2,359 research outputs found
Meaningful aggregation functions mapping ordinal scales into an ordinal scale: a state of the art
We present an overview of the meaningful aggregation functions mapping
ordinal scales into an ordinal scale. Three main classes are discussed, namely
order invariant functions, comparison meaningful functions on a single ordinal
scale, and comparison meaningful functions on independent ordinal scales. It
appears that the most prominent meaningful aggregation functions are lattice
polynomial functions, that is, functions built only on projections and minimum
and maximum operations
Representation of maxitive measures: an overview
Idempotent integration is an analogue of Lebesgue integration where
-maxitive measures replace -additive measures. In addition to
reviewing and unifying several Radon--Nikodym like theorems proven in the
literature for the idempotent integral, we also prove new results of the same
kind.Comment: 40 page
First-order Nilpotent Minimum Logics: first steps
Following the lines of the analysis done in [BPZ07, BCF07] for first-order
G\"odel logics, we present an analogous investigation for Nilpotent Minimum
logic NM. We study decidability and reciprocal inclusion of various sets of
first-order tautologies of some subalgebras of the standard Nilpotent Minimum
algebra. We establish a connection between the validity in an NM-chain of
certain first-order formulas and its order type. Furthermore, we analyze
axiomatizability, undecidability and the monadic fragments.Comment: In this version of the paper the presentation has been improved. The
introduction section has been rewritten, and many modifications have been
done to improve the readability; moreover, numerous references have been
added. Concerning the technical side, some proofs has been shortened or made
more clear, but the mathematical content is substantially the same of the
previous versio
On the semantics of fuzzy logic
AbstractThis paper presents a formal characterization of the major concepts and constructs of fuzzy logic in terms of notions of distance, closeness, and similarity between pairs of possible worlds. The formalism is a direct extension (by recognition of multiple degrees of accessibility, conceivability, or reachability) of the najor modal logic concepts of possible and necessary truth.Given a function that maps pairs of possible worlds into a number between 0 and 1, generalizing the conventional concept of an equivalence relation, the major constructs of fuzzy logic (conditional and unconditioned possibility distributions) are defined in terms of this similarity relation using familiar concepts from the mathematical theory of metric spaces. This interpretation is different in nature and character from the typical, chance-oriented, meanings associated with probabilistic concepts, which are grounded on the mathematical notion of set measure. The similarity structure defines a topological notion of continuity in the space of possible worlds (and in that of its subsets, i.e., propositions) that allows a form of logical “extrapolation” between possible worlds.This logical extrapolation operation corresponds to the major deductive rule of fuzzy logic — the compositional rule of inference or generalized modus ponens of Zadeh — an inferential operation that generalizes its classical counterpart by virtue of its ability to be utilized when propositions representing available evidence match only approximately the antecedents of conditional propositions. The relations between the similarity-based interpretation of the role of conditional possibility distributions and the approximate inferential procedures of Baldwin are also discussed.A straightforward extension of the theory to the case where the similarity scale is symbolic rather than numeric is described. The problem of generating similarity functions from a given set of possibility distributions, with the latter interpreted as defining a number of (graded) discernibility relations and the former as the result of combining them into a joint measure of distinguishability between possible worlds, is briefly discussed
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Concepts and analogies in cybernetics: Mathematical investigations of the role of analogy in concept formation and problem solving; with emphasis for conflict resolution via object and morphism eliminations
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We address two problematic areas of cybernetics; nam. Analogical Problem Solving (APS) and Analogical Learning (AL). Both these human faculties do unquestionably require Intelligence. In addition, we point out that shifting of representations is the main unified theme underlying these two intellectual tasks. We focus our attention on the formulation and clarification of the notion of analogy, which has been loosely treated and used in the literature; and also on its role in shifting of representations.
