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29
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 ..."
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Cited by 67 (5 self)
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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.
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 cases, the ..."
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Cited by 16 (6 self)
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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...
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 ..."
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Cited by 11 (1 self)
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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
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 ..."
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Cited by 10 (2 self)
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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 ..."
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Cited by 8 (2 self)
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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 ..."
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Cited by 8 (4 self)
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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 = ...
New complexity bounds for cylindrical decompositions of subPfaffian sets
"... TarskiSeidenberg principle plays a key role in many applications and algorithms of computer algebra. Moreover it is constructive and some very efficient quantifier elimination algorithms appeared recently. However, TarskiSeidenberg principle is wrong for firstorder theories involving real analyti ..."
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Cited by 4 (2 self)
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TarskiSeidenberg principle plays a key role in many applications and algorithms of computer algebra. Moreover it is constructive and some very efficient quantifier elimination algorithms appeared recently. However, TarskiSeidenberg principle is wrong for firstorder theories involving real analytic functions (e.g. an exponential function). In this case a weaker statement is sometimes true, a possibility to eliminate one sort of quantifiers (either 8 or 9). We construct an algorithm for a cylindrical cell decomposition of a closed cube I n R n compatible with a semianalytic subset S I n , defined by analytic functions from a certain broad finitely defined class (modulo an oracle for deciding emptiness of such sets). In particular the algorithm is able to eliminate one sort of quantifiers from a firstorder formula. The complexity bound of the algorithm is doubly exponential in O(n 2 ).
Neutrosophic logics on NonArchimedean Structures
 Critical Review, Creighton University, USA
"... We present a general way that allows to construct systematically analytic calculi for a large family of nonArchimedean manyvalued logics: hyperrationalvalued, hyperrealvalued, and padic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes ’ ax ..."
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Cited by 2 (0 self)
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We present a general way that allows to construct systematically analytic calculi for a large family of nonArchimedean manyvalued logics: hyperrationalvalued, hyperrealvalued, and padic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes ’ axiom. These logics are built as different extensions of standard manyvalued logics (namely, Lukasiewicz’s, Gödel’s, Product, and Post’s logics). The informal sense of Archimedes ’ axiom is that anything can be measured by a ruler. Also logical multiplevalidity without Archimedes ’ axiom consists in that the set of truth values is infinite and it is not wellfounded and wellordered. We consider two cases of nonArchimedean multivalued logics: the first with manyvalidity in the interval [0, 1] of hypernumbers and the second with manyvalidity in the ring Zp of padic integers. On the base of nonArchimedean valued logics, we construct nonArchimedean valued interval neutrosophic logics by which we can describe neutrality phenomena.
Erlangen Program at Large1: Geometry of Invariants
, 2010
"... doi:10.3842/SIGMA.2010.076 Abstract. This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL 2(R) group. We describe here geometries of corresponding domains. ..."
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Cited by 2 (1 self)
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doi:10.3842/SIGMA.2010.076 Abstract. This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL 2(R) group. We describe here geometries of corresponding domains. The principal rôle is played by Clifford algebras of matching types. In this paper we also generalise the Fillmore–Springer–Cnops construction which describes cycles as points in the extended space. This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach. Key words: analytic function theory; semisimple groups; elliptic; parabolic; hyperbolic; Clifford algebras; complex numbers; dual numbers; double numbers; splitcomplex numbers; Möbius transformations
Pricing European Options Without Probability
, 1995
"... It is well known that in the case where the stock price S t is governed by the equation dS t =S t = dt + oedW t , any European option satisfying weak regularity conditions has a fair price (the BlackScholes formula and its generalizations). We consider the case where no probabilistic assumptions ..."
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Cited by 1 (1 self)
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It is well known that in the case where the stock price S t is governed by the equation dS t =S t = dt + oedW t , any European option satisfying weak regularity conditions has a fair price (the BlackScholes formula and its generalizations). We consider the case where no probabilistic assumptions are made about S t ; instead, we assume that the derivative security D which pays a dividend of (dS t =S t ) 2 (the squared relative increase in the price of S t ) each instant dt is traded in the market. We prove that the "regular" European options have fair prices provided that both S t and D t (the price process of D) are continuous and the fractal dimensions of the graphs of S t and D t satisfy certain inequalities. Intuitively our assumptions are much weaker than the usual assumption dS t =S t = dt + oedW t . Key Words: BlackScholes formula, fractal dimension, pathwise stochastic integral, nonstandard analysis The final version of this paper was prepared for the seminar on the f...