17,747 research outputs found

    Distributive Lattices, Polyhedra, and Generalized Flow

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    A D-polyhedron is a polyhedron PP such that if x,yx,y are in PP then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every D-polyhedron corresponds to a directed graph with arc-parameters, such that every point in the polyhedron corresponds to a vertex potential on the graph. Alternatively, an edge-based description of the point set can be given. The objects in this model are dual to generalized flows, i.e., dual to flows with gains and losses. These models can be specialized to yield some cases of distributive lattices that have been studied previously. Particular specializations are: lattices of flows of planar digraphs (Khuller, Naor and Klein), of α\alpha-orientations of planar graphs (Felsner), of c-orientations (Propp) and of Δ\Delta-bonds of digraphs (Felsner and Knauer). As an additional application we exhibit a distributive lattice structure on generalized flow of breakeven planar digraphs.Comment: 17 pages, 3 figure

    Distributive Laws and Decidable Properties of SOS Specifications

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    Some formats of well-behaved operational specifications, correspond to natural transformations of certain types (for example, GSOS and coGSOS laws). These transformations have a common generalization: distributive laws of monads over comonads. We prove that this elegant theoretical generalization has limited practical benefits: it does not translate to any concrete rule format that would be complete for specifications that contain both GSOS and coGSOS rules. This is shown for the case of labeled transition systems and deterministic stream systems.Comment: In Proceedings EXPRESS/SOS 2014, arXiv:1408.127

    The projective geometry of a group

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    We show that the pair given by the power set and by the "Grassmannian"(set of all subgroups) of an arbitrary group behaves very much like the pair given by a projective space and its dual projective space. More precisely, we generalize several results from preceding joint work with M. Kinyon (arXiv:0903.5441), which concerned abelian groups, to the case of general non-abelian groups. Most notably, pairs of subgroups parametrize torsor and semitorsor structures on the power set. The r\^ole of associative algebras and -pairs from loc. cit. is now taken by analogs of near-rings

    A discussion on the origin of quantum probabilities

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    We study the origin of quantum probabilities as arising from non-boolean propositional-operational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorvian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).Comment: Improved versio

    Coalgebraic Behavioral Metrics

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    We study different behavioral metrics, such as those arising from both branching and linear-time semantics, in a coalgebraic setting. Given a coalgebra α ⁣:XHX\alpha\colon X \to HX for a functor H ⁣:SetSetH \colon \mathrm{Set}\to \mathrm{Set}, we define a framework for deriving pseudometrics on XX which measure the behavioral distance of states. A crucial step is the lifting of the functor HH on Set\mathrm{Set} to a functor H\overline{H} on the category PMet\mathrm{PMet} of pseudometric spaces. We present two different approaches which can be viewed as generalizations of the Kantorovich and Wasserstein pseudometrics for probability measures. We show that the pseudometrics provided by the two approaches coincide on several natural examples, but in general they differ. If HH has a final coalgebra, every lifting H\overline{H} yields in a canonical way a behavioral distance which is usually branching-time, i.e., it generalizes bisimilarity. In order to model linear-time metrics (generalizing trace equivalences), we show sufficient conditions for lifting distributive laws and monads. These results enable us to employ the generalized powerset construction

    Distributivity and deformation of the reals from Tsallis entropy

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    We propose a one-parameter family \ Rq\mathbb{R}_q \ of deformations of the reals, which is motivated by the generalized additivity of the Tsallis entropy. We introduce a generalized multiplication which is distributive with respect to the generalized addition of the Tsallis entropy. These operations establish a one-parameter family of field isomorphisms \ τq\tau_q \ between \ R\mathbb{R} \ and \ Rq\mathbb{R}_q \ through which an absolute value on \ Rq\mathbb{R}_q \ is introduced. This turns out to be a quasisymmetric map, whose metric and measure-theoretical implications are pointed out.Comment: 16 pages, Standard LaTeX2e, To be published in Physica
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