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

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

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 $p(x)=p(x_1,...,x_n)$ is sos-convex if its Hessian $H(x)$ can be factored as $H(x)= M^T(x) M(x)$ with a possibly nonsquare polynomial matrix $M(x)$. It is easy to see that sos-convexity is a sufficient condition for convexity of $p(x)$. 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

Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Groebner Bases

We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the THEOREMA system; some code fragments and sample computations are included

High Frequency of Mosaicism among Patients with Neurofibromatosis Type 1 (NF1) with Microdeletions Caused by Somatic Recombination of the JJAZ1 Gene

Detailed analyses of 20 patients with sporadic neurofibromatosis type 1 (NF1) microdeletions revealed an unexpected high frequency of somatic mosaicism (8/20 [40%]). This proportion of mosaic deletions is much higher than previously anticipated. Of these deletions, 16 were identified by a screen of unselected patients with NF1. None of the eight patients with mosaic deletions exhibited the mental retardation and facial dysmorphism usually associated with NF1 microdeletions. Our study demonstrates the importance of a general screening for NF1 deletions, regardless of a special phenotype, because of a high estimated number of otherwise undetected mosaic NF1 microdeletions. In patients with mosaicism, the proportion of cells with the deletion was 91%–100% in peripheral leukocytes but was much lower (51%–80%) in buccal smears or peripheral skin fibroblasts. Therefore, the analysis of other tissues than blood is recommended, to exclude mosaicism with normal cells in patients with NF1 microdeletions. Furthermore, our study reveals breakpoint heterogeneity. The classic 1.4-Mb deletion was found in 13 patients. These type I deletions encompass 14 genes and have breakpoints in the NF1 low-copy repeats. However, we identified a second major type of NF1 microdeletion, which spans 1.2 Mb and affects 13 genes. This type II deletion was found in 8 (38%) of 21 patients and is mediated by recombination between the JJAZ1 gene and its pseudogene. The JJAZ1 gene, which is completely deleted in patients with type I NF1 microdeletions and is disrupted in deletions of type II, is highly expressed in brain structures associated with learning and memory. Thus, its haploinsufficiency might contribute to mental impairment in patients with constitutional NF1 microdeletions. Conspicuously, seven of the eight mosaic deletions are of type II, whereas only one was a classic type I deletion. Therefore, the JJAZ1 gene is a preferred target of strand exchange during mitotic nonallelic homologous recombination. Although type I NF1 microdeletions occur by interchromosomal recombination during meiosis, our findings imply that type II deletions are mediated by intrachromosomal recombination during mitosis. Thus, NF1 microdeletions acquired during mitotic cell divisions differ from those occurring in meiosis and are caused by different mechanisms