This has created a new line of research where (i) neural networks can be used as a parallel computational model for argumentation and (ii) neural networks can be used to combine argumentation, quantitative reasoning and statistical learning. At the same time, non-standard logic models of argumentation started to emerge. In this paper, we propose a connectionist cognitive model of argumentation that accounts for both standard and non-standard forms of argumentation. The model is shown to be an adequate framework for dealing with standard and non-standard argumentation, including joint-attacks, argument support, ordered attacks, disjunctive attacks, meta-level attacks, self-defeating attacks, argument accrual and uncertainty. We show that the neural cognitive approach offers an adequate way of modelling all of these different aspects of argumentation. We have applied the framework to the modelling of a public prosecution charging decision as part of a real legal decision making case study containing many of the above aspects of argumentation. The results show that the model can be a useful tool in the analysis of legal decision making, including the analysis of what-if questions and the analysis of alternative conclusions. The approach opens up two new perspectives in the short-term: the use of neural networks for computing prevailing arguments efficiently through the propagation in parallel of neuronal activations, and the use of the same networks to evolve the structure of the argumentation network through learning (e.g. to learn the strength of arguments from data).

We study access control policies based on the says operator by introducing a logical framework called Fibred Security Language (FSL) which is able to deal with features like joint responsibility between sets of principals and to identify them by means of first-order formulas. FSL is based on a multimodal logic methodology. We first discuss the main contributions from the expressiveness point of view, we give semantics for the language both for classical and intuitionistic fragment), we then prove that in order to express well-known properties like ‘speaks-for’ or ‘hand-off’, defined in terms of says, we do not need second-order logic (unlike previous approaches) but a decidable fragment of first-order logic suffices. We propose a model-driven study of the says axiomatization by constraining the Kripke models in order to respect desirable security properties, we study how existing access control logics can be translated into FSL and we give completeness for the logic

Goal Directed Proof Theory presents a uniform and coherent methodology for automated deduction in non-classical logics, the relevance of which to computer science is now widely acknowledged. The methodology is based on goal-directed provability. It is a generalization of the logic programming style of deduction, and it is particularly favourable for proof search. The methodology is applied for the first time in a uniform way to a wide range of non-classical systems, covering intuitionistic, intermediate, modal and substructural logics. The book can also be used as an introduction to these logical systems form a procedural perspective. Readership: Computer scientists, mathematicians and philosophers, and anyone interested in the automation of reasoning based on non-classical logics. The book is suitable for self study, its only prerequisite being some elementary knowledge of logic and proof theory.

We show that modal logics characterized by a class of frames satisfying the insertion property are suitable for Reiter's default logic. We refine the canonical fix point construction defined by Marek, Schwarz and Truszczyński for Reiter's default logic and thus we addrress a new paradigm for nonmonotonic logic. In fact, differently from the construction defined by these authors. we show that suitable modal logics for such a construction must indeed contain K D4. When reflexivity is added to the modal logic used for the fix point construction then we come to the Marek Schwarz and Truszczyński framework for Reiter's default logic. Our framework, in fact, is appropriate also to the family of modal logics in between S4 and S4f. If, instead, reflexivity is dropped, then we show that a new family of modal logics is gained, namely the modal logics in between KD4 and KD4Z. The upper bound can be extended to the modal logic KD4LZ whenever the propositional language taken into account is finite

There are several areas in logic where the monotonicity of the consequence relation fails to hold. Roughly these are the traditional non-monotonic systems arising in Artificial Intelligence, numerical non-monotonic systems, resource logics, and the logic of theory change. We are seeking a common axiomatic and semantical approach to the notion of consequence whieh can be specialised to any of the above areas. This paper introduces the notions of structured consequence relation, shift operators and structural connectives, and shows an intrinsic connection between the above areas.