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

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

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

Decidability of Zenoness, Token Liveness and Boundedness of Dense-Timed Petri Nets

We consider 'Timed Petri Nets (TPNs)' : extensions of Petri nets in which each token is equipped with a real-valued clock. We consider the following three verification problems for TPN. (i) 'Zenoness:' whether there is an infinite computation from a given marking which takes only a finite amount of time. We show decidability of zenoness for TPNs, thus solving an open problem from \citeEscrig:etal:TPN. (ii) 'Token liveness:' whether a token is ıt 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.To appear in FSTTCS '04</p

Closed, Open and Robust Timed Networks

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 [AJ03] it was shown that checking safety properties is decidable in the case where each timed process is equipped with a single real-valued clock. In [ADM04], we showed that this is no longer possible if each timed process is equipped with at least two real-valued clocks. In this paper, we study two subclasses of timed networks: 'closed' and 'open' timed networks. In closed timed networks, all clock constraints are non-strict, while in open timed networks, all clock constraints are strict (thus corresponds to syntactic removal of equality testing). We show that the problem becomes decidable for closed timed network, while it remains undecidable for open timed networks. We also consider 'robust' semantics of timed networks by introducing timing fuzziness through semantic removal of equality testing. We show that the problem is undecidable both for closed and open timed networks under the robust semantics.To appear in Infinity '04</p

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

Decidability of Zenoness, Token Liveness and Boundedness of Dense-Timed Petri Nets

We consider 'Timed Petri Nets (TPNs)' : extensions of Petri nets in which each token is equipped with a real-valued clock. We consider the following three verification problems for TPN. (i) 'Zenoness:' whether there is an infinite computation from a given marking which takes only a finite amount of time. We show decidability of zenoness for TPNs, thus solving an open problem from \citeEscrig:etal:TPN. (ii) 'Token liveness:' whether a token is ıt 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.To appear in FSTTCS '04</p

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

We consider Dense-Timed Petri Nets (TPN), an extension of Petri nets in whicheach 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 disabletransitions). 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 showdecidability of zenoness for TPNs, thus solving an open problem from [Escrig etal.]. Furthermore, the related question if there exist arbitrarily fastcomputations from a given marking is also decidable. On the other hand,universal zenoness, i.e., the question if all infinite computations from agiven marking are zeno, is undecidable. (ii) Token liveness: whether a token isalive in a marking, i.e., whether there is a computation from the marking whicheventually consumes the token. We show decidability of the problem by reducingit to the coverability problem, which is decidable for TPNs. (iii) Boundedness:whether the size of the reachable markings is bounded. We consider two versionsof the problem; namely semantic boundedness where only live tokens are takeninto consideration in the markings, and syntactic boundedness where also deadtokens are considered. We show undecidability of semantic boundedness, while weprove that syntactic boundedness is decidable through an extension of theKarp-Miller algorithm.Comment: 61 pages, 18 figure