49,827 research outputs found
Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems
We consider the problem of counting (stable) equilibriums of an important
family of algebraic differential equations modeling multistable biological
regulatory systems. The problem can be solved, in principle, using real
quantifier elimination algorithms, in particular real root classification
algorithms. However, it is well known that they can handle only very small
cases due to the enormous computing time requirements. In this paper, we
present a special algorithm which is much more efficient than the general
methods. Its efficiency comes from the exploitation of certain interesting
structures of the family of differential equations.Comment: 24 pages, 5 algorithms, 10 figure
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
Polynomial Interrupt Timed Automata
Interrupt Timed Automata (ITA) form a subclass of stopwatch automata where
reachability and some variants of timed model checking are decidable even in
presence of parameters. They are well suited to model and analyze real-time
operating systems. Here we extend ITA with polynomial guards and updates,
leading to the class of polynomial ITA (PolITA). We prove the decidability of
the reachability and model checking of a timed version of CTL by an adaptation
of the cylindrical decomposition method for the first-order theory of reals.
Compared to previous approaches, our procedure handles parameters and clocks in
a unified way. Moreover, we show that PolITA are incomparable with stopwatch
automata. Finally additional features are introduced while preserving
decidability
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