Results 1  10
of
54
Constraint Hierarchies and Logic Programming
, 1989
"... Constraint Logic Programming (CLP) is a general scheme for extending logic programming to include constraints. It is parameterized by D, the domain of the constraints. However, CLP(D) languages, as well as most other constraint systems, only allow the programmer to specify constraints that must hold ..."
Abstract

Cited by 67 (5 self)
 Add to MetaCart
(Show Context)
Constraint Logic Programming (CLP) is a general scheme for extending logic programming to include constraints. It is parameterized by D, the domain of the constraints. However, CLP(D) languages, as well as most other constraint systems, only allow the programmer to specify constraints that must hold. In many applications, such as interactive graphics, page layout, and decision support, one needs to express preferences as well as strict requirements. If we wish to make full use of the constraint paradigm, we need ways to represent these defaults and preferences declaratively, as constraints, rather than encoding them in the procedural parts of the language. We describe a scheme for extending CLP(D) to include both required and preferential constraints, with an arbitrary number of strengths of preference. We present some of the theory of such languages, and an algorithm for executing them. To test our ideas, we have implemented an interpreter for an instance of this language scheme with D equal to the reals. We describe our interpreter, and outline some examples of using this language.
Per Evaluating Linguistic Expressions and Functional Fuzzy Theories in Fuzzy Logic
 Computing with Words in Systems Analysis
, 1998
"... In this paper, we introduce a new mathematical model of the meaning of the basic linguistic trichotomy, which are the canonical words \small", \medium " and \big". The model is based on the concept of horizon as elaborated in the Alternative Set Theory. Such a model makes also possibl ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
(Show Context)
In this paper, we introduce a new mathematical model of the meaning of the basic linguistic trichotomy, which are the canonical words \small", \medium " and \big". The model is based on the concept of horizon as elaborated in the Alternative Set Theory. Such a model makes also possible to include naturally the linguistic hedges which form a consistent class of functions. Each linguistic hedge is thus characterized by one number only. Then it is shown that continuous functional dependencies between x and y can be described (precisely or approximately) by the collections of logical formulas of implicative form with predicates interpreted by fuzzy sets with meaning of the basic linguistic trichotomy. It demonstrates the expressive power of modied by linguistic hedges membership functions of fuzzy sets from the basic triplet. 1
Lexicographic probability, conditional probability, and nonstandard probability
 In Theoretical Aspects of Rationality and Knowledge: Proc. Eighth Conference (TARK 2001
, 2001
"... The relationship between Popper spaces (conditional probability spaces that satisfy some regularity conditions), lexicographic probability systems (LPS’s) [Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b], and nonstandard probability spaces (NPS’s) is considered. If coun ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
(Show Context)
The relationship between Popper spaces (conditional probability spaces that satisfy some regularity conditions), lexicographic probability systems (LPS’s) [Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b], and nonstandard probability spaces (NPS’s) is considered. If countable additivity is assumed, Popper spaces and a subclass of LPS’s are equivalent; without the assumption of countable additivity, the equivalence no longer holds. If the state space is finite, LPS’s are equivalent to NPS’s. However, if the state space is infinite, NPS’s are shown to be more general than LPS’s. JEL classification numbers: C70; D80; D81; 1
Calculus and Numerics on LeviCivita Fields
, 1996
"... The formal process of the evaluation of derivatives using some of the various modern methods of computational differentiation can be recognized as an example for the application of conventional "approximate" numerical techniques on a nonarchimedean extension of the real numbers. In many c ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
The formal process of the evaluation of derivatives using some of the various modern methods of computational differentiation can be recognized as an example for the application of conventional "approximate" numerical techniques on a nonarchimedean extension of the real numbers. In many cases, the application of "infinitely small" numbers instead of "small but finite" numbers allows the use of the old numerical algorithm, but now with an error that in a rigorous way can be shown to become infinitely small (and hence irrelevant). While intuitive ideas in this direction have accompanied analysis from the early days of Newton and Leibniz, the first rigorous work goes back to LeviCivita, who introduced a number field that in the past few years turned out to be particularly suitable for numerical problems. While LeviCivita's field appears to be of fundamental importance and simplicity, efforts to introduce advanced concepts of calculus on it are only very new. In this paper, we address s...
