{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume542","volumeNumber":4351,"name":"Dagstuhl Seminar Proceedings, Volume 4351","dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume542"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article909","name":"04351 Abstracts Collection \u2013 Spatial Representation: Discrete vs. Continuous Computational Models","abstract":"From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351\r\n``Spatial Representation: Discrete vs. Continuous Computational Models''\r\nwas held in the International Conference and Research Center (IBFI),\r\nSchloss Dagstuhl.\r\nDuring the seminar, several participants presented their current\r\nresearch, and ongoing work and open problems were discussed. Abstracts of\r\nthe presentations given during the seminar as well as abstracts of\r\nseminar results and ideas are put together in this paper. The first section\r\ndescribes the seminar topics and goals in general.\r\nLinks to extended abstracts or full papers are provided, if available.","keywords":["Domain theory","formal topology","constructive topology","domain representation","space-time","quantum gravity","inverse limit construction","matroid geometry","descriptive set theory","Borel hierarchy","Hausdorff difference hierarchy","Wadge degree partial metric","fractafold","region geometry","oriented projective geometry"],"author":[{"@type":"Person","name":"Kopperman, Ralph","givenName":"Ralph","familyName":"Kopperman"},{"@type":"Person","name":"Panangaden, Prakash","givenName":"Prakash","familyName":"Panangaden"},{"@type":"Person","name":"Smyth, Michael B.","givenName":"Michael B.","familyName":"Smyth"},{"@type":"Person","name":"Spreen, Dieter","givenName":"Dieter","familyName":"Spreen"},{"@type":"Person","name":"Webster, Julian","givenName":"Julian","familyName":"Webster"}],"position":1,"pageStart":1,"pageEnd":24,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kopperman, Ralph","givenName":"Ralph","familyName":"Kopperman"},{"@type":"Person","name":"Panangaden, Prakash","givenName":"Prakash","familyName":"Panangaden"},{"@type":"Person","name":"Smyth, Michael B.","givenName":"Michael B.","familyName":"Smyth"},{"@type":"Person","name":"Spreen, Dieter","givenName":"Dieter","familyName":"Spreen"},{"@type":"Person","name":"Webster, Julian","givenName":"Julian","familyName":"Webster"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article910","name":"04351 Summary \u2013 Spatial Representation: Discrete vs. Continuous Computational Models","abstract":"Topological notions and methods are used in various areas of the physical sciences and engineering, and therefore computer processing of topological data is important. Separate from this, but closely related, are computer science uses of topology: applications to programming language semantics and computing with exact real numbers are important examples. The seminar concentrated on an important approach, which is basic to all these applications, i.e. spatial representation.","keywords":["Domain theory","formal topology","constructive topology","domain representation","space-time","quantum gravity","inverse limit construction","matroid geometry","descriptive set theory","Borel hierarchy","Hausdorff difference hierarchy","Wadge degree","partial metric","fractafold","region geometry"],"author":[{"@type":"Person","name":"Kopperman, Ralph","givenName":"Ralph","familyName":"Kopperman"},{"@type":"Person","name":"Panangaden, Prakash","givenName":"Prakash","familyName":"Panangaden"},{"@type":"Person","name":"Smyth, Michael B.","givenName":"Michael B.","familyName":"Smyth"},{"@type":"Person","name":"Spreen, Dieter","givenName":"Dieter","familyName":"Spreen"},{"@type":"Person","name":"Webster, Julian","givenName":"Julian","familyName":"Webster"}],"position":2,"pageStart":1,"pageEnd":5,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kopperman, Ralph","givenName":"Ralph","familyName":"Kopperman"},{"@type":"Person","name":"Panangaden, Prakash","givenName":"Prakash","familyName":"Panangaden"},{"@type":"Person","name":"Smyth, Michael B.","givenName":"Michael B.","familyName":"Smyth"},{"@type":"Person","name":"Spreen, Dieter","givenName":"Dieter","familyName":"Spreen"},{"@type":"Person","name":"Webster, Julian","givenName":"Julian","familyName":"Webster"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article911","name":"A Cartesian Closed Extension of the Category of Locales","abstract":"We present a Cartesian closed category ELOC of equilocales,\r\nwhich contains the category LOC of locales as a reflective full subcategory.