TY - JOUR AB - Giotto is a high-level programming language for time-triggered control applications. The authors begin with a conceptual overview of its methodology, discuss the Giotto helicopter project, and summarize available Giotto implementations. AU - Henzinger, Thomas A AU - Kirsch, Christoph AU - Sanvido, Marco AU - Pree, Wolfgang ID - 4468 IS - 1 JF - IEEE Control Systems Magazine SN - 1066-033X TI - From control models to real-time code using Giotto VL - 23 ER - TY - CHAP AB - Giotto is a principled, tool-supported design methodology for implementing embedded control systems on platforms of possibly distributed sensors, actuators, CPUs, and networks. Giotto is based on the principle that time-triggered task invocations plus time-triggered mode switches can form the abstract essence of programming real-time control systems. Giotto consists of a programming language with a formal semantics, and a retargetable compiler and runtime library. Giotto supports the automation of control system design by strictly separating platform-independent functionality and timing concerns from platform-dependent scheduling and communication issues. The time-triggered predictability of Giotto makes it particularly suitable for safety-critical applications with hard real-time constraints. We illustrate the platform independence and time-triggered execution of Giotto by coordinating a heterogeneous flock of Intel x86 robots and Lego Mindstorms robots. AU - Henzinger, Thomas A AU - Horowitz, Benjamin AU - Kirsch, Christoph ID - 4465 SN - 9780471234364 T2 - Software-Enabled Control: Information Technology for Dynamical Systems TI - Embedded control systems development with Giotto ER - TY - CONF AB - One source of complexity in the μ-calculus is its ability to specify an unbounded number of switches between universal (AX) and existential (EX) branching modes. We therefore study the problems of satisfiability, validity, model checking, and implication for the universal and existential fragments of the μ-calculus, in which only one branching mode is allowed. The universal fragment is rich enough to express most specifications of interest, and therefore improved algorithms are of practical importance. We show that while the satisfiability and validity problems become indeed simpler for the existential and universal fragments, this is, unfortunately, not the case for model checking and implication. We also show the corresponding results for the alternationfree fragment of the μ-calculus, where no alternations between least and greatest fixed points are allowed. Our results imply that efforts to find a polynomial-time model-checking algorithm for the μ-calculus can be replaced by efforts to find such an algorithm for the universal or existential fragment. AU - Henzinger, Thomas A AU - Kupferman, Orna AU - Majumdar, Ritankar ID - 4466 SN - 9783540008989 T2 - Proceedings of the 9th International Conference on Tools and Algorithms for the Construction and Analysis of Systems TI - On the universal and existential fragments of the mu-calculus VL - 2619 ER - TY - CONF AB - BLAST (the Berkeley Lazy Abstraction Software verification Tool) is a verification system for checking safety properties of C programs using automatic property-driven construction and model checking of software abstractions. Blast implements an abstract-model check-refine loop to check for reachability of a specified label in the program. The abstract model is built on the fly using predicate abstraction. This model is then checked for reachability. If there is no (abstract) path to the specified error label, Blast reports that the system is safe and produces a succinct proof. Otherwise, it checks if the path is feasible using symbolic execution of the program. If the path is feasible, Blast outputs the path as an error trace, otherwise, it uses the infeasibility of the path to refine the abstract model. Blast short-circuits the loop from abstraction to verification to refinement, integrating the three steps tightly through “lazy abstraction” [5]. This integration can offer significant advantages in performance by avoiding the repetition of work from one iteration of the loop to the next. AU - Henzinger, Thomas A AU - Jhala, Ranjit AU - Majumdar, Ritankar AU - Sutre, Grégoire ID - 4467 SN - 9783540401179 T2 - Proceedings of the 10th International SPIN Workshop TI - Software verification with BLAST VL - 2648 ER - TY - CONF AB - We present an algorithm called TAR (“Thread-modular Abstraction Refinement”) for model checking safety properties of concurrent software. The TAR algorithm uses thread-modular assume-guarantee reasoning to overcome the exponential complexity in the control state of multithreaded programs. Thread modularity means that TAR explores the state space of one thread at a time, making assumptions about how the environment can interfere. The TAR algorithm uses counterexample-guided