Characterizing the Common Prior Assumption
 Journal of Economic Theory
, 2002
"... Abstract: Logical characterizations of the common prior assumption (CPA) are investigated. Two approaches are considered. The first is called frame distinguishability, and is similar in spirit to the approaches considered in the economics literature. Results similar to those obtained in the economi ..."
Abstract

Cited by 13 (7 self)
 Add to MetaCart
Abstract: Logical characterizations of the common prior assumption (CPA) are investigated. Two approaches are considered. The first is called frame distinguishability, and is similar in spirit to the approaches considered in the economics literature. Results similar to those obtained in the economics literature are proved here as well, namely, that we can distinguish finite spaces that satisfy the CPA from those that do not in terms of disagreements in expectation. However, it is shown that, for the language used here, no formulas can distinguish infinite spaces satisfying the CPA from those that do not. The second approach considered is that of finding a sound and complete axiomatization. Such an axiomatization is provided; again, the key axiom involves disagreements in expectation. The same axiom system is shown to be sound and complete both in the finite and the infinite case. Thus, the two approaches to characterizing the CPA behave quite differently in the case of infinite spaces. 1
Foundations Of Nonstandard Analysis  A Gentle Introduction to Nonstandard Extemsions
 In Nonstandard analysis (Edinburgh
"... this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field
Order of Magnitude Reasoning in Qualitative Differential Equations
, 1987
"... We present a theory that combines order of magnitude reasoning with envisionment of qualitative differential equations. Such a theory can be used to reason qualitatively about dynamical systems containing parameters of widely varying magnitudes. We present an a mathematical analysis of envisionment ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
We present a theory that combines order of magnitude reasoning with envisionment of qualitative differential equations. Such a theory can be used to reason qualitatively about dynamical systems containing parameters of widely varying magnitudes. We present an a mathematical analysis of envisionment over orders of magnitude, including a complete categorization of adjacent pairs of qualitative states. We show how this theory can be applied to simple problems, we give an algorithm for generating a complete envisionment graph, and we discuss the implementation of this algorithm in a running program.
Complexity Lower Bounds for Computation Trees with Elementary Transcendental Function Gates (Extended Abstract)
 Theoretical Computer Science
, 1996
"... We consider computation trees which admit as gate functions along with the usual arithmetic operations also algebraic or transcendental functions like exp; log; sin; square root (defined in the relevant domains) or much more general, Pfaffian functions. A new method for proving lower bounds on the d ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
(Show Context)
We consider computation trees which admit as gate functions along with the usual arithmetic operations also algebraic or transcendental functions like exp; log; sin; square root (defined in the relevant domains) or much more general, Pfaffian functions. A new method for proving lower bounds on the depth of these trees is developed which allows to prove a lower bound \Omega\Gamma p log N ) for testing membership to a convex polyhedron with N facets of all dimensions, provided that N is large enough. This method differs essentially from the approaches adopted for algebraic computation trees ([1], [4], [26], [13]). 1 Pfaffian computation trees We consider the following computation model, a generalization of the algebraic computation trees (see, e.g., [1], [26]). Definition 1. Pfaffian computation tree T is a tree at every node v of which a Pfaffian function f v in variables X 1 ; : : : ; Xn is attached, which satisfies the following properties. Let f v0 ; : : : ; f v l ; f v l+1 = ...
Numerosities of labelled sets: a new way of counting
 Advances in Math., 173
, 2003
"... Abstract. The notions of “labelled set ” and “numerosity ” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Abstract. The notions of “labelled set ” and “numerosity ” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ultrafilter, hence it is independent of the axioms of ZFC. Introduction. Similarly as cardinals and ordinals, the “numerosities ” we present in this paper originate as an attempt to extending the notion of finite cardinality. By considering suitable “labellings”, we show that a notion of numerosity for (countable) infinite sets can be defined in such a way that the usual properties of finite cardinalities
Alphatheory: an elementary axiomatics for nonstandard analysis
 Expo. Math
"... Abstract. The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal ” number α. The resulting axiomatic system, including a formalization of an interpretation of Cauchy’s idea of infinitesimals, is related to the existence of ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal ” number α. The resulting axiomatic system, including a formalization of an interpretation of Cauchy’s idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The AlphaTheory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals.