Dense-Timed Petri Nets: Checking Zenoness, Token liveness and Boundedness

We consider Dense-Timed Petri Nets (TPN), an extension of Petri nets in which each token is equipped with a real-valued clock and where the semantics is lazy (i.e., enabled transitions need not fire; time can pass and disable transitions). We consider the following verification problems for TPNs. (i) Zenoness: whether there exists a zeno-computation from a given marking, i.e., an infinite computation which takes only a finite amount of time. We show decidability of zenoness for TPNs, thus solving an open problem from [Escrig et al.]. Furthermore, the related question if there exist arbitrarily fast computations from a given marking is also decidable. On the other hand, universal zenoness, i.e., the question if all infinite computations from a given marking are zeno, is undecidable. (ii) Token liveness: whether a token is alive in a marking, i.e., whether there is a computation from the marking which eventually consumes the token. We show decidability of the problem by reducing it to the coverability problem, which is decidable for TPNs. (iii) Boundedness: whether the size of the reachable markings is bounded. We consider two versions of the problem; namely semantic boundedness where only live tokens are taken into consideration in the markings, and syntactic boundedness where also dead tokens are considered. We show undecidability of semantic boundedness, while we prove that syntactic boundedness is decidable through an extension of the Karp-Miller algorithm.Comment: 61 pages, 18 figure

Hierarchical Clustering Using the Arithmetic-Harmonic Cut: Complexity and Experiments

Clustering, particularly hierarchical clustering, is an important method for understanding and analysing data across a wide variety of knowledge domains with notable utility in systems where the data can be classified in an evolutionary context. This paper introduces a new hierarchical clustering problem defined by a novel objective function we call the arithmetic-harmonic cut. We show that the problem of finding such a cut is -hard and -hard but is fixed-parameter tractable, which indicates that although the problem is unlikely to have a polynomial time algorithm (even for approximation), exact parameterized and local search based techniques may produce workable algorithms. To this end, we implement a memetic algorithm for the problem and demonstrate the effectiveness of the arithmetic-harmonic cut on a number of datasets including a cancer type dataset and a corona virus dataset. We show favorable performance compared to currently used hierarchical clustering techniques such as -Means, Graclus and Normalized-Cut. The arithmetic-harmonic cut metric overcoming difficulties other hierarchal methods have in representing both intercluster differences and intracluster similarities

Biomarkers for epithelial ovarian cancers

Epithelial carcinoma of the ovary is one of the most common gynecological malignancies and the fifth most frequent cause of cancer death in women. Currently blood test of advanced epithelial tumors are reflected in a high level of CA 125 antigen. However, it is not a good marker for early stage tumors, and may yield false positives. Clearly, there is a need for better understanding of the molecular pathogenesis of epithelial ovarian cancer, so that new drug targets or biomarkers that facilitate early detection can be identified. This work concentrates on finding genetic markers for three epithelial ovarian tumors, using a simple computational method. We give a small set of genetic markers which are able to distinguish clear cell and mucinous ovarian cancers (13 and 26 genes respectively) from other epithelial ovarian tumors with 100% accuracy. We obtain the genes HNF1-beta (TCF2) and GGT1 as the best markers for the clear cell and CEACAM6 (NCA) as the best marker for mucinous ovarian tumors. We employ a feature selection technique based on minimum probability of error for this purpose. We give a ranking of the important genes responsible for these tumors and validate the results using the leave-one-out cross-validation technique. Using this method, we also agree with the common notion that WT1 is one of the best genes to separate serous ovarian tumors from other epithelial ovarian tumors

Exploratory consensus of hierarchical clusterings for melanoma and breast cancer

Finding subtypes of heterogeneous diseases is the biggest challenge in the area of biology. Often, clustering is used to provide a hypothesis for the subtypes of a heterogeneous disease. However, there are usually discrepancies between the clusterings produced by different algorithms. This work introduces a simple method which provides the most consistent clusters across three different clustering algorithms for a melanoma and a breast cancer data set. The method is validated by showing that the Silhouette, Dunne’s and Davies-Bouldin’s cluster validation indices are better for the proposed algorithm than those obtained by k-means and another consensus clustering algorithm. The hypotheses of the consensus clusters on both the data sets are corroborated by clear genetic markers and 100 percent classification accuracy. In Bittner et al.’s melanoma data set, a previously hypothesized primary cluster is recognized as the largest consensus cluster and a new partition of this cluster into two subclusters is proposed. In van’t Veer et al.’s breast cancer data set, previously proposed “basal” and “luminal A” subtypes are clearly recognized as the two predominant clusters. Furthermore, a new hypothesis is provided about the existence of two subgroups within the “basal” subtype in this data set. The clusters of van’t Veer’s data set is also validated by high classification accuracy obtained in the data set of van de Vijver et al