We describe analogizing situations in a new representational scheme, borrowed from mathematics and modified and extended to cater for our targets. We call it k-structure, closely resembling semantic networks and directed graphs; the main components of it are the so-called objects and morphisms. We argue and substantiate the need for such a representation scheme, by analysing what its constituents stand for and by cataloguing its virtues, the main one being its visual appeal and its mathematical clarity, and by listing its disadvantages when it is compared to other representation systems. Emphasis is also given to its descriptive power and usefulness by implementing it in a number of APS and AL situations. Besides representation issues, attention is paid to intelligence mechanisms which are involved in APS and AL. A cornerstone in APS and a fundamental theme in AL is the 'skeletization of k-structures'. APS is conceived as 'harmonization of skeletons'. The methodology we develop involves techniques which are computer implemented and extensively studied in theoretic terms via a proposed theory for extended k-structures. To name but a few: 1. 'the separation of the context of a concept from the concept itself', based on the ideas of k-opens and k-spaces; 2, 'object and morphism elimination' of a controversial nature; and 3. 'conflict or deadlock or dilemma resolution' which naturally arises in a k-structure interaction. The overall system, is then applied to capture the essence of EVANS' (1963) analogy-type problems and WINSTOM (1970) learning-type situations. In our attempt not to be too informal, we use basic notions and terminology from abstract Algebra, Topology and Category theory. We rather tend to be "non-logical" (analogical) in EVANS' and WINSTON's sense; "non-numeric", in MESAROVIC (1970) terms (we rather deal with abstract conceptual entities); "non-linguistic" (we do not touch natural language); and "non-resolution" oriented, in the sense of BLEDSOE (1977). However, we give hints sometimes about logical deductive axiomatic systems, employing First Order Predicate Calculus (FOPC); and about semiotics, by which we denote syntactic-semantic-pragmatic features of our system and issues of the problem domains it is acting upon. We believe in what we call: shift from the traditional 'Heuristic search paradigm' era to the 'Analogy-paradigm' era underlying Artificial Intelligence and Cybernetics. We justify this merely by listing a number of A. I. works, which employ, in some way or another, the concept of analogy, over the last fifteen years or so, where a noticeable peak is obvious during the last years and especially in 1977. Finally, we hope that if the proposed conceptual framework and techniques developed do not straightforwardly constitute some kind of platform for Artificial Intelligence, at least it would give some insights into and illuminate our understanding of the two most fundamental faculties the human brain is occupied with; namely problem solving and learning
Facets and Levels of Mathematical Abstraction
International audienceMathematical abstraction is the process of considering and manipulating operations, rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived from the content connected to particular applications. There is no one single way of performing mathematical abstraction. The term "abstraction" does not name a unique procedure but a general process, which goes many ways that are mostly simultaneous and intertwined ; in particular, the process does not amount only to logical subsumption. I will consider comparatively how philosophers consider abstraction and how mathematicians perform it, with the aim to bring to light the fundamental thinking processes at play, and to illustrate by significant examples how much intricate and multi-leveled may be the combination of typical mathematical techniques which include axiomatic method, invarianceprinciples, equivalence relations and functional correspondences.L'abstraction mathématique consiste en la considération et la manipulation d'opérations, règles et concepts indépendamment du contenu dont les nantissent des applications particulières et du rapport qu'ils peuvent avoir avec les phénomènes et les circonstances du monde réel. L'abstraction mathématique emprunte diverses voies. Le terme " abstraction " ne désigne pasune procédure unique, mais un processus général où s'entrecroisent divers procédés employés successivement ou simultanément. En particulier, l'abstraction mathématique ne se réduit pas à la subsomption logique. Je vais étudier comparativement en quels termes les philosophes expliquent l'abstraction et par quels moyens les mathématiciens la mettent en oeuvre. Je voudrais parlà mettre en lumière les principaux processus de pensée en jeu et illustrer par des exemples divers niveaux d'intrication de techniques mathématiques récurrentes, qui incluent notamment la méthode axiomatique, les principes d'invariance, les relations d'équivalence et les correspondances fonctionnelles
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