\r\nThe embedding of LOC into ELOC preserves products and all exponentials of exponentiable locales.","keywords":["Locale","Cartesian closed category"],"author":{"@type":"Person","name":"Heckmann, Reinhold","givenName":"Reinhold","familyName":"Heckmann"},"position":3,"pageStart":1,"pageEnd":20,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Heckmann, Reinhold","givenName":"Reinhold","familyName":"Heckmann"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article912","name":"A Category of Discrete Closure Spaces","abstract":"Discrete systems such as sets, monoids, groups are familiar categories. \r\nThe internal strucutre of the latter two is defined by an algebraic operator. \r\nIn this paper we describe the internal structure of the base set by a closure operator. We illustrate the role of such closure in convex geometries and partially ordered sets and thus suggestthe wide applicability of closure systems.\r\nNext we develop the ideas of closed and complete functions over closure spaces. These can be used to establish criteria for asserting \r\nthat \"the closure of a functional image under $f$ is equal to the functional image of the closure\". Functions with these properties can be treated as categorical morphisms. Finally, the category \"CSystem\" of closure systems is shown to be cartesian closed.","keywords":["Category","closure","antimatroid","function"],"author":{"@type":"Person","name":"Pfaltz, John L.","givenName":"John L.","familyName":"Pfaltz"},"position":4,"pageStart":1,"pageEnd":16,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Pfaltz, John L.","givenName":"John L.","familyName":"Pfaltz"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.4","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article913","name":"A domain of spacetime intervals in general relativity","abstract":"We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. This implies that from only a countable\r\ndense set of events and the causality relation, it\r\nis possible to reconstruct a globally hyperbolic\r\nspacetime in a purely order theoretic manner. The\r\nultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains.\r\n\r\nWe obtain a mathematical setting in which one\r\ncan study causality independently of geometry\r\nand differentiable structure, and which also\r\nsuggests that spacetime emanates from\r\nsomething discrete.","keywords":["Causality","spacetime","global hyperbolicity","interval domains","bicontinuous posets","spacetime topology"],"author":[{"@type":"Person","name":"Martin, Keye","givenName":"Keye","familyName":"Martin"},{"@type":"Person","name":"Panangaden, Prakash","givenName":"Prakash","familyName":"Panangaden"}],"position":5,"pageStart":1,"pageEnd":28,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Martin, Keye","givenName":"Keye","familyName":"Martin"},{"@type":"Person","name":"Panangaden, Prakash","givenName":"Prakash","familyName":"Panangaden"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article914","name":"A geometry of information, I: Nerves, posets and differential forms","abstract":"The main theme of this workshop is 'Spatial Representation: Continuous vs. Discrete'. Spatial representation has two contrasting but interacting aspects (i) representation \\emph{of} spaces' and (ii) representation \\emph{by} spaces. In this paper we will examine two aspects that are common to both interpretations of the theme, namely nerve constructions and refinement. Representations change, data changes, spaces change. We will examine the possibility of a 'differential geometry' of spatial representations of both types, and in the sequel give an algebra of differential forms that has the potential to handle the dynamical aspect of such a geometry. We will discuss briefly a conjectured class of spaces, generalising the Cantor set which would seem ideal as a test-bed for the set of tools we are developing.","keywords":["Chu spaces","nerves","differential forms"],"author":[{"@type":"Person","name":"Gratus, Jonathan","givenName":"Jonathan","familyName":"Gratus"},{"@type":"Person","name":"Porter, Timothy","givenName":"Timothy","familyName":"Porter"}],"position":6,"pageStart":1,"pageEnd":0,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Gratus, Jonathan","givenName":"Jonathan","familyName":"Gratus"},{"@type":"Person","name":"Porter, Timothy","givenName":"Timothy","familyName":"Porter"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article915","name":"A geometry of information, II: Sorkin models, and biextensional collapses","abstract":"In this second part of our contribution to the workshop, we look in more detail at the Sorkin model, its relationship to constructions in Chu space theory, and then compare it with the Nerve constructions given in the first part.","keywords":["Chu space","Sorkin model","Nerve"],"author":[{"@type":"Person","name":"Gratus, Jonathan","givenName":"Jonathan","familyName":"Gratus"},{"@type":"Person","name":"Porter, Timothy","givenName":"Timothy","familyName":"Porter"}],"position":7,"pageStart":1,"pageEnd":0,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Gratus, Jonathan","givenName":"Jonathan","familyName":"Gratus"},{"@type":"Person","name":"Porter, Timothy","givenName":"Timothy","familyName":"Porter"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article916","name":"Auxiliary relations and sandwich theorems","abstract":"A well-known topological theorem due to Kat\\v etov states:\r\n\r\nSuppose $(X,\\tau)$ is a normal topological space, and let $f:X\\to[0,1]$ be upper semicontinuous, $g:X\\to[0,1]$ be lower semicontinuous, and $f\\leq g$. Then there is a continuous $h:X\\to[0,1]$ such that $f\\leq h\\leq g$.