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Page 791 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
The bibliography at the end applies to Part II only.” 2602b:03071 03C10 68w30 Weispfenning, Volker (D-PASSMI; Passau) Mixed real-integer linear quantifier elimination.  ...  “Applications include a characterization of 7-definable subsets of the real line, and the modeling of parametric mixed integer linear optimization, of continuous phenomena with periodicity, and the simulation  ... 

A Quantifier Elimination Algorithm for Linear Real Arithmetic [article]

David Monniaux
2008 arXiv   pre-print
We propose a new quantifier elimination algorithm for the theory of linear real arithmetic.  ...  The quantifier elimination algorithm presented in the paper is compared, on examples arising from program analysis problems, to several other implementations, all of which cannot solve some of the examples  ...  Regarding the mixed integer / real problems, the Lira tool implements quantifier elimination using a weak form of Büchi automata matching the b-ary expression of the integers or reals, where b is an arbitrary  ... 
arXiv:0803.1575v2 fatcat:luxi4q7wifeibb2z62ip2fq4ze

A Quantifier Elimination Algorithm for Linear Real Arithmetic [chapter]

David Monniaux
2008 Lecture Notes in Computer Science  
We propose a new quantifier elimination algorithm for the theory of linear real arithmetic.  ...  The quantifier elimination algorithm presented in the paper is compared, on examples arising from program analysis problems and on random examples, to several other implementations, all of which cannot  ...  Regarding the mixed integer / real problems, the Lira tool implements quantifier elimination using a weak form of Büchi automata matching the b-ary expression of the integers or reals, where b is an arbitrary  ... 
doi:10.1007/978-3-540-89439-1_18 fatcat:y2eqvag35fhojc77awt3qgmyxm

First-Order Mixed Integer Linear Programming [article]

Geoffrey Gordon, Sue Ann Hong, Miroslav Dudik
2012 arXiv   pre-print
Mixed integer linear programming (MILP) is a powerful representation often used to formulate decision-making problems under uncertainty.  ...  We establish formal methods for reasoning about first order programs, including a sound and complete lifted inference procedure for integer first order programs.  ...  For our purposes, a particularly convenient solver is based on Gomory cuts for integer linear programs [Gomory, 1958] , along with their generalization to mixed-integer programs.  ... 
arXiv:1205.2644v1 fatcat:khd4vkwffze6poeaacfdu3mr6i

Complexity and uniformity of elimination in Presburger arithmetic

Volker Weispfenning
1997 Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97  
The results are inspired by experimental implementations of bounded quantifier elimination that have solved non-trivial application problems e.g. in paramet ric integer programming.  ...  By weakening the concept of quantifier elimination slightly to bounded quantifier elimination, we show, however, that the upper and lower bound for quantifier elimination in PA can be lowered by exactly  ...  [WX95] has solved among others (not totally unimodular) mixed integer programming problems with up to 11 real and 4 integer variables and 16 constraints in less than 2 minutes on a SUN Spare-10.  ... 
doi:10.1145/258726.258746 fatcat:zgrvwkcnlvbehk2jpeepisfuei

An Online Proof-Producing Decision Procedure for Mixed-Integer Linear Arithmetic [chapter]

Sergey Berezin, Vijay Ganesh, David L. Dill
2003 Lecture Notes in Computer Science  
This work augments CVC with a decision procedure for the theory of mixed integer linear arithmetic based on the Omega-test [Pug91] extended to be online and proof producing.  ...  In practical examples, however, arithmetic constraints are often mixed with constraints from other theories like the theory of arrays, Boolean satisfiability (SAT), bit-vectors, etc.  ...  The choice of Omega-test over other algorithms for solving mixed-integer linear arithmetic problems (simplex, interior point method [BT97] , earlier versions of Fourier-Motzkin elimination [Wil76] ,  ... 
doi:10.1007/3-540-36577-x_38 fatcat:p6ye4ep2pjbtzpu4ffb77hnizi

Ordered Sets in the Calculus of Data Structures [chapter]

Viktor Kuncak, Ruzica Piskac, Philippe Suter
2010 Lecture Notes in Computer Science  
We describe a new member of the Calculus of Data Structures: a quantifier-free fragment that supports 1) boolean algebra of finite and infinite sets of real numbers, 2) linear arithmetic over real numbers  ...  , 3) formulas that can restrict chosen set or element variables to range over integers (providing, among others, the power of mixed integer arithmetic and sets of integers), 4) the cardinality operators  ...  We thank Robin Steiger and Utkarsh Upadhyay who have, in the meantime, implemented a decision procedure for finite sets of integers with the cardinality operator and made it more efficient.  ... 
doi:10.1007/978-3-642-15205-4_5 fatcat:ymd6d2zy5rb7jpqeucoyrnhoha

A Survey of Satisfiability Modulo Theory [article]

