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17
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.
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.
Linear programming and the simplex method
 Notices of the AMS, 54(3):364–369`. George Spanoudakis, Christos Kloukinas, Khaled Mahbub
, 2007
"... This exposition of linear programming and the simplex method is intended as a companion piece to the article in this issue on the life and work of George B. Dantzig in which the impact and significance of this particular achievement are described. It is now nearly sixty years since Dantzig’s origina ..."
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Cited by 5 (0 self)
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This exposition of linear programming and the simplex method is intended as a companion piece to the article in this issue on the life and work of George B. Dantzig in which the impact and significance of this particular achievement are described. It is now nearly sixty years since Dantzig’s original discovery [3] opened up this whole new area of mathematics. The subject is now widely taught throughout the world at the level of an advanced undergraduate course. The pages to follow are an attempt at a capsule presentation of the material that might be covered in three or four lectures in such a course. Linear Programming The subject of linear programming can be defined quite concisely. It is concerned with the problem of maximizing or minimizing a linear function whose variables are required to satisfy a system of linear constraints, a constraint being a linear equation or inequality. The subject might more appropriately be called linear optimization. Problems of this sort come up in a natural and quite elementary way in many contexts but especially in problems of economic planning. Here are two popular examples. The Diet Problem A list of foods is given and the object is to prescribe amounts of each food so as to provide a meal that has preassigned amounts of various nutrients such as calories, vitamins, proteins, starch, David Gale is professor of mathematics at the University of California, Berkeley. His email address is
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)