Conservative extension concepts for nonmonotonic knowledge bases

Conservative extensions of logical theories play an important role in software engineering. They provide a formal basis for program refinement and guarantee the integrity and transparency of modules and objects. This paper provides a detailed analysis of conservative extension concepts in the context of nonmonotonic knowledge bases, in particular default theories. Since there are different approaches to nonmonotonic reasoning based on different strategies for dealing with multiple extensions, we define several alternative refinement concepts and study their interrelationships. We also show that refinement is well behaved with respect to strong stratification, a technique for reducing computational effort in default reasoning. © 2000 John Wiley & Sons, Inc.     

ConceptGraph: A Formal Model for Interpretation and Reasoning During Visual Analysis

In order to discuss the kinds of reasoning a visualization supports and the conclusions that can be drawn within the analysis context, a theoretical framework is needed that enables a formal treatment of the reasoning process. Such a model needs to encompass three stages of the visualization pipeline: encoding, decoding and interpretation. The encoding details how data are transformed into a visualization and what can be seen in the visualization. The decoding explains how humans construct graphical contexts inside the depicted visualization and how they interpret them assigning meaning to displayed structures according to a formal reasoning strategy. In the presented model, we adapt and combine theories for the different steps into a unified formal framework such that the analysis process is modelled as an assignment of meaning to displayed structures according to a formal reasoning strategy. Additionally, we propose the ConceptGraph, a combined graph-based representation of the finite-state transducers resulting from the three stages, that can be used to formalize and understand the reasoning process. We apply the new model to several visualization types and investigate reasoning strategies for various tasks.     

Towards a Formal Account of Reasoning about Evidence: Argumentation Schemes and Generalisations

This paper studies the modelling of legal reasoning about evidence within general theories of defeasible reasoning and argumentation. In particular, Wigmore's method for charting evidence and its use by modern legal evidence scholars is studied in order to give a formal underpinning in terms of logics for defeasible argumentation. Two notions turn out to be crucial, viz. argumentation schemes and empirical generalisations.     

Fuzzy set-based methods in instance-based reasoning

A formal framework of instance-based prediction is presented in which the generalization beyond experience is founded on the concepts of similarity and possibility. The underlying extrapolation principle is formalized within the framework of fuzzy rules. Thus, instance-based reasoning can be realized as fuzzy set-based approximate reasoning. More precisely, our model makes use of so-called possibility rules. These rules establish a relation between the concepts of similarity and possibility, which takes the uncertain character of similarity-based inference into account: inductive inference is possibilistic in the sense that predictions take the form of possibility distributions on the set of outcomes, rather than precise (deterministic) estimations. The basic model is extended by means of fuzzy set-based modeling techniques. This extension provides the basis for incorporating domain-specific (expert) knowledge. Thus, our approach favors a view of instance-based reasoning according to which the user interacts closely with the system     

Scientific Theories of Computational Systems in Model Checking

Model checking, a prominent formal method used to predict and explain the behaviour of software and hardware systems, is examined on the basis of reflective work in the philosophy of science concerning the ontology of scientific theories and model-based reasoning. The empirical theories of computational systems that model checking techniques enable one to build are identified, in the light of the semantic conception of scientific theories, with families of models that are interconnected by simulation relations. And the mappings between these scientific theories and computational systems in their scope are analyzed in terms of suitable specializations of the notions of model of experiment and model of data. Furthermore, the extensively mechanized character of model-based reasoning in model checking is highlighted by a comparison with proof procedures adopted by other formal methods in computer science. Finally, potential epistemic benefits flowing from the application of model checking in other areas of scientific inquiry are emphasized in the context of computer simulation studies of biological information processing.     

Premise Selection for Mathematics by Corpus Analysis and Kernel Methods

Smart premise selection is essential when using automated reasoning as a tool for large-theory formal proof development. This work develops learning-based premise selection in two ways. First, a fine-grained dependency analysis of existing high-level formal mathematical proofs is used to build a large knowledge base of proof dependencies, providing precise data for ATP-based re-verification and for training premise selection algorithms. Second, a new machine learning algorithm for premise selection based on kernel methods is proposed and implemented. To evaluate the impact of both techniques, a benchmark consisting of 2078 large-theory mathematical problems is constructed, extending the older MPTP Challenge benchmark. The combined effect of the techniques results in a 50 % improvement on the benchmark over the state-of-the-art Vampire/SInE system for automated reasoning in large theories.     