David Monniaux
2016 arXiv   pre-print
Satisfiability modulo theory (SMT) consists in testing the satisfiability of first-order formulas over linear integer or real arithmetic, or other theories.  ...  We also cover quantifiers, Craig interpolants, polynomial arithmetic, and how SMT solvers are used in automated software analysis.  ...  In most cases (but not always), the solutions of the formula to be tested for satisfiability correspond linear real arithmetic LRA linear integer arithmetic LIA linear mixed integer and real arithmetic  ... 
arXiv:1606.04786v1 fatcat:q3rzmjpndvdghfekeqe6vgkbq4

Satisfiability checking and symbolic computation

E. Ábrahám, P. Fontaine, S. Forrest, A. Griggio, D. Kroening, W. M. Seiler, T. Sturm, J. Abbott, B. Becker, A. M. Bigatti, M. Brain, B. Buchberger (+3 others)
2017 ACM Communications in Computer Algebra  
non-linear real and integer arithmetic is still in its infancy.  ...  There exist techniques for equality logic with uninterpreted functions, array theory, bit-vector arithmetic and quantifier-free linear real and integer arithmetic; but the development for quantifier-free  ... 
doi:10.1145/3055282.3055285 fatcat:57th32bctrdwvmmds644xchqsy

Complexity of linear relaxations in integer programming [article]

Gennadiy Averkov, Matthias Schymura
2020 arXiv   pre-print
Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding rc(X) and its variant rc_Q(X), restricting the descriptions of X to  ...  This parameter was introduced by Kaibel Weltge (2015) and captures the complexity of linear descriptions of X without using auxiliary variables.  ...  We first develop quantifier elimination for a special mixed-integer version of quantified boolean combinations of linear inequalities.  ... 
arXiv:2003.07817v1 fatcat:iyjooahm4netxerxmi6vyrw3rq

Page 3785 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
The decision procedure is explicit and implementable; it is based on mixed real-integer linear elimination, the symbolic test point method, elementary analysis, and Lindemann’s theorem.  ...  The problems are formalized by formulas with arbi- trary quantified linear variables and a block of quantifiers with respect to mixed linear-transcendental variables.  ... 

A Survey of Satisfiability Modulo Theory [chapter]

David Monniaux
2016 Lecture Notes in Computer Science  
Satisfiability modulo theory (SMT) consists in testing the satisfiability of first-order formulas over linear integer or real arithmetic, or other theories.  ...  We also cover quantifiers, Craig interpolants, polynomial arithmetic, and how SMT solvers are used in automated software analysis.  ...  In most cases (but not always), the solutions of the formula to be tested for satisfiability correspond linear real arithmetic LRA linear integer arithmetic LIA linear mixed integer and real arithmetic  ... 
doi:10.1007/978-3-319-45641-6_26 fatcat:iqkvvrxgrnd6xgpqo3kenx6l7u

Bit-Vector Interpolation and Quantifier Elimination by Lazy Reduction

Peter Backeman, Philipp Rummer, Aleksandar Zeljic
2018 2018 Formal Methods in Computer Aided Design (FMCAD)  
We present a new approach to bitvector interpolation, as well as bit-vector quantifier elimination (QE), that works by lazy translation of bit-vector constraints to unbounded arithmetic.  ...  The lazy encoding is complemented with a set of native proof rules for bit-vector equations and non-linear (polynomial) constraints, this way minimising the number of cases a solver has to consider.  ...  real arithmetic [1] , non-linear real arithmetic [9] , Presburger arithmetic [10] , [4] , [11] , and arrays [12] , [13] , [14] .  ... 
doi:10.23919/fmcad.2018.8603023 dblp:conf/fmcad/BackemanRZ18 fatcat:rwqtgkmrtrdmvlvsra4ewf4f4e

On Decidability Within the Arithmetic of Addition and Divisibility [chapter]

Marius Bozga, Radu Iosif
2005 Lecture Notes in Computer Science  
For this form, called L (1) | in the paper, we show the existence of a quantifier elimination procedure which always leads to formulas of Presburger arithmetic.  ...  The L (1) | , ∃L ( * ) | fragments were inspired by a real application in the field of program verification.  ...  The block quantifier elimination can be now performed along the same lines of the universal mixed case, discussed in the previous.  ... 
doi:10.1007/978-3-540-31982-5_27 fatcat:psi4hkrktvcprf7edpkndgfmmq

The Elimination of Integer Variables

H. P. Williams
1992 Journal of the Operational Research Society  
If all x; are members of R we have a Linear Programme (LP). If all are members of Z we have a Pure Integer Programme (PIP). Otherwise, we have a Mixed Integer Programme (MIP).  ...  For a Linear Programming (LP) model it is always possible to eliminate variables to produce another LP model.  ... 
doi:10.1057/jors.1992.65 fatcat:dkogfv5tivh5vftn66vazk23qq
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