45,409 research outputs found
Revisiting Numerical Pattern Mining with Formal Concept Analysis
In this paper, we investigate the problem of mining numerical data in the
framework of Formal Concept Analysis. The usual way is to use a scaling
procedure --transforming numerical attributes into binary ones-- leading either
to a loss of information or of efficiency, in particular w.r.t. the volume of
extracted patterns. By contrast, we propose to directly work on numerical data
in a more precise and efficient way, and we prove it. For that, the notions of
closed patterns, generators and equivalent classes are revisited in the
numerical context. Moreover, two original algorithms are proposed and used in
an evaluation involving real-world data, showing the predominance of the
present approach
Cech Closure Spaces: A Unified Framework for Discrete Homotopy
Motivated by constructions in topological data analysis and algebraic
combinatorics, we study homotopy theory on the category of Cech closure spaces
, the category whose objects are sets endowed with a Cech closure
operator and whose morphisms are the continuous maps between them. We introduce
new classes of Cech closure structures on metric spaces, graphs, and simplicial
complexes, and we show how each of these cases gives rise to an interesting
homotopy theory. In particular, we show that there exists a natural family of
Cech closure structures on metric spaces which produces a non-trivial homotopy
theory for finite metric spaces, i.e. point clouds, the spaces of interest in
topological data analysis. We then give a Cech closure structure to graphs and
simplicial complexes which may be used to construct a new combinatorial (as
opposed to topological) homotopy theory for each skeleton of those spaces. We
further show that there is a Seifert-van Kampen theorem for closure spaces, a
well-defined notion of persistent homotopy, and an associated interleaving
distance. As an illustration of the difference with the topological setting, we
calculate the fundamental group for the circle, `circular graphs', and the
wedge of circles endowed with different closure structures. Finally, we produce
a continuous map from the topological circle to `circular graphs' which, given
the appropriate closure structures, induces an isomorphism on the fundamental
groups.Comment: Incorporated referee comments, 41 page
Quantale Modules and their Operators, with Applications
The central topic of this work is the categories of modules over unital
quantales. The main categorical properties are established and a special class
of operators, called Q-module transforms, is defined. Such operators - that
turn out to be precisely the homomorphisms between free objects in those
categories - find concrete applications in two different branches of image
processing, namely fuzzy image compression and mathematical morphology
Between quantum logic and concurrency
We start from two closure operators defined on the elements of a special kind
of partially ordered sets, called causal nets. Causal nets are used to model
histories of concurrent processes, recording occurrences of local states and of
events. If every maximal chain (line) of such a partially ordered set meets
every maximal antichain (cut), then the two closure operators coincide, and
generate a complete orthomodular lattice. In this paper we recall that, for any
closed set in this lattice, every line meets either it or its orthocomplement
in the lattice, and show that to any line, a two-valued state on the lattice
can be associated. Starting from this result, we delineate a logical language
whose formulas are interpreted over closed sets of a causal net, where every
line induces an assignment of truth values to formulas. The resulting logic is
non-classical; we show that maximal antichains in a causal net are associated
to Boolean (hence "classical") substructures of the overall quantum logic.Comment: In Proceedings QPL 2012, arXiv:1407.842
Interval-based Synthesis
We introduce the synthesis problem for Halpern and Shoham's modal logic of
intervals extended with an equivalence relation over time points, abbreviated
HSeq. In analogy to the case of monadic second-order logic of one successor,
the considered synthesis problem receives as input an HSeq formula phi and a
finite set Sigma of propositional variables and temporal requests, and it
establishes whether or not, for all possible evaluations of elements in Sigma
in every interval structure, there exists an evaluation of the remaining
propositional variables and temporal requests such that the resulting structure
is a model for phi. We focus our attention on decidability of the synthesis
problem for some meaningful fragments of HSeq, whose modalities are drawn from
the set A (meets), Abar (met by), B (begins), Bbar (begun by), interpreted over
finite linear orders and natural numbers. We prove that the fragment ABBbareq
is decidable (non-primitive recursive hard), while the fragment AAbarBBbar
turns out to be undecidable. In addition, we show that even the synthesis
problem for ABBbar becomes undecidable if we replace finite linear orders by
natural numbers.Comment: In Proceedings GandALF 2014, arXiv:1408.556
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