\r\n\r\nWe show a version of this theorem for many posets with auxiliary relations. In particular, if $P$ is a Scott domain and $f,g:P\\to[0,1]$ are such that $f\\leq g$, and $f$ is lower continuous and $g$ Scott continuous, then for some $h$, $f\\leq h\\leq g$ and $h$ is both Scott and lower continuous.\r\n\r\nAs a result, each Scott continuous function from $P$ to $[0,1]$, is the sup of the functions below it which are both Scott and lower continuous.","keywords":["Adjoint","auxiliary relation","continuous poset","pairwise completely regular (and pairwise normal) bitopological space","upper (lower) semicontinuous Urysohn relation"],"author":[{"@type":"Person","name":"God, Chris","givenName":"Chris","familyName":"God"},{"@type":"Person","name":"Jung, Achim","givenName":"Achim","familyName":"Jung"},{"@type":"Person","name":"Knight, Robin","givenName":"Robin","familyName":"Knight"},{"@type":"Person","name":"Kopperman, Ralph","givenName":"Ralph","familyName":"Kopperman"}],"position":8,"pageStart":1,"pageEnd":4,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"God, Chris","givenName":"Chris","familyName":"God"},{"@type":"Person","name":"Jung, Achim","givenName":"Achim","familyName":"Jung"},{"@type":"Person","name":"Knight, Robin","givenName":"Robin","familyName":"Knight"},{"@type":"Person","name":"Kopperman, Ralph","givenName":"Ralph","familyName":"Kopperman"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.8","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article917","name":"Compactness in apartness spaces?","abstract":"A major problem in the constructive theory of apartness spaces is that of finding a good notion of compactness. Such a notion should (i) reduce to ``complete plus totally bounded'' for uniform spaces and (ii) classically be equivalent to the usual Heine-Borel-Lebesgue property for the apartness topology. The constructive counterpart of the smallest uniform structure compatible with a given apartness, while not constructively a uniform structure, offers a possible solution to the compactness-definition problem. That counterpart turns out to be interesting in its own right, and reveals some additional properties of an apartness that may have uses elsewhere in the theory.","keywords":["Apartness","constructive","compact uniform space"],"author":[{"@type":"Person","name":"Bridges, Douglas","givenName":"Douglas","familyName":"Bridges"},{"@type":"Person","name":"Ishihara, Hajime","givenName":"Hajime","familyName":"Ishihara"},{"@type":"Person","name":"Schuster, Peter","givenName":"Peter","familyName":"Schuster"},{"@type":"Person","name":"Vita, Luminita S.","givenName":"Luminita S.","familyName":"Vita"}],"position":9,"pageStart":1,"pageEnd":7,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bridges, Douglas","givenName":"Douglas","familyName":"Bridges"},{"@type":"Person","name":"Ishihara, Hajime","givenName":"Hajime","familyName":"Ishihara"},{"@type":"Person","name":"Schuster, Peter","givenName":"Peter","familyName":"Schuster"},{"@type":"Person","name":"Vita, Luminita S.","givenName":"Luminita S.","familyName":"Vita"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.9","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article918","name":"Continued Radicals","abstract":"A nested radical with terms $a_1, a_2, \\ldots , a_N$ is an expression of form $\\sqrt{a_N + \\cdots + \\sqrt{a_2 + \\sqrt{a_1}}}$. The limit \r\nas $N$ approaches infinity of such an expression, if it exists, \r\nis called a continued radical. We consider the set of real\r\nnumbers $S(M)$ representable as a continued radical whose terms $a_1, a_2, \\ldots$ are all from a finite set $M$ of nonnegative real numbers. We give conditions on the set $M$ for $S(M)$ to be (a) an interval, and (b) homeomorphic to the Cantor set.","keywords":"Continued radical","author":[{"@type":"Person","name":"Johnson, Jamie","givenName":"Jamie","familyName":"Johnson"},{"@type":"Person","name":"Richmond, Tom","givenName":"Tom","familyName":"Richmond"}],"position":10,"pageStart":1,"pageEnd":4,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Johnson, Jamie","givenName":"Jamie","familyName":"Johnson"},{"@type":"Person","name":"Richmond, Tom","givenName":"Tom","familyName":"Richmond"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.10","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article919","name":"Continuous Semantics for Termination Proofs","abstract":"We prove a general strong normalization theorem for higher type\r\nrewrite systems based on Tait's strong computability predicates and a\r\nstrictly continuous domain-theoretic semantics. The theorem applies to\r\nextensions of Goedel's system $T$, but also to various forms of bar\r\nrecursion for which termination was hitherto unknown.","keywords":["Higher-order term rewriting","termination","domain theory"],"author":{"@type":"Person","name":"Berger, Ulrich","givenName":"Ulrich","familyName":"Berger"},"position":11,"pageStart":1,"pageEnd":19,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Berger, Ulrich","givenName":"Ulrich","familyName":"Berger"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.11","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article920","name":"Deadlocks and Dihomotopy in Mutual Exclusion Models","abstract":"Parallel processes in concurrency theory can be modelled in a geometric framework. A convenient model are the Higher Dimensional Automata of V. Pratt and E. Goubault with cubical complexes as their mathematical description. More abstract models are given by (locally) partially ordered topological spaces, the directed ($d$-spaces) of\r\nM.Grandis and the flows of P. Gaucher. All models invite to use or modify ideas from algebraic topology, notably homotopy.\r\n\r\nIn specific semaphore models for mutual exclusion, we have developed methods and algorithms that can detect deadlocks and unsafe regions and give information about essentially different schedules using higher dimensional ``geometric'' representations of the state space and executions (directed paths) along it.","keywords":["Mutual exclusion","deadlock detection","dihomotopy"],"author":{"@type":"Person","name":"Raussen, Martin","givenName":"Martin","familyName":"Raussen"},"position":12,"pageStart":1,"pageEnd":8,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Raussen, Martin","givenName":"Martin","familyName":"Raussen"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.12","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article921","name":"Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again","abstract":"The geometric models of concurrency - Dijkstra's PV-models and V. Pratt's Higher Dimensional Automata - \r\nrely on a translation of discrete or algebraic information to geometry. \r\nIn both these cases, the translation is the geometric realisation of a semi cubical complex, \r\nwhich is then a locally partially ordered space, an lpo space. \r\nThe aim is to use the algebraic topology machinery, suitably adapted to the fact \r\nthat there is a preferred time direction. \r\nThen the results - for instance dihomotopy classes of dipaths, which model \r\nthe number of inequivalent computations should be used on the discrete model and give the corresponding discrete objects.\r\n \r\nWe prove that this is in fact the case for the models considered: \r\nEach dipath is dihomottopic to a combinatorial dipath \r\nand if two combinatorial dipaths are dihomotopic, then they are combinatorially equivalent.\r\n Moreover, the notions of dihomotopy (LF., E. Goubault, M. Raussen) \r\nand d-homotopy (M. Grandis) are proven to be equivalent for these models \r\n- hence the Van Kampen theorem is available for dihomotopy.\r\n\r\nFinally we give an idea of how many spaces have a local po-structure given by cubes.\r\n The answer is, that any cubicalized space has such a structure \r\nafter at most one subdivision. \r\nIn particular, all triangulable spaces have a cubical local po-structure.","keywords":["Cubical Complex","Higher Dimensional Automaton","Ditopology"],"author":{"@type":"Person","name":"Fajstrup, Lisbeth","givenName":"Lisbeth","familyName":"Fajstrup"},"position":13,"pageStart":1,"pageEnd":3,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Fajstrup, Lisbeth","givenName":"Lisbeth","familyName":"Fajstrup"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.13","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article922","name":"Discrete classical vs. continuous quantum data in abstract quantum mechanics","abstract":"``Quantum'' stands for for the concepts (both operational and formal)\r\nwhich had to be added to classical physics in order to understand\r\notherwise unexplainable observed phenomena such as the structure of\r\nthe spectral lines in atomic spectra. While the basic part of\r\nclassical mechanics deals with the (essentially) reversible\r\ndynamics, quantum required adding the notions of ``measurement'' and\r\n(possibly non-local) ``correlations'' to the discussion. Crucially,\r\nall this comes with a ``probabilistic calculus''. The corresponding\r\nmathematical formalism was considered to have reached maturity in\r\n[von Neumann 1932], but there are some manifest problems with that\r\nformalism:\r\n\r\n(i) While measurements are applied to physical systems, application\r\nof their formal counterpart (i.e. a self-adjoint linear operator) to\r\nthe vector representing that state of the system in no way reflects\r\nhow the state changes during the act of measurement. Analogously,\r\nthe composite of two self-adjoint operators has no physical\r\nsignificance while in practice measurements can be effectuated\r\nsequentially. More generally, the formal types in von Neumann's\r\nformalism do not reflect the nature of the corresponding underlying\r\nconcept at all!\r\n\r\n(ii) Part of the problem regarding the measurements discussed above\r\nis that in the von Neumann formalism there is no place for storage,\r\nmanipulation and exchange of the classical data obtained from\r\nmeasurements. Protocols such as quantum teleportation involving\r\nthese cannot be given a full formal description.\r\n\r\n(iii) The behavioral properties of quantum entanglement which for\r\nexample enable continuous data exchange using only finitary\r\ncommunication are hidden in the formalism.\r\n\r\nIn [Abramsky and Coecke 2004] we addressed all these problems, and in\r\naddition provided a purely categorical axiomatization of quantum\r\nmechanics. The concepts of the abstract quantum mechanics are\r\nformulated relative to a strongly compact closed category with\r\nbiproducts (of which the category FdHilb of finite dimensional\r\nHilbert spaces and linear maps is an example). Preparations,\r\nmeasurements, either destructive or not, classical data exchange are\r\nall morphisms in that category, and their types fully reflect their\r\nkinds. Correctness properties of standard quantum protocols can be\r\nabstractly proven.\r\n\r\nSurprisingly, in this seemingly purely qualitative setting even the\r\nquantitative Born rule arises, that is the rule which tells you how\r\nto calculate the probabilities. Indeed, each such category has as\r\nendomorphism Hom of the tensor unit an abelian semiring of\r\n`scalars', and a special subset of these scalars will play the role\r\nof weights: each scalar induces a natural transformation which\r\npropagates through physical processes, and when a `state' undergoes\r\na `measurement', the composition of the corresponding morphisms\r\ngives rise to the weight. Hence the probabilistic weights live\r\nwithin the category of processes.\r\n\r\nJ. von Neumann. Mathematische Grundlagen der Quantenmechanik.\r\nSpringer-Verlag (1932). English translation in Mathematical\r\nFoundations of Quantum Mechanics. Princeton University Press (1955).\r\n\r\nS. Abramsky and B. Coecke. A categorical semantics of quantum\r\nprotocols. In the proceedings of LiCS'04 (2004). An extended version\r\nis available at arXiv:quant-ph\/0402130 A more reader friendly\r\nversion entitled `Quantum information flow, concretely, abstractly'\r\nis at http:\/\/www.vub.ac.be\/CLEA\/Bob\/Papers\/QPL.pdf","keywords":["Category theory","strong compact closure","quantum information-flow"],"author":[{"@type":"Person","name":"Abramsky, Samson","givenName":"Samson","familyName":"Abramsky"},{"@type":"Person","name":"Coecke, Bob","givenName":"Bob","familyName":"Coecke"}],"position":14,"pageStart":1,"pageEnd":21,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Abramsky, Samson","givenName":"Samson","familyName":"Abramsky"},{"@type":"Person","name":"Coecke, Bob","givenName":"Bob","familyName":"Coecke"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.14","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article923","name":"Dyadic Subbases and Representations of Topological Spaces","abstract":"We explain topological properties of the embedding-based approach to\r\ncomputability on topological spaces. With this approach, he considered\r\na special kind of embedding of a topological space into Plotkin's\r\n$T^\\omega$, which is the set of infinite sequences of $T = \\{0,1,\\bot \\}$.\r\nWe show that such an embedding can also be characterized by a dyadic\r\nsubbase, which is a countable subbase $S = (S_0^0, S_0^1, S_1^0, S_1^1, \\ldots)$ such that $S_n^j$ $(n = 0,1,2,\\ldots; j = 0,1$ are regular open\r\nand $S_n^0$ and $S_n^1$ are exteriors of each other. We survey properties\r\nof dyadic subbases which are related to efficiency properties of the\r\nrepresentation corresponding to the embedding.","keywords":["Dyadic subbase","embedding","computation over topological spaces","Plotkin's $T^\\omega$"],"author":{"@type":"Person","name":"Tsuiki, Hideki","givenName":"Hideki","familyName":"Tsuiki"},"position":15,"pageStart":1,"pageEnd":8,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Tsuiki, Hideki","givenName":"Hideki","familyName":"Tsuiki"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.15","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article924","name":"Integrating Topology and Geometry for Macro-Molecular Simulations","abstract":"Emerging macro-molecular simulations, such as supercoiling \r\nof DNA and protein unfolding, have an opportunity to profit from \r\ntwo decades of experience with geometric models within computer-aided\r\ngeometric design (CAGD). For CAGD, static models are often sufficient,\r\nwhile form and function are inextricably related in biochemistry, resulting\r\nin greater attention to critical topological characteristics of these\r\ndynamic models. The greater emphasis upon dynamic change in macro-molecular\r\nsimulations imposes increased demands for faithful integration\r\nof topology and geometry, as well as much stricter requirements\r\nfor computational efficiency. This article presents transitions from the\r\nCAGD domain to meet the greater fidelity and performance demands \r\nfor macro-molecular simulations.","keywords":["Computational topology","spline","approximation"],"author":[{"@type":"Person","name":"Moore, Edward L. F.","givenName":"Edward L. F.","familyName":"Moore"},{"@type":"Person","name":"Peters, Thomas J.","givenName":"Thomas J.","familyName":"Peters"},{"@type":"Person","name":"Ferguson, David R.","givenName":"David R.","familyName":"Ferguson"},{"@type":"Person","name":"Stewart, Neil F.","givenName":"Neil F.","familyName":"Stewart"}],"position":16,"pageStart":1,"pageEnd":0,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Moore, Edward L. F.","givenName":"Edward L. F.","familyName":"Moore"},{"@type":"Person","name":"Peters, Thomas J.","givenName":"Thomas J.","familyName":"Peters"},{"@type":"Person","name":"Ferguson, David R.","givenName":"David R.","familyName":"Ferguson"},{"@type":"Person","name":"Stewart, Neil F.","givenName":"Neil F.","familyName":"Stewart"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.16","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article925","name":"On Maximality of Compact Topologies","abstract":"Using some advanced properties of the de Groot dual and some generalization of the Hofmann-Mislove theorem, we solve in the positive the question of D. E. Cameron: Is every compact topology contained in some maximal compact topology?","keywords":["de Groot dual","compact saturated set","wide Scott open filter","maximal compact topology"],"author":{"@type":"Person","name":"Kovar, Martin","givenName":"Martin","familyName":"Kovar"},"position":17,"pageStart":1,"pageEnd":10,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Kovar, Martin","givenName":"Martin","familyName":"Kovar"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.17","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article926","name":"The Construction of Finer Compact Topologies","abstract":"It is well known that each locally compact strongly sober topology is contained in a compact Hausdorff topology; just take the supremum of its topology with its dual topology. On the other hand, examples of compact topologies are known that do not have a finer compact Hausdorff topology.\r\nThis led to the question (first explicitly formulated by D.E. Cameron) whether each compact topology is contained in a compact topology with respect to which all compact sets are closed. (For the obvious reason these spaces are called maximal compact in the literature.)\r\nWhile this major problem remains open, we present several partial solutions to the question in our talk. For instance we show that each compact topology is contained in a compact topology with respect to which convergent sequences have unique limits. In fact each compact topology is contained in a compact topology with respect to which countable compact sets are closed. Furthermore we note that each compact sober T_1-topology is contained in a maximal compact topology and that each sober compact T_1-topology which is locally compact or sequential is the infimum of a family of maximal compact topologies.","keywords":["Maximal compact","KC-space","sober","US-space","locally compact","sequential","sequentially compact"],"author":[{"@type":"Person","name":"K\u00fcnzi, Hans-Peter A.","givenName":"Hans-Peter A.","familyName":"K\u00fcnzi"},{"@type":"Person","name":"Zypen, Dominic van der","givenName":"Dominic van der","familyName":"Zypen"}],"position":18,"pageStart":1,"pageEnd":5,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"K\u00fcnzi, Hans-Peter A.","givenName":"Hans-Peter A.","familyName":"K\u00fcnzi"},{"@type":"Person","name":"Zypen, Dominic van der","givenName":"Dominic van der","familyName":"Zypen"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.18","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article927","name":"The de Groot dual for general collections of sets","abstract":"A topology is de Groot dual of another topology, if it has a closed base consisting of all its compact saturated sets. Until 2001 it was an unsolved problem of J. Lawson and M. Mislove whether the sequence of iterated dualizations of a topological space is finite. In this paper we generalize the author's original construction to an arbitrary family instead of a topology. Among other results we prove that for any family $\\C\\subseteq 2^X$ it holds $\\C^{dd}=\\C^{dddd}$. We also show similar identities for some other similar and topology-related structures.","keywords":["Saturated set","dual topology","compactness operator"],"author":{"@type":"Person","name":"Kovar, Martin","givenName":"Martin","familyName":"Kovar"},"position":19,"pageStart":1,"pageEnd":8,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Kovar, Martin","givenName":"Martin","familyName":"Kovar"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.19","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article928","name":"The Hofmann-Mislove Theorem for general posets","abstract":"In this paper we attempt to find and investigate the most general class of posets which satisfy a properly generalized version of the Hofmann-Mislove theorem. For that purpose, we generalize and study some notions (like compactness, the Scott topology, Scott open filters, prime elements, the spectrum etc.), and adjust them for use in general posets. Then we characterize the posets satisfying the Hofmann-Mislove theorem by the relationship between the generalized Scott closed prime subsets and the generalized prime elements of the poset. The theory become classic for distributive lattices. Remark that the topologies induced on the generalized spectra in general need not be\r\nsober.","keywords":["Posets","generalized Scott topology","Scott open filters","(filtered) compactness","saturated"],"author":{"@type":"Person","name":"Kovar, Martin","givenName":"Martin","familyName":"Kovar"},"position":20,"pageStart":1,"pageEnd":16,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Kovar, Martin","givenName":"Martin","familyName":"Kovar"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.20","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article929","name":"The Hofmann-Mislove Theorem for general topological structures","abstract":"In this paper we prove a modification of Hofmann-Mislove theorem for a topological structure similar to the minusspaces of de Groot, in which the empty set \"need not be open\". This will extend, in a slightly relaxed form, the validity of the classical Hofmann-Mislove theorem also to some of those spaces, whose underlying topology need not be (quasi-) sober.","keywords":["Compact saturated set","Scott open filter","(quasi-) sober space"],"author":{"@type":"Person","name":"Kovar, Martin","givenName":"Martin","familyName":"Kovar"},"position":21,"pageStart":1,"pageEnd":9,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Kovar, Martin","givenName":"Martin","familyName":"Kovar"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.21","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"},{"@type":"ScholarlyArticle","@id":"#article930","name":"What do partial metrics represent?","abstract":"Partial metrics were introduced in 1992\r\nas a metric to allow the distance of a point from\r\nitself to be non zero. This notion of self distance, designed to extend\r\nmetrical concepts to Scott topologies as used\r\nin computing, has little intuition for the mainstream Hausdorff topologist.\r\nThe talk will show that a partial metric over a set can be represented by a metric over that set with a so-called 'base point'.\r\nThus we establish that a partial metric is essentially a structure combining both a metric space and a skewed view of that space from the base point. From this we can deduce what it is that partial metrics are really all about.","keywords":["Metric","partial metric","base point"],"author":[{"@type":"Person","name":"Kopperman, Ralph","givenName":"Ralph","familyName":"Kopperman"},{"@type":"Person","name":"Matthews, Steve","givenName":"Steve","familyName":"Matthews"},{"@type":"Person","name":"Pajoohesh, Homeira","givenName":"Homeira","familyName":"Pajoohesh"}],"position":22,"pageStart":1,"pageEnd":4,"dateCreated":"2005-04-22","datePublished":"2005-04-22","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kopperman, Ralph","givenName":"Ralph","familyName":"Kopperman"},{"@type":"Person","name":"Matthews, Steve","givenName":"Steve","familyName":"Matthews"},{"@type":"Person","name":"Pajoohesh, Homeira","givenName":"Homeira","familyName":"Pajoohesh"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04351.22","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume542"}]}