The Complexity of Elementary Problems in Archimedean Ordered Groups
EUROCAL '85 Research Contributions from the European Conference on Computer Algebra-Volume 2
Weitere zum Erfüllungsproblem polynomial äquivalente kombinatorische Aufgaben
Komplexität von Entscheidungsproblemen, Ein Seminar
The complexity of elementary algebra and geometry
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Precise bounds for presburger arithmetic and the reals with addition: Preliminary report
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
A bibliography of quantifier elimination for real closed fields
Journal of Symbolic Computation
Quantifier elimination for formulas constrained by quadratic equations
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
Quantifier elimination for real algebra—the cubic case
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
Computational geometry: a retrospective
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Uniform quantifier elimination and constraint query processing
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Complexity and uniformity of elimination in Presburger arithmetic
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Guarded expressions in practice
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Approaches to parallel quantifier elimination
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
Mixed real-integer linear quantifier elimination
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Journal of the ACM (JACM)
Deciding linear-trigonometric problems
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Testing Positiveness of Polynomials
Journal of Automated Reasoning
A New Approach for Automatic Theorem Proving in Real Geometry
Journal of Automated Reasoning
Solving Geometric Problems with Real Quantifier Elimination
ADG '98 Proceedings of the Second International Workshop on Automated Deduction in Geometry
Continuous First-Order Constraint Satisfaction
AISC '02/Calculemus '02 Proceedings of the Joint International Conferences on Artificial Intelligence, Automated Reasoning, and Symbolic Computation
A generic projection operator for partial cylindrical algebraic decomposition
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Computer algebra handbook
Non-linear loop invariant generation using Gröbner bases
Proceedings of the 31st ACM SIGPLAN-SIGACT symposium on Principles of programming languages
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
Convergent approximate solving of first-order constraints by approximate quantifiers
ACM Transactions on Computational Logic (TOCL)
Model-Theoretic Methods in Combined Constraint Satisfiability
Journal of Automated Reasoning
Some formal tools for analyzing quantum automata
Theoretical Computer Science - In honour of Professor Christian Choffrut on the occasion of his 60th birthday
Interpolation for data structures
Proceedings of the 14th ACM SIGSOFT international symposium on Foundations of software engineering
Efficient solving of quantified inequality constraints over the real numbers
ACM Transactions on Computational Logic (TOCL)
Multiple object semilinear motion planning
Journal of Symbolic Computation
A Decision Procedure for Linear "Big O" Equations
Journal of Automated Reasoning
The complexity of quantifier elimination and cylindrical algebraic decomposition
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Proof Synthesis and Reflection for Linear Arithmetic
Journal of Automated Reasoning
Parametric Linear Arithmetic over Ordered Fields in Isabelle/HOL
Proceedings of the 9th AISC international conference, the 15th Calculemas symposium, and the 7th international MKM conference on Intelligent Computer Mathematics
IJCAR '08 Proceedings of the 4th international joint conference on Automated Reasoning
A Folding Algorithm for Eliminating Existential Variables from Constraint Logic Programs
ICLP '08 Proceedings of the 24th International Conference on Logic Programming
Quantifier Elimination via Functional Composition
CAV '09 Proceedings of the 21st International Conference on Computer Aided Verification
Proceedings of the 2009 conference on Symbolic numeric computation
Effective Quantifier Elimination for Presburger Arithmetic with Infinity
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
A Folding Rule for Eliminating Existential Variables from Constraint Logic Programs
Fundamenta Informaticae - Advances in Computational Logic (CIL C08)
SyNRAC: a Maple-package for solving real algebraic constraints
ICCS'03 Proceedings of the 1st international conference on Computational science: PartI
Black-box/white-box simplification and applications to quantifier elimination
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Parametric quantified SAT solving
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Journal of Automated Reasoning
Supporting global numerical optimization of rational functions by generic symbolic convexity tests
CASC'10 Proceedings of the 12th international conference on Computer algebra in scientific computing
CASC'10 Proceedings of the 12th international conference on Computer algebra in scientific computing
A constraint satisfaction framework for executing perceptions and actions in diagrammatic reasoning
