63 research outputs found
Finding polynomial loop invariants for probabilistic programs
Quantitative loop invariants are an essential element in the verification of
probabilistic programs. Recently, multivariate Lagrange interpolation has been
applied to synthesizing polynomial invariants. In this paper, we propose an
alternative approach. First, we fix a polynomial template as a candidate of a
loop invariant. Using Stengle's Positivstellensatz and a transformation to a
sum-of-squares problem, we find sufficient conditions on the coefficients.
Then, we solve a semidefinite programming feasibility problem to synthesize the
loop invariants. If the semidefinite program is unfeasible, we backtrack after
increasing the degree of the template. Our approach is semi-complete in the
sense that it will always lead us to a feasible solution if one exists and
numerical errors are small. Experimental results show the efficiency of our
approach.Comment: accompanies an ATVA 2017 submissio
Sharper and Simpler Nonlinear Interpolants for Program Verification
Interpolation of jointly infeasible predicates plays important roles in
various program verification techniques such as invariant synthesis and CEGAR.
Intrigued by the recent result by Dai et al.\ that combines real algebraic
geometry and SDP optimization in synthesis of polynomial interpolants, the
current paper contributes its enhancement that yields sharper and simpler
interpolants. The enhancement is made possible by: theoretical observations in
real algebraic geometry; and our continued fraction-based algorithm that rounds
off (potentially erroneous) numerical solutions of SDP solvers. Experiment
results support our tool's effectiveness; we also demonstrate the benefit of
sharp and simple interpolants in program verification examples
Certification of Bounds of Non-linear Functions: the Templates Method
The aim of this work is to certify lower bounds for real-valued multivariate
functions, defined by semialgebraic or transcendental expressions. The
certificate must be, eventually, formally provable in a proof system such as
Coq. The application range for such a tool is widespread; for instance Hales'
proof of Kepler's conjecture yields thousands of inequalities. We introduce an
approximation algorithm, which combines ideas of the max-plus basis method (in
optimal control) and of the linear templates method developed by Manna et al.
(in static analysis). This algorithm consists in bounding some of the
constituents of the function by suprema of quadratic forms with a well chosen
curvature. This leads to semialgebraic optimization problems, solved by
sum-of-squares relaxations. Templates limit the blow up of these relaxations at
the price of coarsening the approximation. We illustrate the efficiency of our
framework with various examples from the literature and discuss the interfacing
with Coq.Comment: 16 pages, 3 figures, 2 table
Transcriptome bioinformatic analysis identifies potential therapeutic mechanism of pentylenetetrazole in down syndrome
<p>Abstract</p> <p>Background</p> <p>Pentylenetetrazole (PTZ) has recently been found to ameliorate cognitive impairment in rodent models of Down syndrome (DS). The mechanism underlying PTZ's therapeutic effect in DS is however not clear. Microarray profiling has previously reported differential expression, both up- and down-regulation, of genes in DS. Given this, transcriptomic data related to PTZ treatment, if available, could be used to understand the drug's therapeutic mechanism in DS. No such mammalian data however exists. Nevertheless, a <it>Drosophila </it>model inspired by PTZ induced kindling plasticity in rodents has recently been described. Microarray profiling has shown PTZ's downregulatory effect on gene expression in the fly heads.</p> <p>Methods</p> <p>In a comparative transcriptomics approach, I have analyzed the available microarray data in order to identify potential therapeutic mechanism of PTZ in DS. In the analysis, summary data of up- and down-regulated genes reported in human DS studies and of down-regulated genes reported in the <it>Drosophila </it>model has been used.</p> <p>Results</p> <p>I find that transcriptomic correlate of chronic PTZ in <it>Drosophila </it>counteracts that of DS. Genes downregulated by PTZ significantly over-represent genes upregulated in DS and under-represent genes downregulated in DS. Further, the genes which are common in the downregulated and upregulated DS set show enrichment for MAP kinase pathway.</p> <p>Conclusion</p> <p>My analysis suggests that downregulation of MAP kinase pathway may mediate therapeutic effect of PTZ in DS. Existing evidence implicating MAP kinase pathway in DS supports this observation.</p
On the Generation of Positivstellensatz Witnesses in Degenerate Cases
One can reduce the problem of proving that a polynomial is nonnegative, or
more generally of proving that a system of polynomial inequalities has no
solutions, to finding polynomials that are sums of squares of polynomials and
satisfy some linear equality (Positivstellensatz). This produces a witness for
the desired property, from which it is reasonably easy to obtain a formal proof
of the property suitable for a proof assistant such as Coq. The problem of
finding a witness reduces to a feasibility problem in semidefinite programming,
for which there exist numerical solvers. Unfortunately, this problem is in
general not strictly feasible, meaning the solution can be a convex set with
empty interior, in which case the numerical optimization method fails.
