This paper presents Newton, a branch and prune algorithm used to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists of enforcing at each node of the search tree a unique local consistency condition, called box-consistency, which approximates the notion of arc-consistency well known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the Hansen-Sengupta narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with state-of-the-art continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.

Interpolation based automatic abstraction is a powerful and robust technique for the automated analysis of hardware and software systems. Its use has however been limited to controldominated applications because of a lack of algorithms for computing interpolants for data structures used in software programs. We present efficient procedures to construct interpolants for the theories of arrays, sets, and multisets using the reduction approach for obtaining decision procedures for complex data structures. The approach taken is that of reducing the theories of such data structures to the theories of equality and linear arithmetic for which efficient interpolating decision procedures exist. This enables interpolation based techniques to be applied to proving properties of programs that manipulate these data structures.

Severe congenital neutropenia (SCN) is an inborn disorder of granulopoiesis. Like most other bone marrow failure syndromes, it is associated with a marked propensity to transform into a myelodysplastic syndrome (MDS) or acute leukemia, with a cumulative rate of transformation to MDS/leukemia that exceeds 20%. The genetic (and/or epigenetic) changes that contribute to malignant transformation in SCN are largely unknown. In this study, we performed mutational profiling of 14 genes previously implicated in leukemogenesis using 14 MDS/leukemia samples from patients with SCN. We used high-throughput exon-based resequencing of whole-genome-amplified genomic DNA with a semiautomated method to detect mutations. The sensitivity and specificity of the sequencing pipeline was validated by determining the frequency of mutations in these 14 genes using 188 de novo AML samples. As expected, mutations of tyrosine kinase genes (FLT3, KIT, and JAK2) were common in de novo AML, with a cumulative frequency of 30%. In contrast, no mutations in these genes were detected in the SCN samples; instead, mutations of CSF3R, encoding the G-CSF receptor, were common. These data support the hypothesis that mutations of CSF3R may provide the "activated tyrosine kinase signal" that is thought to be important for leukemogenesis. (Blood. 2007;110: [1648][1649][1650][1651][1652][1653][1654][1655]

This paper presents a method for automatically generating all polynomial invariants in simple loops. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Based on this connection, a fixpoint procedure using operations on ideals and Gröbner basis constructions is proposed for finding all polynomial invariants. Most importantly, it is proved that the procedure terminates in at most m + 1 iterations, where m is the number of program variables. The proof relies on showing that the irreducible components of the varieties associated with the ideals generated by the procedure either remain the same or increase their dimension at every iteration of the fixpoint procedure. This yields a correct and complete algorithm for inferring conjunctions of polynomial equalities as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover non-trivial invariants for several examples to illustrate the power of the technique.1 If the field K is algebraically closed then IV(I) = Rad(I), the radical of I, which is the set of polynomials p such that there exists k ∈ N satisfying p k ∈ I.

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A new algorithm for computing a comprehensive Gröbner system of a parametric polynomial ideal over k [U ][X] is presented. This algorithm generates fewer branches (segments) compared to Suzuki and Sato's algorithm as well as Nabeshima's algorithm, resulting in considerable efficiency. As a result, the algorithm is able to compute comprehensive Gröb-ner systems of parametric polynomial ideals arising from applications which have been beyond the reach of other well known algorithms. The starting point of the new algorithm is Weispfenning's algorithm with a key insight by Suzuki and Sato who proposed computing first a Gröbner basis of an ideal over k[U, X] before performing any branches based on parametric constraints. Based on Kalkbrener's results about stability and specialization of Gröbner basis of ideals, the proposed algorithm exploits the result that along any branch in a tree corresponding to a comprehensive Gröb-ner system, it is only necessary to consider one polynomial for each nondivisible leading power product in k(U ) [X] with the condition that the product of their leading coefficients is not 0; other branches correspond to the cases where this product is 0. In addition, for dealing with a disequality parametric constraint, a probabilistic check is employed for radical membership test of an ideal of parametric constraints. This is in contrast to a general expensive check based on Rabinovitch's trick using a new variable as in Nabeshima's algorithm. The proposed algorithm has been implemented in Magma and experimented with a number of examples from different applications. Its performance (vis a vie number of branches and execution timings) has been compared with the Suzuki-Sato's algorithm and Nabeshima's speed-up algorithm. The algorithm has been successfully used to solve the famous P3P problem from computer vision.

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