An argumentation framework in default logic

This article presents a formal theory about nontrivial reasoning with inconsistent information, applicable, among other things, to defeasible reasoning. The theory, which is inspired by a formal analysis of legal argument, is based on the idea that inconsistency tolerant reasoning is more than revising an unstructural set of premises; rather it should be regarded as constructing and comparing arguments for incompatible conclusions. This point of view gives rise to two important observations, both pointing at some flaws of other theories. The first is that arguments should be compared as they are constructed, viz. step-by-step, while the second observation is that a knowledge representation language is needed with a defeasible conditional, since the material implication gives rise to arguments which are not constructed in actual reasoning. Accordingly, a nonmonotonic logic, default logic, is chosen as the formalism underlying the argumentation framework. The general structure of the framework allows for any standard for comparing pairs of arguments; in this study two such standards are investigated, based on specificity and on orderings of the premises.     

Process and Policy: Resource-Bounded NonDemonstrative Reasoning

This paper investigates the appropriateness of formal dialectics as a basis for nonmonotonic reasoning and defeasible reasoning that takes computational limits seriously. Rules that can come into conflict should be regarded as policies, which are inputs to deliberative processes. Dialectical protocols are appropriate for such deliberations when resources are bounded and search is serial.
AI, it is claimed here, is now perfectly positioned to correct many misconceptions about reasoning that have resulted from mathematical logic's enormous success in this century: among them, (1) that all reasons are demonstrative, (2) that rational belief is constrained, not constructed, and (3) that process and disputation are not essential to reasoning. AI mainly provides new impetus to formalize the alternative (but older) conception of reasoning, and AI provides mechanisms with which to create compelling formalism that describes the control of processes.
The technical contributions here are: the partial justification of dialectic based on controlling search; the observation that nonmonotonic reasoning can be subsumed under certain kinds of dialectics; the portrayal of inference in knowledge bases as policy reasoning; the review of logics of dialogue and proposed extensions; and the preformal and initial formal discussion of aspects and variations of dialectical systems with nondemonstrative reasons.     

An improved factor based approach to precedential constraint

In this article I argue for rule-based, non-monotonic theories of common law judicial reasoning and improve upon one such theory offered by Horty and Bench-Capon. The improvements reveal some of the interconnections between formal theories of judicial reasoning and traditional issues within jurisprudence regarding the notions of the ratio decidendi and obiter dicta. Though I do not purport to resolve the long-standing jurisprudential issues here, it is beneficial for theorists both of legal philosophy and formalizing legal reasoning to see where the two projects interact.     

On case-based learnability of languages

Case-based reasoning is deemed an important technology to alleviate the bottleneck of knowledge acquisition in Artificial Intelligence (AI). In case-based reasoning, knowledge is represented in the form of particular cases with an appropriate similarity measure rather than any form of rules. The case-based reasoning paradigm adopts the view that an Al system is dynamically changing during its life-cycle which immediately leads to learning considerations. Within the present paper, we investigate the problem of case-based learning of indexable classes of formal languages. Prior to learning considerations, we study the problem of case-based representability and show that every indexable class is case-based representable with respect to a fixed similarity measure. Next, we investigate several models of case-based learning and systematically analyze their strengths as well as their limitations. Finally, the general approach to case-based learnability of indexable classes of formal languages is prototypically applied to so-called containmet decision lists, since they seem particularly tailored to case-based knowledge processing.     

Reasoning about pictures and similarity retrieval for image information systems based on SK-set knowledge representation

Spatial reasoning and similarity retrieval are two important functions of any image information system. Good spatial knowledge representation for images is necessary to adequately support these two functions. In this paper, we propose a new spatial knowledge representation, called the SK-set based on morphological skeleton theories. Spatial reasoning algorithms which achieve more accurate results by directly analysing skeletons are described. SK-set facilitates browsing and progressive visualization. We also define four new types of similarity measures and propose a similarity retrieval algorithm for performing image retrieval. Moreover, using SK-set as a spatial knowledge representation will reduce the storage space required by an image database significantly.     