Previously published methods thus assumed strict feasibility; we propose a
workaround for this difficulty. We implemented our method and illustrate its
use with examples, including extractions of proofs to Coq.Comment: To appear in ITP 201
A convex polynomial that is not sos-convex
A multivariate polynomial is sos-convex if its Hessian
can be factored as with a possibly nonsquare
polynomial matrix . It is easy to see that sos-convexity is a sufficient
condition for convexity of . Moreover, the problem of deciding
sos-convexity of a polynomial can be cast as the feasibility of a semidefinite
program, which can be solved efficiently. Motivated by this computational
tractability, it has been recently speculated whether sos-convexity is also a
necessary condition for convexity of polynomials. In this paper, we give a
negative answer to this question by presenting an explicit example of a
trivariate homogeneous polynomial of degree eight that is convex but not
sos-convex. Interestingly, our example is found with software using sum of
squares programming techniques and the duality theory of semidefinite
optimization. As a byproduct of our numerical procedure, we obtain a simple
method for searching over a restricted family of nonnegative polynomials that
are not sums of squares.Comment: 15 page
Fenofibrate Reduces Mortality and Precludes Neurological Deficits in Survivors in Murine Model of Japanese Encephalitis Viral Infection
Background: Japanese encephalitis (JE), the most common form of viral encephalitis occurs periodically in endemic areas leading to high mortality and neurological deficits in survivors. It is caused by a flavivirus, Japanese encephalitis virus (JEV), which is transmitted to humans through mosquitoes. No effective cure exists for reducing mortality and morbidity caused by JEV infection, which is primarily due to excessive inflammatory response. Fenofibrate, a peroxisome proliferator-activated receptor-a (PPARa) agonist is known to resolve inflammation by repressing nuclear factor-kB (NF-kB) and enhancing transcription of anti-oxidant and anti-inflammatory genes. In addition, fenofibrate also up-regulates a class of proteins, cytochrome P4504Fs (Cyp4fs), which are involved in detoxification of the potent pro-inflammatory eicosanoid, leukotriene B4 (LTB4) to 20-hydroxy LTB4. Methodology/Principal Findings: The neuroprotective effect of fenofibrate was examined using in vitro (BV-2 microglial cell line) and in vivo (BALB/c mice) models of JEV infection. Mice were treated with fenofibrate for 2 or 4 days prior to JEV exposure. Pretreatment with fenofibrate for 4 but not 2 days reduced mortality by 80 % and brain LTB4 levels decreased concomitantly with the induction of Cyp4f15 and 4f18, which catalyze detoxification of LTB4 through hydroxylation. Expression of cytokines and chemokine decreased significantly as did microglial activation and replication of the JEV virus. Conclusions/Significance: Fenofibrate confers neuroprotection against Japanese encephalitis, in vivo, in mouse model o
Concentration- and schedule-dependent effects of chemotherapy on the angiogenic potential and drug sensitivity of vascular endothelial cells
Computing sum of squares decompositions with rational coefficients
AbstractSum of squares (SOS) decompositions for nonnegative polynomials are usually computed numerically, using convex optimization solvers. Although the underlying floating point methods in principle allow for numerical approximations of arbitrary precision, the computed solutions will never be exact. In many applications such as geometric theorem proving, it is of interest to obtain solutions that can be exactly verified. In this paper, we present a numeric–symbolic method that exploits the efficiency of numerical techniques to obtain an approximate solution, which is then used as a starting point for the computation of an exact rational result. We show that under a strict feasibility assumption, an approximate solution of the semidefinite program is sufficient to obtain a rational decomposition, and quantify the relation between the numerical error versus the rounding tolerance needed. Furthermore, we present an implementation of our method for the computer algebra system Macaulay 2
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