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35
Belief Functions and Default Reasoning
, 2000
"... We present a new approach to deal with default information based on the theory of belief functions. Our semantic structures, inspired by Adams' epsilon semantics, are epsilonbelief assignments, where mass values are either close to 0 or close to 1. In the first part of this paper, we show that t ..."
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Cited by 38 (3 self)
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We present a new approach to deal with default information based on the theory of belief functions. Our semantic structures, inspired by Adams' epsilon semantics, are epsilonbelief assignments, where mass values are either close to 0 or close to 1. In the first part of this paper, we show that these structures can be used to give a uniform semantics to several popular nonmonotonic systems, including Kraus, Lehmann and Magidor's system P, Pearl's system Z, Brewka's preferred subtheories, Geffner's conditional entailment, Pinkas' penalty logic, possibilistic logic and the lexicographic approach. In the second part, we use epsilonbelief assignments to build a new system, called LCD, and show that this system correctly addresses the wellknown problems of specificity, irrelevance, blocking of inheritance, ambiguity, and redundancy.
Weak theories of nonstandard arithmetic and analysis
 Reverse Mathematics
, 2001
"... Abstract. A general method of interpreting weak highertype theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomialtime ..."
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Cited by 7 (5 self)
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Abstract. A general method of interpreting weak highertype theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomialtime computable arithmetic. A means of formalizing basic real analysis in such theories is sketched. §1. Introduction. Nonstandard analysis, as developed by Abraham Robinson, provides an elegant paradigm for the application of metamathematical ideas in mathematics. The idea is simple: use modeltheoretic methods to build rich extensions of a mathematical structure, like secondorder arithmetic or a universe of sets; reason about what is true in these enriched structures;
Proofplanning Nonstandard Analysis
 IN THE 7TH INTERNATIONAL SYMPOSIUM ON AI AND MATHEMATICS
, 2002
"... This paper presents work carried out in the Clam proofplanner (Richardson et al. 00) on automating mathematical proofs using induction and nonstandard analysis. The central idea is to show that the proofs we present are wellsuited to proofplanning, due to their shared common structure. The theor ..."
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Cited by 6 (3 self)
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This paper presents work carried out in the Clam proofplanner (Richardson et al. 00) on automating mathematical proofs using induction and nonstandard analysis. The central idea is to show that the proofs we present are wellsuited to proofplanning, due to their shared common structure. The theorems presented in this paper belong to standard analysis, and have been proved using induction and techniques from nonstandard analysis. We rst give an overview of the proofplanning paradigm, giving a brief exposition of rippling as a heuristic for guiding rewriting. We then present the basic notions of nonstandard analysis and explain our axiomatisation. We then go on to explain the theorems we intend to prove and sketch their proofs. Finally we show the parts of the proofs which have been planned automatically in Clam and draw some conclusions from the work completed so far
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Cited by 6 (0 self)
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
Dealing With ZeroTime Transitions in Axiom Systems
 INFCTRL: Information and Computation (formerly Information and Control
, 1999
"... In the modelization of timedependent systems it is often useful to use the abstraction of zerotime transitions, i.e., changes of system state that occur in a time that can be neglected with respect to the whole dynamics of system evolution. Such an abstraction, however, sometimes generates critica ..."
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Cited by 2 (1 self)
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In the modelization of timedependent systems it is often useful to use the abstraction of zerotime transitions, i.e., changes of system state that occur in a time that can be neglected with respect to the whole dynamics of system evolution. Such an abstraction, however, sometimes generates critical situations in the formal system analysis. This may lead to limitations or unnatural use of such formal analysis. In this paper we present an approach that keeps the intuitive appealing of the zerotime transition abstraction yet maintaining simplicity and generality in its use. The approach is based on considering zerotime transitions as occurring in an infinitesimal, yet nonnull time. The adopted notation is borrowed from nonstandard analysis. The approach is illustrated through Petri nets as a case of state machines and TRIO as a case of logicbased assertion language, but it can be easily applied to any formal system dealing with states, time, and transitions.
Theory Extension in ACL2(r)
"... Abstract. ACL2(r) is a modified version of the theorem prover ACL2 that adds support for the irrational numbers using nonstandard analysis. It has been used to prove basic theorems of analysis, as well as the correctness of the implementation of transcendental functions in hardware. This paper pres ..."
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Cited by 2 (1 self)
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Abstract. ACL2(r) is a modified version of the theorem prover ACL2 that adds support for the irrational numbers using nonstandard analysis. It has been used to prove basic theorems of analysis, as well as the correctness of the implementation of transcendental functions in hardware. This paper presents the logical foundations of ACL2(r). These foundations are also used to justify significant enhancements to ACL2(r). 1.
Ultrapowers as Sheaves on a Category of Ultrafilters
, 2001
"... In 1993 I. Moerdijk presented a new model of nonstandard arithmetic in the topos of sheaves on a category of filters, Sh($\mathbb{F}$). This was later extended by E. Palmgren to a model of nonstandard analysis. The model in particular makes use of the sheaves ${}^*S$, which at any filter $\mathcal{F ..."
