60 research outputs found
Program Verification in the presence of complex numbers, functions with branch cuts etc
In considering the reliability of numerical programs, it is normal to "limit
our study to the semantics dealing with numerical precision" (Martel, 2005). On
the other hand, there is a great deal of work on the reliability of programs
that essentially ignores the numerics. The thesis of this paper is that there
is a class of problems that fall between these two, which could be described as
"does the low-level arithmetic implement the high-level mathematics". Many of
these problems arise because mathematics, particularly the mathematics of the
complex numbers, is more difficult than expected: for example the complex
function log is not continuous, writing down a program to compute an inverse
function is more complicated than just solving an equation, and many algebraic
simplification rules are not universally valid.
The good news is that these problems are theoretically capable of being
solved, and are practically close to being solved, but not yet solved, in
several real-world examples. However, there is still a long way to go before
implementations match the theoretical possibilities
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
ACL2 Proofs of Nonlinear Inequalities with Imandra
We present a proof-producing integration of ACL2 and Imandra for proving
nonlinear inequalities. This leverages a new Imandra interface exposing its
nonlinear decision procedures. The reasoning takes place over the reals, but
the proofs produced are valid over the rationals and may be run in both ACL2
and ACL2(r). The ACL2 proofs Imandra constructs are extracted from
Positivstellensatz refutations, a real algebraic analogue of the
Nullstellensatz, and are found using convex optimization.Comment: In Proceedings ACL2-2023, arXiv:2311.0837
A Lazy SMT-Solver for a Non-Linear Subset of Real Algebra
There are several methods for the synthesis and analysis of hybrid
systems that require efficient algorithms and tools for satisfiability
checking. For analysis, e.g., bounded model checking describes
counterexamples of a fixed length by logical formulas, whose
satisfiability corresponds to the existence of such a counterexample.
As an example for parameter synthesis, we can state the correctness of
a parameterized system by a logical formula; the solution set of
the formula gives us possible safe instances of the parameters.
For discrete systems, which can be described by propositional logic
formulas, SAT-solvers can be used for the satisfiability checks. For
hybrid systems, having mixed discrete-continuous behavior, SMT-solvers
are needed. SMT-solving extends SAT with theories, and has its main
focus on linear arithmetic, which is sufficient to handle, e.g.,
linear hybrid systems. However, there are only few solvers for
more expressive but still decidable logics like the
first-order theory of the reals with addition and multiplication --
real algebra. Since the synthesis and analysis of non-linear
hybrid systems requires such a powerful logic, we need efficient
SMT-solvers for real algebra. Our goal is to develop such an
SMT-solver for the real algebra, which is both complete and
efficient
Determinantal sets, singularities and application to optimal control in medical imagery
Control theory has recently been involved in the field of nuclear magnetic
resonance imagery. The goal is to control the magnetic field optimally in order
to improve the contrast between two biological matters on the pictures.
Geometric optimal control leads us here to analyze mero-morphic vector fields
depending upon physical parameters , and having their singularities defined by
a deter-minantal variety. The involved matrix has polynomial entries with
respect to both the state variables and the parameters. Taking into account the
physical constraints of the problem, one needs to classify, with respect to the
parameters, the number of real singularities lying in some prescribed
semi-algebraic set. We develop a dedicated algorithm for real root
classification of the singularities of the rank defects of a polynomial matrix,
cut with a given semi-algebraic set. The algorithm works under some genericity
assumptions which are easy to check. These assumptions are not so restrictive
and are satisfied in the aforementioned application. As more general strategies
for real root classification do, our algorithm needs to compute the critical
loci of some maps, intersections with the boundary of the semi-algebraic
domain, etc. In order to compute these objects, the determinantal structure is
exploited through a stratifi-cation by the rank of the polynomial matrix. This
speeds up the computations by a factor 100. Furthermore, our implementation is
able to solve the application in medical imagery, which was out of reach of
more general algorithms for real root classification. For instance,
computational results show that the contrast problem where one of the matters
is water is partitioned into three distinct classes
Synthesizing Switching Controllers for Hybrid Systems by Continuous Invariant Generation
We extend a template-based approach for synthesizing switching controllers
for semi-algebraic hybrid systems, in which all expressions are polynomials.
This is achieved by combining a QE (quantifier elimination)-based method for
generating continuous invariants with a qualitative approach for predefining
templates. Our synthesis method is relatively complete with regard to a given
family of predefined templates. Using qualitative analysis, we discuss
heuristics to reduce the numbers of parameters appearing in the templates. To
avoid too much human interaction in choosing templates as well as the high
computational complexity caused by QE, we further investigate applications of
the SOS (sum-of-squares) relaxation approach and the template polyhedra
approach in continuous invariant generation, which are both well supported by
efficient numerical solvers
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