3,293 research outputs found
A lower bound for the complexity of linear optimization from a quantifier-elimination point of view
We discuss the impact of data structures in quantifier elimination.
We analyze the arithmetic complexity of the feasibility problem in
linear optimization theory as a quantifier-elimination problem. For
the case of polyhedra defined by halfspaces in we prove
that, if dense representation is used to code polynomials, any
quantifier-free formula expressing the set of parameters describing
nonempty polyhedra has size
Bounds on the Automata Size for Presburger Arithmetic
Automata provide a decision procedure for Presburger arithmetic. However,
until now only crude lower and upper bounds were known on the sizes of the
automata produced by this approach. In this paper, we prove an upper bound on
the the number of states of the minimal deterministic automaton for a
Presburger arithmetic formula. This bound depends on the length of the formula
and the quantifiers occurring in the formula. The upper bound is established by
comparing the automata for Presburger arithmetic formulas with the formulas
produced by a quantifier elimination method. We also show that our bound is
tight, even for nondeterministic automata. Moreover, we provide optimal
automata constructions for linear equations and inequations
Binary Reachability of Timed Pushdown Automata via Quantifier Elimination and Cyclic Order Atoms
We study an expressive model of timed pushdown automata extended with modular and fractional clock constraints. We show that the binary reachability relation is effectively expressible in hybrid linear arithmetic with a rational and an integer sort. This subsumes analogous expressibility results previously known for finite and pushdown timed automata with untimed stack. As key technical tools, we use quantifier elimination for a fragment of hybrid linear arithmetic and for cyclic order atoms, and a reduction to register pushdown automata over cyclic order atoms
FMplex: A Novel Method for Solving Linear Real Arithmetic Problems
In this paper we introduce a novel quantifier elimination method for
conjunctions of linear real arithmetic constraints. Our algorithm is based on
the Fourier-Motzkin variable elimination procedure, but by case splitting we
are able to reduce the worst-case complexity from doubly to singly exponential.
The adaption of the procedure for SMT solving has strong correspondence to the
simplex algorithm, therefore we name it FMplex. Besides the theoretical
foundations, we provide an experimental evaluation in the context of SMT
solving
An Instantiation-Based Approach for Solving Quantified Linear Arithmetic
This paper presents a framework to derive instantiation-based decision
procedures for satisfiability of quantified formulas in first-order theories,
including its correctness, implementation, and evaluation. Using this framework
we derive decision procedures for linear real arithmetic (LRA) and linear
integer arithmetic (LIA) formulas with one quantifier alternation. Our
procedure can be integrated into the solving architecture used by typical SMT
solvers. Experimental results on standardized benchmarks from model checking,
static analysis, and synthesis show that our implementation of the procedure in
the SMT solver CVC4 outperforms existing tools for quantified linear
arithmetic
A Survey of Satisfiability Modulo Theory
Satisfiability modulo theory (SMT) consists in testing the satisfiability of
first-order formulas over linear integer or real arithmetic, or other theories.
In this survey, we explain the combination of propositional satisfiability and
decision procedures for conjunctions known as DPLL(T), and the alternative
"natural domain" approaches. We also cover quantifiers, Craig interpolants,
polynomial arithmetic, and how SMT solvers are used in automated software
analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest,
Romania. 201
Adapting Real Quantifier Elimination Methods for Conflict Set Computation
The satisfiability problem in real closed fields is decidable. In the context
of satisfiability modulo theories, the problem restricted to conjunctive sets
of literals, that is, sets of polynomial constraints, is of particular
importance. One of the central problems is the computation of good explanations
of the unsatisfiability of such sets, i.e.\ obtaining a small subset of the
input constraints whose conjunction is already unsatisfiable. We adapt two
commonly used real quantifier elimination methods, cylindrical algebraic
decomposition and virtual substitution, to provide such conflict sets and
demonstrate the performance of our method in practice
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