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12
A Logic for Reasoning about Probabilities
 Information and Computation
, 1990
"... We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable ( ..."
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Cited by 214 (21 self)
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We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by DempsterShafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satistiability is NPcomplete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields. ( 1990 Academic Press. Inc 1.
Reasoning about Online Algorithms with Weighted Automata
, 2008
"... We describe an automatatheoretic approach for the competitive analysis of online algorithms. Our approach is based on weighted automata, which assign to each input word a cost in IR ≥0. By relating the “unbounded look ahead ” of optimal offline algorithms with nondeterminism, and relating the “no l ..."
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Cited by 9 (7 self)
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We describe an automatatheoretic approach for the competitive analysis of online algorithms. Our approach is based on weighted automata, which assign to each input word a cost in IR ≥0. By relating the “unbounded look ahead ” of optimal offline algorithms with nondeterminism, and relating the “no look ahead ” of online algorithms with determinism, we are able to solve problems about the competitive ratio of online algorithms, and the memory they require, by reducing them to questions about determinization and approximated determinization of weighted automata. 1
The linear separability problem: Some testing methods
 IEEE TNN
, 2006
"... The notion of linear separability is used widely in machine learning research. Learning algorithms that use this concept to learn include neural networks (Single Layer Perceptron and Recursive Deterministic Perceptron), and kernel machines (Support Vector Machines). This paper presents an overview ..."
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Cited by 6 (2 self)
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The notion of linear separability is used widely in machine learning research. Learning algorithms that use this concept to learn include neural networks (Single Layer Perceptron and Recursive Deterministic Perceptron), and kernel machines (Support Vector Machines). This paper presents an overview of several of the methods for testing linear separability between two classes. The methods are divided into four groups: those based on linear programming, those based on computational geometry, one based on neural networks, and one based on quadratic programming. The Fisher linear discriminant method is also presented. A section on the quantification of the complexity of classification problems is included.
SylvesterGallai theorem and metric betweenness
, 2002
"... Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines fro ..."
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Cited by 4 (0 self)
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Sylvester conjectured in 1893 and Gallai proved some forty years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the SylvesterGallai theorem generalizes as follows: in every finite metric space, there is a line consisting of either two points or all the points of the space. Then we present slight evidence in support of this rash conjecture and finally we discuss the underlying ternary relation of metric betweenness. 1 The SylvesterGallai theorem Sylvester (1893) proposed the following problem: Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all line in the same right line.
Verifying The ‘Consistency’ Of Shading Patterns And 3D Structures
, 1993
"... The problem of interpreting images in terms of their shading and reflectance components has traditionally been addressed as an early vision task in a simple 2D Mondrian domain. Recently it has been appreciated that in a 3D world, such conventional approaches are inadequate; more sophisticated strate ..."
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Cited by 2 (0 self)
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The problem of interpreting images in terms of their shading and reflectance components has traditionally been addressed as an early vision task in a simple 2D Mondrian domain. Recently it has been appreciated that in a 3D world, such conventional approaches are inadequate; more sophisticated strategies are required. One such strategy has been proposed by Sinha [22, 25], who has addressed the problem as a midlevel vision task rather than as a purely lowlevel one. Sinha suggested that a key computation that needs to be performed for interpreting images acquired in a 3D domain is the verification of the consistency of image shading patterns and the likely 3D structure of the scene. This is the problem we have addressed in the present paper. Considerations of robustness and generality have prompted us to discard available quantitative techniques in favor of a qualitative one. The two prime attributes of our technique are its use of qualitative comparisons of graylevels instead of their precise absolute measurements and also its doing away with the need of an exact prespecification of the surface reflectance function. We show that this idea lends itself naturally to a linearprogramming solution technique and that results obtained with some sample images are in conformity with human perception.
Characterizing and Reasoning about Probabilistic and NonProbabilistic Expectation
, 2007
"... Expectation is a central notion in probability theory. The notion of expectation also makes sense for other notions of uncertainty. We introduce a propositional logic for reasoning about expectation, where the semantics depends on the underlying representation of uncertainty. We give sound and compl ..."
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Expectation is a central notion in probability theory. The notion of expectation also makes sense for other notions of uncertainty. We introduce a propositional logic for reasoning about expectation, where the semantics depends on the underlying representation of uncertainty. We give sound and complete axiomatizations for the logic in the case that the underlying representation is (a) probability, (b) sets of probability measures, (c) belief functions, and (d) possibility measures. We show that this logic is more expressive than the corresponding logic for reasoning about likelihood in the case of sets of probability measures, but equiexpressive in the case of probability, belief, and possibility. Finally, we show that satisfiability for these logics is NPcomplete, no harder than satisfiability for propositional logic.