A framework for checking proofs naturally

We propose a logical framework, called Natural Framework (NF), which supports formal reasoning about computation and logic (CAL) on a computer. NF is based on a theory of Judgments and Derivations. NF is designed by observing how working mathematical theories are created and developed. Our observation is that the notions of judgments and derivations are the two fundamental notions used in any mathematical activity. We have therefore developed a theory of judgments and derivations and designed a framework in which the theory provides a uniform and common play ground on which various mathematical theories can be defined as derivation games and can be played, namely, can write and check proofs. NF is equipped with a higher-order intuitionistic logic and derivations (proofs) are described following Gentzen’s natural deduction style. NF is part of an interactive computer environment CAL and it is also referred to as NF/CAL. CAL is written in Emacs Lisp and it is run within a special buffer of the Emacs editor. CAL consists of user interface, a general purpose parser and a checker for checking proofs of NF derivation games. NF/CAL system has been successfully used as an education system for teaching CAL for undergraduate students for about 8 years. We will give an overview of the NF/CAL system both from theoretical and practical sides.     

A model of legal reasoning with cases incorporating theories and values

Reasoning with cases has been a primary focus of those working in AI and law who have attempted to model legal reasoning. In this paper we put forward a formal model of reasoning with cases which captures many of the insights from that previous work. We begin by stating our view of reasoning with cases as a process of constructing, evaluating and applying a theory. Central to our model is a view of the relationship between cases, rules based on cases, and the social values which justify those rules. Having given our view of these relationships, we present our formal model of them, and explain how theories can be constructed, compared and evaluated. We then show how previous work can be described in terms of our model, and discuss extensions to the basic model to accommodate particular features of previous work. We conclude by identifying some directions for future work.     

Vivid: A framework for heterogeneous problem solving

We introduce Vivid, a domain-independent framework for mechanized heterogeneous reasoning that combines diagrammatic and symbolic representation and inference. The framework is presented in the form of a family of denotational proof languages (DPLs). We present novel formal structures, called named system states, that are specifically designed for modeling potentially underdetermined diagrams. These structures allow us to deal with incomplete information, a pervasive feature of heterogeneous problem solving. We introduce a notion of attribute interpretations that enables us to interpret first-order relational signatures into named system states, and develop a formal semantic framework based on 3-valued logic. We extend the assumption-base semantics of DPLs to accommodate diagrammatic reasoning by introducing general inference mechanisms for the valid extraction of information from diagrams, and for the incorporation of sentential information into diagrams. A rigorous big-step operational semantics is given, on the basis of which we prove that the framework is sound. We present examples of particular instances of Vivid in order to solve a series of problems, and discuss related work.     

Normative conflicts in legal reasoning

This article proposes a formal analysis of a fundamental aspect of legal reasoning: dealing with normative conflicts. Firstly, examples are illustrated concerning the dynamics of legal systems, the application of rules and exceptions, and the semantic indeterminacy of legal sources. Then two approaches to cope with conflicting information are presented: the preferred theories of Brewka, and the belief change functions of Alchourrón, Gärdenfors, and Makinson. The relations between those approaches are closely examined, and some aspects of a model of reasoning with normative conflicts are outlined. Since this model takes into account an ordering of the involved regulations, criteria to order legal norms are finally specified.     

Credible Case-Based Inference Using Similarity Profiles

In this paper, we propose a method for retrieving promising candidate solutions in case-based problem solving. Our method, referred to as credible case-based inference, makes use of so-called similarity profiles as a formal model of the key hypothesis underlying case-based reasoning (CBR), namely, the assumption that similar problems have similar solutions. Proceeding from this formalization, it becomes possible to derive theoretical properties of the corresponding inference scheme in a rigorous way. In particular, it can be shown that, under mild technical conditions, a set of candidates covers the true solution with high probability. Thus, the approach supports an important subtask in CBR, namely, to generate potential solutions for a new target problem in a sound manner and hence contributes to the methodical foundations of CBR. Due to its generality, it can be employed for different types of performance tasks and can easily be integrated in existing CBR systems.     

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In this paper, we propose a method for retrieving promising candidate solutions in case-based problem solving. Our method, referred to as credible case-based inference, makes use of so-called similarity profiles as a formal model of the key hypothesis underlying case-based reasoning (CBR), namely, the assumption that similar problems have similar solutions. Proceeding from this formalization, it becomes possible to derive theoretical properties of the corresponding inference scheme in a rigorous way. In particular, it can be shown that, under mild technical conditions, a set of candidates covers the true solution with high probability. Thus, the approach supports an important subtask in CBR, namely, to generate potential solutions for a new target problem in a sound manner and hence contributes to the methodical foundations of CBR. Due to its generality, it can be employed for different types of performance tasks and can easily be integrated in existing CBR systems