Journal of Artificial Intelligence Research
Solution formulas for cubic equations without or with constraints
Journal of Symbolic Computation
Verification and synthesis using real quantifier elimination
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Automatic verification of the adequacy of models for families of geometric objects
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
Virtual substitution for SMT-solving
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Differential dynamic logics: automated theorem proving for hybrid systems
Differential dynamic logics: automated theorem proving for hybrid systems
Quantifier elimination for statistical problems
UAI'99 Proceedings of the Fifteenth conference on Uncertainty in artificial intelligence
ICCS'05 Proceedings of the 5th international conference on Computational Science - Volume Part III
A logic for reasoning about evidence
UAI'03 Proceedings of the Nineteenth conference on Uncertainty in Artificial Intelligence
Verifying mixed real-integer quantifier elimination
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
An algebraic approach for the unsatisfiability of nonlinear constraints
CSL'05 Proceedings of the 19th international conference on Computer Science Logic
Quantifier elimination for constraint logic programming
CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
Efficient preprocessing methods for quantifier elimination
CASC'06 Proceedings of the 9th international conference on Computer Algebra in Scientific Computing
New domains for applied quantifier elimination
CASC'06 Proceedings of the 9th international conference on Computer Algebra in Scientific Computing
ICCS'06 Proceedings of the 6th international conference on Computational Science - Volume Part II
On hierarchical reasoning in combinations of theories
IJCAR'10 Proceedings of the 5th international conference on Automated Reasoning
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Mutual exclusion by interpolation
FLOPS'12 Proceedings of the 11th international conference on Functional and Logic Programming
SMT-RAT: an SMT-compliant nonlinear real arithmetic toolbox
SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
Interpolation-Based height analysis for improving a recurrence solver
FOPARA'11 Proceedings of the Second international conference on Foundational and Practical Aspects of Resource Analysis
A Folding Rule for Eliminating Existential Variables from Constraint Logic Programs
Fundamenta Informaticae - Advances in Computational Logic (CIL C08)
Weak integer quantifier elimination beyond the linear case
CASC'07 Proceedings of the 10th international conference on Computer Algebra in Scientific Computing
Robust stability for parametric linear ODEs
CASC'07 Proceedings of the 10th international conference on Computer Algebra in Scientific Computing
Computing hopf bifurcations in chemical reaction networks using reaction coordinates
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Theoretical Computer Science
A symbiosis of interval constraint propagation and cylindrical algebraic decomposition
CADE'13 Proceedings of the 24th international conference on Automated Deduction
Note: On the complexity of quantified linear systems
Theoretical Computer Science
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We consider linear problems in fields, ordered fields, discretely valued fields (with finite residue field or residue field of characteristic zero) and fields with finitely many independent orderings and discrete valuations. Most of the fields considered will be of characteristic zero. Formally, linear statements about these structures (with parameters) are given by formulas of the respective first-order language, in which all bound variables occur only linearly. We study symbolic algorithms (linear elimination procedures) that reduce linear formulas to linear formulas of a very simple form, i.e. quantifier-free linear formulas, and algorithms (linear decision procedures) that decide whether a given linear sentence holds in all structures of the given class. For all classes of fields considered, we find linear elimination procedures that run in double exponential space and time. As a consequence, we can show that for fields (with one or several discrete valuations), linear statements can be transferred from characteristic zero to prime characteristic p, provided p is double exponential in the length of the statement. (For similar bounds in the non-linear case, see Brown, 1978.) We find corresponding linear decision procedures in the Berman complexity classes @?c@?NSTA(*,2^c^n,dn) for d = 1, 2. In particular, all hese procedures run in exponential space. The technique employed is quantifier elimination via Skolem terms based on Ferrante & Rackoff (1975). Using ideas of Fischer & Rabin (1974), Berman (1977), Furer (1982), we establish lower bounds for these problems showing that our upper bounds are essentially tight. For linear formulas with a bounded number of quantifiers all our algorithms run in polynomial time. For linear formulas of bounded quantifier alternation most of the algorithms run in time 2^O^(^n^^^k^) for fixed k.