predicate-abstraction refinement to overcome the usually infinite complexity in the data state of C programs. A successive approximation scheme automatically infers the necessary precision on data variables as well as suitable environment assumptions. The scheme is novel in that transition relations are approximated from above, while at the same time environment assumptions are approximated from below. In our software verification tool BLAST we have implemented a fully automatic race checker for multithreaded C programs which is based on the TAR algorithm. This tool has verified a wide variety of commonly used locking idioms, including locking schemes that are not amenable to existing dynamic and static race checkers such as ERASER or WARLOCK. AU - Henzinger, Thomas A AU - Jhala, Ranjit AU - Majumdar, Ritankar AU - Qadeer, Shaz ID - 4463 SN - 9783540405245 T2 - Proceedings of the 15th International Conference on Computer Aided Verification TI - Thread-modular abstraction refinement VL - 2725 ER - TY - CONF AB - A major hurdle in the algorithmic verification and control of systems is the need to find suitable abstract models, which omit enough details to overcome the state-explosion problem, but retain enough details to exhibit satisfaction or controllability with respect to the specification. The paradigm of counterexample-guided abstraction refinement suggests a fully automatic way of finding suitable abstract models: one starts with a coarse abstraction, attempts to verify or control the abstract model, and if this attempt fails and the abstract counterexample does not correspond to a concrete counterexample, then one uses the spurious counterexample to guide the refinement of the abstract model. We present a counterexample-guided refinement algorithm for solving ω-regular control objectives. The main difficulty is that in control, unlike in verification, counterexamples are strategies in a game between system and controller. In the case that the controller has no choices, our scheme subsumes known counterexample-guided refinement algorithms for the verification of ω-regular specifications. Our algorithm is useful in all situations where ω-regular games need to be solved, such as supervisory control, sequential and program synthesis, and modular verification. The algorithm is fully symbolic, and therefore applicable also to infinite-state systems. AU - Henzinger, Thomas A AU - Jhala, Ranjit AU - Majumdar, Ritankar ID - 4462 SN - 9783540404934 T2 - Proceedings of the 30th International Colloquium on Automata, Languages and Programming TI - Counterexample-guided control VL - 2719 ER - TY - CONF AB - We introduce the paradigm of schedule-carrying code (SCC). A hard real-time program can be executed on a given platform only if there exists a feasible schedule for the real-time tasks of the program. Traditionally, a scheduler determines the existence of a feasible schedule according to some scheduling strategy. With SCC, a compiler proves the existence of a feasible schedule by generating executable code that is attached to the program and represents its schedule. An SCC executable is a real-time program that carries its schedule as code, which is produced once and can be revalidated and executed with each use. We evaluate SCC both in theory and practice. In theory, we give two scenarios, of nonpreemptive and distributed scheduling for Giotto programs, where the generation of a feasible schedule is hard, while the validation of scheduling instructions that are attached to the programs is easy. In practice, we implement SCC and show that explicit scheduling instructions can reduce the scheduling overhead up to 35% and can provide an efficient, flexible, and verifiable means for compiling Giotto programs on complex architectures, such as the TTA. AU - Henzinger, Thomas A AU - Kirsch, Christoph AU - Matic, Slobodan ID - 4464 SN - 9783540202233 T2 - Proceedings of the 3rd International Conference on Embedded Software TI - Schedule-carrying code VL - 2855 ER - TY - JOUR AB - Symbolic model checking, which enables the automatic verification of large systems, proceeds by calculating expressions that represent state sets. Traditionally, symbolic model-checking tools are based on back- ward state traversal; their basic operation is the function pre, which, given a set of states, returns the set of all predecessor states. This is because specifiers usually employ formalisms with future-time modalities, which are naturally evaluated by iterating applications of pre. It has been shown experimentally that symbolic model checking can perform significantly better if it is based, instead, on forward state traversal; in this case, the basic operation is the function post, which, given a set of states, returns the set of all successor states. This is because forward state traversal can ensure that only parts of the state space that are reachable from an initial state and relevant for the satisfaction or violation of the specification are explored; that is, errors can be detected as soon as possible. In this paper, we investigate which specifications can be checked by symbolic forward state traversal. We formulate the problems of symbolic backward and forward model checking by means of two μ-calculi. The pre-μ calculus is based on the pre operation, and the post-μ calculus is based on the post operation. These two μ-calculi induce query logics, which augment fixpoint expressions with a boolean emptiness query. Using query logics, we are able to relate and compare the symbolic backward and forward approaches. In particular, we prove that all ω-regular (linear-time) specifications can be expressed as post-μ queries, and therefore checked using symbolic forward state traversal. On the other hand, we show that there are simple branching-time specifications that cannot be checked in this way. AU - Henzinger, Thomas A AU - Kupferman, Orna AU - Qadeer, Shaz ID - 4460 IS - 3 JF - Formal Methods in System Design SN - 0925-9856 TI - From pre-historic to post-modern symbolic model checking VL - 23 ER - TY - JOUR AB - Giotto provides an abstract programmer's model for the implementation of embedded control systems with hard real-time constraints. A typical control application consists of periodic software tasks together with a mode-switching logic for enabling and disabling tasks. Giotto specifies time-triggered sensor readings, task invocations, actuator updates, and mode switches independent of any implementation platform. Giotto can be annotated with platform constraints such as task-to-host mappings, and task and communication schedules. The annotations are directives for the Giotto compiler, but they do not alter the functionality and timing of a Giotto program. By separating the platform-independent from the platform-dependent concerns, Giotto enables a great deal of flexibility in choosing control platforms as well as a great deal of automation in the validation and synthesis of control software. The time-triggered nature of Giotto achieves timing predictability, which makes Giotto particularly suitable for safety-critical applications. AU - Henzinger, Thomas A AU - Horowitz, Benjamin AU - Kirsch, Christoph ID - 4469 IS - 1 JF - Proceedings of the IEEE SN - 0018-9219 TI - Giotto: A time-triggered language for embedded programming VL - 91 ER - TY - JOUR AB - Mosaic hybrid zones arise when ecologically differentiated taxa hybridize across a network of habitat patches. Frequent interbreeding across a small-scale patchwork can erode species differences that might have been preserved in a clinal hybrid zone. In particular, the rapid breakdown of neutral divergence sets an upper limit to the time for which differences at marker loci can persist. We present here a case study of a mosaic hybrid zone between the fire-bellied toads Bombina bombina and B. variegata (Anura: Discoglossidae) near Apahida in Romania. In our 20 × 20 km study area, we detected no evidence of a clinal transition but found a strong association between aquatic habitat and mean allele frequencies at four molecular markers. In particular, pure populations of B. bombina in ponds appear to cause massive introgression into the surrounding B. variegata gene pool found in temporary aquatic sites. Nevertheless, the genetic structure of these hybrid populations was remarkably similar to those of a previously studied transect near Pescenica (Croatia), which had both clinal and mosaic features: estimates of heterozygote deficit and linkage disequilibrium in each country are similar. In Apahida, the observed strong linkage disequilibria should stem from an imperfect habitat preference that guides most (but not all) adults into the habitats to which they are adapted. In the absence of a clinal structure, the inferred migration rate between habitats implies that associations between selected loci and neutral markers should break down rapidly. Although plausible selection strengths can maintain differentiation at those loci adapting the toads to either permanent or temporary breeding sites, the divergence at neutral markers must be transient. The hybrid zone may be approaching a state in which the gene pools are homogenized at all but the selected loci, not dissimilar from an early stage of sympatric divergence. AU - Vines, Timothy AU - Kohler, S C AU - Thiel, M AU - Ghira, Ioan AU - Sands, T R AU - Maccallum, Catriona AU - Barton, Nicholas H AU - Nürnberger, Beate ID - 4338 IS - 8 JF - Evolution SN - 0014-3820 TI - On the maintenance of reproductive isolation in a mosaic hybrid zone between the toads Bombina bombina and B. variegata VL - 57 ER - TY - JOUR AB - The phylogeny of Crocodylia offers an unusual twist on the usual molecules versus morphology story. The true gharial (Gavialis gangeticus) and the false gharial (Tomistoma schlegelii), as their common names imply, have appeared in all cladistic morphological analyses as distantly related species, convergent upon a similar morphology. In contrast, all previous molecular studies have shown them to be sister taxa. We present the first phylogenetic study of Crocodylia using a nuclear gene. We cloned and sequenced the c-myc proto-oncogene from Alligator mississippiensis to facilitate primer design and then sequenced an 1,100-base pair fragment that includes both coding and noncoding regions and informative indels for one species in each extant crocodylian genus and six avian outgroups. Phylogenetic analyses using parsimony, maximum likelihood, and Bayesian inference all strongly agreed on the same tree, which is identical to the tree found in previous molecular analyses: Gavialis and Tomistoma are sister taxa and together are the sister group of Crocodylidae. Kishino-Hasegawa tests rejected the morphological tree in favor of the molecular tree. We excluded long-branch attraction and variation in base composition among taxa as explanations for this topology. To explore the causes of discrepancy between molecular and morphological estimates of crocodylian phylogeny, we examined puzzling features of the morphological data using a priori partitions of the data based on anatomical regions and investigated the effects of different coding schemes for two obvious morphological similarities of the two gharials. AU - Harshman, John AU - Huddleston, Christopher AU - Bollback, Jonathan P AU - Parsons, Thomas AU - Braun, Michael ID - 4350 IS - 3 JF - Systematic Biology SN - 0039-7989 TI - True and false gharials: A nuclear gene phylogeny of crocodylia VL - 52 ER - TY - JOUR AB - Many questions in evolutionary biology are best addressed by comparing traits in different species. Often such studies involve mapping characters on phylogenetic trees. Mapping characters on trees allows the nature, number, and timing of the transformations to be identified. The parsimony method is the only method available for mapping morphological characters on phylogenies. Although the parsimony method often makes reasonable reconstructions of the history of a character, it has a number of limitations. These limitations include the inability to consider more than a single change along a branch on a tree and the uncoupling of evolutionary time from amount of character change. We extended a method described by Nielsen (2002, Syst. Biol. 51:729-739) to the mapping of morphological characters under continuous-time Markov models and demonstrate here the utility of the method for mapping characters on trees and for identifying character correlation. AU - Huelsenbeck, John AU - Nielsen, Rasmus AU - Bollback, Jonathan P ID - 4348 IS - 2 JF - Systematic Biology SN - 0039-7989 TI - Stochastic mapping of morphological characters VL - 52 ER - TY - JOUR AB - Chromosomal rearrangements can promote reproductive isolation by reducing recombination along a large section of the genome. We model the effects of the genetic barrier to gene flow caused by a chromosomal rearrangement on the rate of accumulation of postzygotic isolation genes in parapatry. We find that, if reproductive isolation is produced by the accumulation in parapatry of sets of alleles compatible within but incompatible across species, chromosomal rearrangements are far more likely to favor it than classical genetic barriers without chromosomal changes. New evidence of the role of chromosomal rearrangements in parapatric speciation suggests that postzygotic isolation is often due to the accumulation of such incompatibilities. The model makes testable qualitative predictions about the genetic signature of speciation. AU - Navarro, Arcadio AU - Barton, Nicholas H ID - 4254 IS - 3 JF - Evolution; International Journal of Organic Evolution SN - 0014-3820 TI - Accumulating postzygotic isolation genes in parapatry: a new twist on chromosomal speciation VL - 57 ER - TY - JOUR AB - Variation within a species may be structured both geographically and by genetic background. We review the effects of such structuring on neutral variants, using a framework based on the coalescent process. Short-term effects of sex differences and age structure can be averaged out using fast timescale approximations, allowing a simple general treatment of effective population size and migration. We consider the effects of geographic structure on variation within and between local populations, first in general terms, and then for specific migration models. We discuss the close parallels between geographic structure and stable types of genetic structure caused by selection, including balancing selection and background selection. The effects of departures from stability, such as selective sweeps and population bottlenecks, are also described. Methods for distinguishing population history from the effects of ongoing gene flow are discussed. We relate the theoretical