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Cited by 1 (1 self)
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In 1993 I. Moerdijk presented a new model of nonstandard arithmetic in the topos of sheaves on a category of filters, Sh($\mathbb{F}$). This was later extended by E. Palmgren to a model of nonstandard analysis. The model in particular makes use of the sheaves ${}^*S$, which at any filter $\mathcal{F}$ is the reduced power of the set $S$ over $\mathcal{F}$, ${}^*S(\mathcal{F})$. The details of this will be given in section 1.3. Before this, in section 1.1, we will give a short background to the subject of sheaves and logic and, in section 1.2, some preliminaries. In this paper we focus our attention on the sheaves on the subcategory of ultrafilters, Sh($\mathbb{U}$). The category $\mathbb{U}$ will be discussed in section 2. The sheaves of the form ${}^*S$ now, at an ultrafilter $\mathcal{U}$, represents the ultrapower of $S$ over $\mathcal{U}$, ${}^*S(\mathcal{U})$. More details on the sheaves over $\mathbb{U}$ can be found in section 3. In section 4 we study the internal logic in the topos of sheaves, which is classic since Sh($\mathbb{U}$) is an atomic topos. We prove that this logic does not coincide with the logic in any of the ultrapowers ${}^*S(\mathcal{U})$. The category of ultrafilters has a strong connection with ultrafilters under the RudinKeisler ordering, for instance we have $\mathcal{U} \leq \mathcal{V}$ if and only if $\textup{Hom}_{\mathbb{U}}(\mathcal{V}, \mathcal{U}) ot = \emptyset$. In the paper we define the RudinKeisler ordering on Sh($\mathbb{U}$) and study the consequences of it in our setting. In the paper we investigate the properties of Sh($\mathbb{U}$). We establish two transfer principles: external transfer, which is corresponding to {\L}o{\'s} theorem, and an internal transfer principle. We show that the topos theoretic axiom of choice does not hold in Sh($\mathbb{U}$) but establish some weak form of it and also prove some other properties similar to results proved by Palmgren about Sh($\mathbb{F}$). In section 5 we show that the topos can be used to model Nelson's internal set theory (IST). IST is an axiomatic approach to nonstandard analysis, which adds to ZFC a undefined unary predicate St($x$), for the standard sets, and axioms relating the standard and nonstandard sets.
Thomson's Lamp is Dysfunctional
, 1997
"... James Thomson envisaged a lamp which would be turned on for 1 minute, off for 1/2 minute, on for 1/4 minute, etc. ad infinitum. He asked whether the lamp would be on or off at the end of 2 minutes. Use of "internal set theory" (a version of nonstandard analysis), developed by Edward Nelson, shows Th ..."
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Cited by 1 (0 self)
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James Thomson envisaged a lamp which would be turned on for 1 minute, off for 1/2 minute, on for 1/4 minute, etc. ad infinitum. He asked whether the lamp would be on or off at the end of 2 minutes. Use of "internal set theory" (a version of nonstandard analysis), developed by Edward Nelson, shows Thomson's lamp is chimerical; its copy within set theory yields a contradiction. The demonstration extends to placing restrictions on other "infinite tasks": Zeno's paradoxes of motion; Kant's First Antinomy; and MalamentHogarth spacetimes in General Relativity. Critique of infinite tasks yields an analysis of motion and space & time; at some scale, motion would appear staccato and the latter pair would appear granular. The critique also shows necessary existence of some degree of "physical law". The suitability of internal set theory for analyzing phenomena is examined, using a paper by Bridger (1 997) to frame the discussion. Dr. William 1. McLaughlin 301170s Jet Propulsion Laboratory 4800...
On approximation of topological algebraic systems by finite ones
, 2003
"... We introduce and discuss a definition of approximation of a topological algebraic system A by finite algebraic systems of some class K. For the case of a discrete algebraic system this definition is equivalent to the wellknown definition of a local embedding of an algebraic system A in a class K of ..."
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We introduce and discuss a definition of approximation of a topological algebraic system A by finite algebraic systems of some class K. For the case of a discrete algebraic system this definition is equivalent to the wellknown definition of a local embedding of an algebraic system A in a class K of algebraic systems. According to this definition A is locally embedded in K iff it is a subsystem of an ultraproduct of some systems in K. We obtain a similar characterization of approximation of a locally compact system A by systems in K. We inroduce the bounded formulas of the signature of A and their approximations similar to those introduced by C.W.Henson [8] for Banach spaces. We prove that a positive bounded formula ϕ holds in A if all precise enough approximations of ϕ hold in all precise enough approximations of A. We prove that a locally compact field cannot be approximated by finite associative rings (not necessary commutative). Finite approximations of the field R can be concedered as computer systems for reals. Thus, it is impossible to construct a computer arithmetic for reals that is an associative ring. 1
AVERAGING THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS AND RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
, 2006
"... Abstract. We prove averaging theorems for ordinary differential equations and retarded functional differential equations. Our assumptions are weaker than those required in the results of the existing literature. Usually, we require that the nonautonomous differential equation and the autonomous aver ..."
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Abstract. We prove averaging theorems for ordinary differential equations and retarded functional differential equations. Our assumptions are weaker than those required in the results of the existing literature. Usually, we require that the nonautonomous differential equation and the autonomous averaged equation are locally Lipschitz and that the solutions of both equations exist on some interval. We extend this result to the case of vector fields which are continuous in the spatial variable uniformly with respect to time and without any assumption on the interval of existence of the solutions of the nonautonmous differential equation. Our results are formulated in classical mathematics. Their proofs use nonstandard analysis. 1.