Two examples of hypergraph edgecoloring, and their connections with other topics in Combinatorics
, 2002
"... This thesis consists of two independent parts, whose common root is the notion of hypergraph edgecoloring. In the first part we deal with a class of nondirected hypergraphs. A particular kind of edgecoloring is defined and studied in Chapter 2. The analysis of such coloring leads to a different t ..."
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Cited by 1 (1 self)
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This thesis consists of two independent parts, whose common root is the notion of hypergraph edgecoloring. In the first part we deal with a class of nondirected hypergraphs. A particular kind of edgecoloring is defined and studied in Chapter 2. The analysis of such coloring leads to a different topic in Chapter 3, namely total weight orders over monomials of fixed degree. We remark that our results on weight orders are not related to Chapter 2. Indeed, the edgecoloring has provided nothing more than a motivation for subsequently focusing on the main topic. Nevertheless, some combinatorial properties of these hypergraphs seemed nice to us. This is the reason why we have dedicated the whole Chapter 2 to them. On the contrary, the second notion of edgecoloring has a fundamental role in Part II. In this context, every edge of a hypergraph consists of a tail (a subset of the vertices) and a head (one vertex). Following the current terminology, edges and vertices are renamed as arcs and nodes respectively, whereas the hypergraph is said to be directed. We define a notion of coloring for the arcs. Our definition is an extension of the existing notion of arccoloring for directed graphs. We enlight some relationships between the incidence structure and the coloring properties. In particular, we analyze the question of minimizing the number of colors. We are led to consider some combinatorial properties of adjacency matrices for hypergraphs. Such matrices generalize the intuitive concept of a wall made of bricks. In our rephrasing, coloring the arcs corresponds to adequately coloring each brick of the wall. 1 Contents I Greedy edgecoloring for a class of hypergraphs, as an ingenuous introduction to total weight orders over monomials of fixed degree 3
COMBINATORIAL PROPERTIES OF FOURIERMOTZKIN ELIMINATION ∗
"... Abstract. FourierMotzkin elimination is a classical method for solving linear inequalities in which one variable is eliminated in each iteration. This method is considered here as a matrix operation and properties of this operation are established. In particular, the focus is on situations where th ..."
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Abstract. FourierMotzkin elimination is a classical method for solving linear inequalities in which one variable is eliminated in each iteration. This method is considered here as a matrix operation and properties of this operation are established. In particular, the focus is on situations where this matrix operation preserves combinatorial matrices (defined here as (0, 1, −1)matrices). Key words. Linear inequalities, FourierMotzkin elimination, Network matrices. AMS subject classifications. 05C50, 15A39, 90C27. 1. Introduction. FourierMotzkin elimination is a computational method that may be seen as a generalization of Gaussian elimination. The method is used for finding one, or even all, solutions to a given linear system of inequalities Ax ≤ b where A ∈ R m,n and b ∈ R m. Here vector inequality is to be interpreted componentwise.
Combinatorial Aspects of Total Weight Orders over Monomials of Fixed Degree
, 2001
"... Abstract. Among all the restrictions of weight orders to the subsets of monomials with a fixed degree, we consider those that yield a total order. Furthermore, we assume that each weight vector consists of an increasing tuple of weights. Every restriction, which is shown to be achieved by some monom ..."
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Abstract. Among all the restrictions of weight orders to the subsets of monomials with a fixed degree, we consider those that yield a total order. Furthermore, we assume that each weight vector consists of an increasing tuple of weights. Every restriction, which is shown to be achieved by some monomial order, is interpreted as a suitable linearization of the poset arising by the intersection of all the weight orders. In the case of three variables, an enumeration is provided. For a higher number of variables, we show a necessary condition for obtaining such restrictions, using deducibility rules applied to homogeneous inequalities. The logarithmic version of this approach is deeply related to classical results of Farkas type, on systems of linear inequalities. Finally, we analyze the linearizations determined by sequences of prime numbers and provide some connections with topics in arithmetic.
THEOREMS OF ALTERNATIVES FOR SYSTEMS OF LINEAR CONSTRAINTS
"... Here we consider systems of linear constraints, consisting of equations or inequalities or both. A feasible solution of a system is a vector which satis es all the constraints in the system. If a feasible solution exists, the system is said to be feasible. The system is said to be infeasible if ther ..."
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Here we consider systems of linear constraints, consisting of equations or inequalities or both. A feasible solution of a system is a vector which satis es all the constraints in the system. If a feasible solution exists, the system is said to be feasible. The system is said to be infeasible if there exists no feasible solution for it. A typical theorem of alternatives shows that corresponding to any given system of linear constraints, system I, there is another associated system of linear constraints, system II, based on the same data, satisfying the property that one of the systems among I, II is feasible i the other is infeasible. These theorems of alternatives are very useful for deriving optimality conditions for many optimization problems. First consider systems consisting of linear equations only. The fundamental inconsistent equation is 0=1 (1) consider the following system of equations x1 + x2 + x3 = 2;x1; x2; x3 =;1: (2)