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16
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 (19 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.
Parametric Linear and Quadratic Optimization by Elimination
 UNIVERSITÄT PASSAU
, 1994
"... We propose a new elimination method for linear and quadratic optimization involving parametric coefficients. In comparison to the classical FourierMotzkin method that is of doubly exponential worstcase complexity our method is singly exponential in the worst case. Moreover it applies also to the ..."
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Cited by 22 (7 self)
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We propose a new elimination method for linear and quadratic optimization involving parametric coefficients. In comparison to the classical FourierMotzkin method that is of doubly exponential worstcase complexity our method is singly exponential in the worst case. Moreover it applies also to the minimization of a quadratic objective functions without convexity hypothesis under linear constraints, and to objective functions with arbitrary parametric coefficients. For problems with additive parameters the method is worstcase optimal. Examples computed in a REDUCEimplementation confirm the superiority of the method over FourierMotzkin and its applicability to problems of interesting size.
Simulation and Optimization by Quantifier Elimination
 Journal of Symbolic Computation
, 1996
"... Quantifier elimination in real algebra is a fascinating problem that stimulated methods from logic, algebra, real algebraic geometry, analysis, complexity theory and other tools from computer science in an effort to obtain `efficient' solutions (see (Renegar 1992, Collins & Hong 1991) and the refere ..."
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Cited by 20 (2 self)
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Quantifier elimination in real algebra is a fascinating problem that stimulated methods from logic, algebra, real algebraic geometry, analysis, complexity theory and other tools from computer science in an effort to obtain `efficient' solutions (see (Renegar 1992, Collins & Hong 1991) and the references given there). Up to quite recently the problems that could be solved by quantifier elimination software were exclusively academic, designed to test the strength and weaknesses of certain algorithms and implementations in comparison to others. Meanwhile some researchers (see e.g. (Liska & Steinberg 1993, Liska & Steinberg 1994, Dorato et al. 1995)) have pointed out that some highly nontrivial problems in various branches of applied mathematics can be phrased as quantifier elimination problems, that are at the verge of automatic solvability for today's elimination methods. The present paper deals with a very restricted quantifier elimination problem, viz. the elimina...
Quantifierelimination for the firstorder theory of boolean algebras with linear cardinality constraints
 In Proc. Advances in Databases and Information Systems (ADBIS’04), volume 3255 of LNCS
, 2004
"... Abstract. We present for the firstorder theory of atomic Boolean algebras of sets with linear cardinality constraints a quantifier elimination algorithm. In the case of atomic Boolean algebras of sets, this is a new generalization of Boole’s wellknown variable elimination method for conjunctions o ..."
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Cited by 9 (0 self)
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Abstract. We present for the firstorder theory of atomic Boolean algebras of sets with linear cardinality constraints a quantifier elimination algorithm. In the case of atomic Boolean algebras of sets, this is a new generalization of Boole’s wellknown variable elimination method for conjunctions of Boolean equality constraints. We also explain the connection of this new logical result with the evaluation of relational calculus queries on constraint databases that contain Boolean linear cardinality constraints. 1
Verifying mixed realinteger quantifier elimination
 IJCAR 2006, LNCS 4130
, 2006
"... We present a formally verified quantifier elimination procedure for the first order theory over linear mixed realinteger arithmetics in higherorder logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for lin ..."
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Cited by 8 (5 self)
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We present a formally verified quantifier elimination procedure for the first order theory over linear mixed realinteger arithmetics in higherorder logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for linear real arithmetics.
Proof synthesis and reflection for linear arithmetic. Submitted
, 2006
"... This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in ta ..."
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Cited by 6 (5 self)
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This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in tactic style, i.e. by a proofproducing functional program, and once by reflection, i.e. by computations inside the logic rather than in the metalanguage. Both formalizations are highly generic because they make only minimal assumptions w.r.t. the underlying logical system and theorem prover. An implementation in Isabelle/HOL shows that the reflective approach is between one and two orders of magnitude faster. 1
ON THE DEVELOPMENT OF OPTIMIZATION THEORY
 THE AMERICAN MATHEMATICAL MONTHLY, 87 (1980), PP. 527{542.
, 1980
"... ..."
Polyhedral Approaches to Mixed Integer Linear Programming
, 2008
"... This survey presents tools from polyhedral theory that are used in integer programming. It applies them to the study of valid inequalities for mixed integer linear sets, such as Gomory’s mixed integer cuts. ..."
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Cited by 5 (1 self)
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This survey presents tools from polyhedral theory that are used in integer programming. It applies them to the study of valid inequalities for mixed integer linear sets, such as Gomory’s mixed integer cuts.
Extended Formulations in Combinatorial Optimization
, 2009
"... This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By ”perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequ ..."
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Cited by 3 (0 self)
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This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By ”perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called ”compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, MinkowskiWeyl’s theorem, Balas ’ theorem for the union of polyhedra, Yannakakis ’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans ’ result on the size of an extended formulation for the permutahedron, and the FaenzaKaibel extended formulation for orbitopes. Supported by the Progetto di Eccellenza 20082009 of the Fondazione Cassa di Risparmio di Padova e
Numerical Analysis in the Twentieth Century
 in Numerical Analysis: Historical Developments in the 20th Century, C. Brezinski e L. Wuytack, Editors, North–Holland
, 2001
"... This paper attracted much attention while a similar result obtained by William Karush in his Master's Thesis in 1939 [154] under the supervision of Lawrence M. Graves at the University of Chicago and by Fritz John (19101995) in 1948 [147] were almost totally ignored (John's paper was even rejected) ..."
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Cited by 3 (0 self)
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This paper attracted much attention while a similar result obtained by William Karush in his Master's Thesis in 1939 [154] under the supervision of Lawrence M. Graves at the University of Chicago and by Fritz John (19101995) in 1948 [147] were almost totally ignored (John's paper was even rejected)