Theory of linear and integer programming
Theory of linear and integer programming
An algorithm for linear programming which requires O(((m+n)n2+(m+n)1.5n)L) arithmetic operations
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
The complexity of linear problems in fields
Journal of Symbolic Computation
Real quantifier elimination is doubly exponential
Journal of Symbolic Computation
Computational difficulties of bilevel linear programming
Operations Research
Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
The computational complexity of multi-level linear programs
Annals of Operations Research - Special issue on hierarchical optimization
On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM (JACM)
REDLOG: computer algebra meets computer logic
ACM SIGSAM Bulletin
Journal of the ACM (JACM)
Composite model-checking: verification with type-specific symbolic representations
ACM Transactions on Software Engineering and Methodology (TOSEM)
A New Approach for Automatic Theorem Proving in Real Geometry
Journal of Automated Reasoning
A new polynomial-time algorithm for linear programming
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
Efficient solving of quantified inequality constraints over the real numbers
ACM Transactions on Computational Logic (TOCL)
On a decision procedure for quantified linear programs
Annals of Mathematics and Artificial Intelligence
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In this paper, we explore the computational complexity of the conjunctive fragment of the first-order theory of linear arithmetic. Quantified propositional formulas of linear inequalities with (k-1) quantifier alternations are log-space complete in @S"k^P or @P"k^P depending on the initial quantifier. We show that when we restrict ourselves to quantified conjunctions of linear inequalities, i.e., quantified linear systems, the complexity classes collapse to polynomial time. In other words, the presence of universal quantifiers does not alter the complexity of the linear programming problem, which is known to be in P. Our result reinforces the importance of sentence formats from the perspective of computational complexity.