3,408 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
On Generalized Records and Spatial Conjunction in Role Logic
We have previously introduced role logic as a notation for describing
properties of relational structures in shape analysis, databases and knowledge
bases. A natural fragment of role logic corresponds to two-variable logic with
counting and is therefore decidable. We show how to use role logic to describe
open and closed records, as well the dual of records, inverse records. We
observe that the spatial conjunction operation of separation logic naturally
models record concatenation. Moreover, we show how to eliminate the spatial
conjunction of formulas of quantifier depth one in first-order logic with
counting. As a result, allowing spatial conjunction of formulas of quantifier
depth one preserves the decidability of two-variable logic with counting. This
result applies to two-variable role logic fragment as well. The resulting logic
smoothly integrates type system and predicate calculus notation and can be
viewed as a natural generalization of the notation for constraints arising in
role analysis and similar shape analysis approaches.Comment: 30 pages. A version appears in SAS 200
A Survey Of State-Feedback Simultaneous Stabilization Techniques
This paper surveys the control theory literature having to do with the simultaneous stabilization of countably finite sets of systems in the state space domain. Design methods based upon control parameterization linear equation solution and linear matrix inequalities are discussed. The roles of nonlinear programming and convex programming techniques are included as is a brief description of the applicability of software based quantifier elimination techniques
On Termination of Integer Linear Loops
A fundamental problem in program verification concerns the termination of
simple linear loops of the form x := u ; while Bx >= b do {x := Ax + a} where x
is a vector of variables, u, a, and c are integer vectors, and A and B are
integer matrices. Assuming the matrix A is diagonalisable, we give a decision
procedure for the problem of whether, for all initial integer vectors u, such a
loop terminates. The correctness of our algorithm relies on sophisticated tools
from algebraic and analytic number theory, Diophantine geometry, and real
algebraic geometry. To the best of our knowledge, this is the first substantial
advance on a 10-year-old open problem of Tiwari (2004) and Braverman (2006).Comment: Accepted to SODA1
Quantifier elimination in quasianalytic structures via non-standard analysis
The paper is a continuation of our earlier article where we developed a
theory of active and non-active infinitesimals and intended to establish
quantifier elimination in quasianalytic structures. That article, however, did
not attain full generality, which refers to one of its results, namely the
theorem on an active infinitesimal, playing an essential role in our
non-standard analysis. The general case was covered in our subsequent preprint,
which constitutes a basis for the approach presented here. We also provide a
quasianalytic exposition of the results concerning rectilinearization of terms
and of definable functions from our earlier research. It will be used to
demonstrate a quasianalytic structure corresponding to a Denjoy-Carleman class
which, unlike the classical analytic structure, does not admit quantifier
elimination in the language of restricted quasianalytic functions augmented
merely by the reciprocal function. More precisely, we construct a plane
definable curve, which indicates both that the classical theorem by J. Denef
and L. van den Dries as well as \L{}ojasiewicz's theorem that every subanalytic
curve is semianalytic are no longer true for quasianalytic structures. Besides
rectilinearization of terms, our construction makes use of some theorems on
power substitution for Denjoy-Carleman classes and on non-extendability of
quasianalytic function germs. The last result relies on Grothendieck's
factorization and open mapping theorems for (LF)-spaces. Note finally that this
paper comprises our earlier preprints on the subject from May 2012.Comment: Final version, 36 pages. arXiv admin note: substantial text overlap
with arXiv:1310.130
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