Even though many attempts have been made to define the boundary between decidable and undecidable hybrid systems, the affair is far from being resolved. More and more low dimensional systems are being shown to be undecidable with respect to reachability, and many open problems in between are being discovered. In this paper, we present various two dimensional hybrid systems for which the reachability problem is undecidable. We show their undecidability by simulating Minsky machines. Their proximity to the decidability frontier is understood by inspecting the most parsimonious constraints necessary to make reachability over these automata decidable. We also show that for other two dimensional systems, the reachability question remains unanswered, by proving that it is as hard as the reachability problem for piecewise affine maps on the real line, which is a well known open problem.
We describe an efficient algorithm to construct genome wide haplotype restriction maps of an individual by aligning single molecule DNA fragments collected with Optical Mapping technology. Using this algorithm and small amount of genomic material, we can construct the parental haplotypes for each diploid chromosome for any individual, one from the father and the other from the mother. Since such haplotype maps reveal the polymorphisms due to single nucleotide differences (SNPs) and small insertions and deletions (RFLPs), they are useful in association studies, studies involving genomic instabilities in cancer, and genetics. For instance, such haplotype restriction maps of individuals in a population can be used in association studies to locate genes responsible for genetics diseases with relatively low cost and high throughput.If the underlying problem is formulated as a combinatorial optimization problem, it can be shown to be NP-complete (a special case of K-population problem). But by effectively exploiting the structure of the underlying error processes and using a novel analog of the Baum-Welch algorithm for HMM models, we devise a probabilistic algorithm with a time complexity that is linear in the number of markers.The algorithms were tested by constructing the first genome wide haplotype restriction map of the microbe T. Pseudoana, as well as constructing a haplotype restriction map of a 120 Megabase region of Human chromosome 4. The frequency of false positives and false negatives was estimated using simulated data. The empirical results were found very promising.
We collaborate in a research program aimed at creating a rigorous framework, experimental infrastructure, and computational environment for understanding, experimenting with, manipulating, and modifying a diverse set of fundamental biological processes at multiple scales and spatio-temporal modes. The novelty of our research is based on an approach that (i) requires coevolution of experimental science and theoretical techniques and (ii) exploits a certain universality in biology guided by a parsimonious model of evolutionary mechanisms operating at the genomic level and manifesting at the proteomic, transcriptomic, phylogenic, and other higher levels. Our current program in "systems biology" endeavors to marry large-scale biological experiments with the tools to ponder and reason about large, complex, and subtle natural systems. To achieve this ambitious goal, ideas and concepts are combined from many different fields: biological experimentation, applied mathematical modeling, computational reasoning schemes, and large-scale numerical and symbolic simulations. From a biological viewpoint, the basic issues are many: (i) understanding common and shared structural motifs among biological processes; (ii) modeling biological noise due to interactions among a small number of key molecules or loss of synchrony; (iii) explaining the robustness of these systems in spite of such noise; and (iv) cataloging multistatic behavior and adaptation exhibited by many biological processes.
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