results to observed patterns of variation in natural populations. AU - Charlesworth, Brian AU - Charlesworth, Deborah AU - Barton, Nicholas H ID - 4257 JF - Annual Review of Ecology and Systematics SN - 1543-592X TI - The effects of genetic and geographic structure on neutral variation VL - 34 ER - TY - JOUR AB - Artificial Life models may shed new light on the long-standing challenge for evolutionary biology of explaining the origins of complex organs. Real progress on this issue, however, requires Artificial Life researchers to take seriously the tools and insights from population genetics. AU - Barton, Nicholas H AU - Zuidema, Willem ID - 4256 IS - 16 JF - Current Biology SN - 0960-9822 TI - The erratic path towards complexity VL - 13 ER - TY - JOUR AB - Humans and their closest evolutionary relatives, the chimpanzees, differ in ∼1.24% of their genomic DNA sequences. The fraction of these changes accumulated during the speciation processes that have separated the two lineages may be of special relevance in understanding the basis of their differences. We analyzed human and chimpanzee sequence data to search for the patterns of divergence and polymorphism predicted by a theoretical model of speciation. According to the model, positively selected changes should accumulate in chromosomes that present fixed structural differences, such as inversions, between the two species. Protein evolution was more than 2.2 times faster in chromosomes that had undergone structural rearrangements compared with colinear chromosomes. Also, nucleotide variability is slightly lower in rearranged chromosomes. These patterns of divergence and polymorphism may be, at least in part, the molecular footprint of speciation events in the human and chimpanzee lineages. AU - Navarro, Arcadio AU - Barton, Nicholas H ID - 4255 IS - 5617 JF - Science SN - 0036-8075 TI - Chromosomal speciation and molecular divergence -- Accelerated evolution in rearranged chromosomes VL - 300 ER - TY - JOUR AB - During vertebrate gastrulation, highly coordinated cellular rearrangements lead to the formation of the three germ layers, ectoderm, mesoderm and endoderm. In zebrafish, silberblick (slb)/wnt11 regulates normal gastrulation movements by activating a signalling pathway similar to the Frizzled-signalling pathway, which establishes epithelial planar cell polarity (PCP) in Drosophila. However, the cellular mechanisms by which slb/wnt11 functions during zebrafish gastrulation are still unclear. Using high-resolution two-photon confocal imaging followed by computer-assisted reconstruction and motion analysis, we have analysed the movement and morphology of individual cells in three dimensions during the course of gastrulation. We show that in slb-mutant embryos, hypoblast cells within the forming germ ring have slower, less directed migratory movements at the onset of gastrulation. These aberrant cell movements are accompanied by defects in the orientation of cellular processes along the individual movement directions of these cells. We conclude that slb/wnt11-mediated orientation of cellular processes plays a role in facilitating and stabilising movements of hypoblast cells in the germ ring, thereby pointing at a novel function of the slb/wnt11 signalling pathway for the regulation of migratory cell movements at early stages of gastrulation. AU - Ulrich, Florian AU - Concha, Miguel AU - Heid, Paul AU - Voss, Ed AU - Witzel, Sabine AU - Roehl, Henry AU - Tada, Masazumi AU - Wilson, Stephen AU - Adams, Richard AU - Soll, David AU - Heisenberg, Carl-Philipp J ID - 4146 IS - 22 JF - Development SN - 1011-6370 TI - Slb/Wnt11 controls hypoblast cell migration and morphogenesis at the onset of zebrafish gastrulation VL - 130 ER - TY - JOUR AB - Background: During vertebrate gastrulation, cell polarization and migration are core components in the cellular rearrangements that lead to the formation of the three germ layers, ectoderm, mesoderm, and endoderm. Previous studies have implicated the Wnt/planar cell polarity (PCP) signaling pathway in controlling cell morphology and movement during gastrulation. However, cell polarization and directed cell migration are reduced but not completely abolished in the absence of Wnt/PCP signals; this observation indicates that other signaling pathways must be involved. Results: We show that Phosphoinositide 3-Kinases (PI3Ks) are required at the onset of zebrafish gastrulation in mesendodermal cells for process formation and cell polarization. Platelet Derived Growth Factor (PDGF) functions upstream of PI3K, while Protein Kinase B (PKB), a downstream effector of PI3K activity, localizes to the leading edge of migrating mesendodermal cells. In the absence of PI3K activity, PKB localization and cell polarization are strongly reduced in mesendodermal cells and are followed by slower but still highly coordinated and directed movements of these cells. Conclusions: We have identified a novel role of a signaling pathway comprised of PDGF, PI3K, and PKB in the control of morphogenetic cell movements during gastrulation. Furthermore, our findings provide insight into the relationship between cell polarization and directed cell migration at the onset of zebrafish gastrulation. AU - Montero, Juan AU - Kilian, Beate AU - Chan, Joanne AU - Bayliss, Peter AU - Heisenberg, Carl-Philipp J ID - 4169 IS - 15 JF - Current Biology SN - 0960-9822 TI - Phosphoinositide 3-kinase is required for process outgrowth and cell polarization of gastrulating mesendodermal cells VL - 13 ER - TY - JOUR AB - Wnt genes play important roles in regulating patterning and morphogenesis during vertebrate gastrulation. In zebrafish, slb/wnt11 is required for convergence and extension movements, but not cell fate specification during gastrulation. To determine if other Wnt genes functionally interact with slb/wnt11, we analysed the role of ppt/wnt5 during zebrafish gastrulation. ppt/wnt5 is maternally provided and zygotically expressed at all stages during gastrulation. The analysis of ppt mutant embryos reveals that Ppt/Wnt5 regulates cell elongation and convergent extension movements in posterior regions of the gastrula, while its function in more anterior regions is largely redundant to that of Slb/Wnt11. Frizzled-2 functions downstream of ppt/wnt5, indicating that it might act as a receptor for Ppt/Wnt5 in this process. The characterisation of the role of Ppt/Wnt5 provides insight into the functional diversity of Wnt genes in regulating vertebrate gastrulation movements. (C) 2003 Elsevier Science Ireland Ltd. All rights reserved. AU - Kilian, Beate AU - Mansukoski, Hannu AU - Barbosa, Filipa AU - Ulrich, Florian AU - Tada, Masazumi AU - Heisenberg, Carl-Philipp J ID - 4185 IS - 4 JF - Mechanisms of Development SN - 0925-4773 TI - The role of Ppt/Wnt5 in regulating cell shape and movement during zebrafish gastrulation VL - 120 ER - TY - JOUR AB - Computing the volume occupied by individual atoms in macromolecular structures has been the subject of research for several decades. This interest has grown in the recent years, because weighted volumes are widely used in implicit solvent models. Applications of the latter in molecular mechanics simulations require that the derivatives of these weighted volumes be known. In this article, we give a formula for the volume derivative of a molecule modeled as a space-filling diagram made up of balls in motion. The formula is given in terms of the weights, radii, and distances between the centers as well as the sizes of the facets of the power diagram restricted to the space-filling diagram. Special attention is given to the detection and treatment of singularities as well as discontinuities of the derivative. AU - Edelsbrunner, Herbert AU - Koehl, Patrice ID - 3992 IS - 5 JF - PNAS SN - 0027-8424 TI - The weighted-volume derivative of a space-filling diagram VL - 100 ER - TY - CONF AB - We introduce relaxed scheduling as a paradigm for mesh maintenance and demonstrate its applicability to triangulating a skin surface in R-3. AU - Edelsbrunner, Herbert AU - Üngör, Alper ID - 3999 SN - 9783540207764 T2 - Proceedings of the Japanese Conference on Discrete and Computational Geometry TI - Relaxed scheduling in dynamic skin triangulation VL - 2866 ER - TY - CONF AB - We combine topological and geometric methods to construct a multi-resolution data structure for functions over two-dimensional domains. Starting with the Morse-Smale complex, we construct a topological hierarchy by progressively canceling critical points in pairs. Concurrently, we create a geometric hierarchy by adapting the geometry to the changes in topology. The data structure supports mesh traversal operations similarly to traditional multi-resolution representations. AU - Bremer, Peer AU - Edelsbrunner, Herbert AU - Hamann, Bernd AU - Pascucci, Valerio ID - 3997 SN - 0780381203 T2 - Proceedings of the 14th IEEE Conference on Visualization TI - A multi-resolution data structure for two-dimensional Morse-Smale functions ER - TY - JOUR AB - Recent studies show that signaling through integrin receptors is required for normal cell movements during Xenopus gastrulation. Integrins function in this process by modulating the activity of cadherin adhesion molecules within tissues undergoing convergence and extension movements. AU - Montero, Juan AU - Heisenberg, Carl-Philipp J ID - 4168 IS - 2 JF - Developmental Cell SN - 1534-5807 TI - Adhesive crosstalk in gastrulation VL - 5 ER - TY - CHAP AB - We give analytic inclusion-exclusion formulas for the area and perimeter derivatives of a union of finitely many disks in the plane. AU - Cheng, Ho AU - Edelsbrunner, Herbert ID - 3991 SN - 9783540005797 T2 - Computer Science in Perspective: Essays Dedicated to Thomas Ottmann TI - Area and perimeter derivatives of a union of disks VL - 2598 ER -