Model Checking Parameterized Timed Systems

In recent years, there has been much advancement in the area of verification of infinite-state systems. A system can have an infinite state-space due to unbounded data structures such as counters, clocks, stacks, queues, etc. It may also be infinite-state due to parameterization, i.e., the possibility of having an arbitrary number of components in the system. For parameterized systems, we are interested in checking correctness of all the instances in one verification step. In this thesis, we consider systems which contain both sources of infiniteness, namely: (a) real-valued clocks and (b) parameterization. More precisely, we consider two models: (a) the timed Petri net (TPN) model, which is an extension of the classical Petri net model; and (b) the timed network (TN) model in which an arbitrary number of timed automata run in parallel. We consider verification of safety properties for timed Petri nets using forward analysis. Since forward analysis is necessarily incomplete, we provide a semi-algorithm augmented with an acceleration technique in order to make it terminate more often on practical examples. Then we consider a number of problems which are generalisations of the corresponding ones for timed automata and Petri nets. For instance, we consider zenoness where we check the existence of an infinite computation with a finite duration. We also consider two variants of boundedness problem: syntactic boundedness in which both live and dead tokens are considered and semantic boundedness where only live tokens are considered. We show that the former problem is decidable while the latter is not. Finally, we show undecidability of LTL model checking both for dense and discrete timed Petri nets. Next we consider timed networks. We show undecidability of safety properties in case each component is equipped with two or more clocks. This result contrasts previous decidability result for the case where each component has a single clock. Also ,we show that the problem is decidable when clocks range over the discrete time domain. This decidability result holds when the processes have any finite number of clocks. Furthermore, we outline the border between decidability and undecidability of safety for TNs by considering several syntactic and semantic variants

Multi-clock timed networks

Abstract. We consider verification of safety properties for parameterized systems of timed processes, so called timed networks. A timed network consists of a finite state process, called a controller, and an arbitrary set of identical timed processes. In a previous work, we showed that checking safety properties is decidable in the case where each timed process is equipped with a single real-valued clock. It was left open whether the result could be extended to multi-clock timed networks. We show that the problem becomes undecidable when each timed process has two clocks. On the other hand, we show that the problem is decidable when clocks range over a discrete time domain. This decidability result holds when processes have any finite number of clocks.

Hierarchical clustering, languages and cancer

In this paper, we introduce a novel objective function for the hierarchical clustering of data from distance matrices, a very relevant task in Bioinformatics. To test the robustness of the method, we test it in two areas: (a) the problem of deriving a phylogeny of languages and (b) subtype cancer classification from microarray data. For comparison purposes, we also consider both the use of ultrametric trees (generated via a two-phase evolutionary approach that creates a large number of hypothesis trees, and then takes a consensus), and the best-known results from the literature. We used a dataset of measured 'separation time' among 84 Indo-European languages. The hierarchy we produce agrees very well with existing data about these languages across a wide range of levels, and it helps to clarify and raise new hypothesis about the evolution of these languages. Our method also generated a classification tree for the different cancers in the NCI60 microarray dataset (comprising gene expression data for 60 cancer cell lines). In this case, the method seems to support the current belief about the heterogeneous nature of the ovarian, breast and non-small-lung cancer, as opposed to the relative homogeneity of other types of cancer. However, our method reveals a close relationship of the melanoma and CNS cell-lines. This is in correspondence with the fact that metastatic melanoma first appears in central nervous system (CNS)

Closed, Open, and Robust Timed Networks

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Investigating the change of the hierarchical pattern of gene expression in the normal and Parkinson's brain using a combinatorial optimization based unsupervised clustering method

Previous works on Parkinson's disease (PD) mainly focused on genes differentially expressed between the anterior and the posterior sections of the brains of a normal mouse and the one with PD. However, no work has been done in finding a hierarchical pattern of gene expression between the different regions of a brain. Such a hierarchy is useful to locate genetic specializations within a normal brain, thus in analyzing how brain infirmities affect these specializations. We use a recently proposed method of robust hierarchical clustering using arithmeticharmonic cut to construct the hierarchical relation between different regions of the brain. Then, we show how similar regions of the normal and PD brain differ in gene expressions, indicating a functional variation due to Parkinson's disease in a few high-level clusters of brain regions