31,245 research outputs found
Classical Structures Based on Unitaries
Starting from the observation that distinct notions of copying have arisen in
different categorical fields (logic and computation, contrasted with quantum
mechanics) this paper addresses the question of when, or whether, they may
coincide. Provided all definitions are strict in the categorical sense, we show
that this can never be the case. However, allowing for the defining axioms to
be taken up to canonical isomorphism, a close connection between the classical
structures of categorical quantum mechanics, and the categorical property of
self-similarity familiar from logical and computational models becomes
apparent.
The required canonical isomorphisms are non-trivial, and mix both typed
(multi-object) and untyped (single-object) tensors and structural isomorphisms;
we give coherence results that justify this approach.
We then give a class of examples where distinct self-similar structures at an
object determine distinct matrix representations of arrows, in the same way as
classical structures determine matrix representations in Hilbert space. We also
give analogues of familiar notions from linear algebra in this setting such as
changes of basis, and diagonalisation.Comment: 24 pages,7 diagram
BOOL-AN: A method for comparative sequence analysis and phylogenetic reconstruction
A novel discrete mathematical approach is proposed as an additional tool for molecular systematics which does not require prior statistical assumptions concerning the evolutionary process. The method is based on algorithms generating mathematical representations directly from DNA/RNA or protein sequences, followed by the output of numerical (scalar or vector) and visual characteristics (graphs). The binary encoded sequence information is transformed into a compact analytical form, called the Iterative Canonical Form (or ICF) of Boolean functions, which can then be used as a generalized molecular descriptor. The method provides raw vector data for calculating different distance matrices, which in turn can be analyzed by neighbor-joining or UPGMA to derive a phylogenetic tree, or by principal coordinates analysis to get an ordination scattergram. The new method and the associated software for inferring phylogenetic trees are called the Boolean analysis or BOOL-AN
Computing k-Modal Embeddings of Planar Digraphs
Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks.
First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k>2 and show that the problem is NP-complete for planar digraphs of maximum degree Delta <= k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo [https://doi.org/10.1007/978-3-319-73915-1_37], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [https://doi.org/10.7155/jgaa.00437], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size. Second, motivated by the recently-introduced planar L-drawings of planar digraphs [https://doi.org/10.1007/978-3-319-73915-1_36], which require the computation of a 4-modal embedding, we focus our attention on k=4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to Delta, by providing a linear-time algorithm for planar digraphs with Delta <= 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable - a result of independent interest that improves on Porschen et al. [https://doi.org/10.1007/978-3-540-24605-3_14]. Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k=4 and that this value of k is best possible for the digraphs in this family
Effective Marking Equivalence Checking in Systems with Dynamic Process Creation
The starting point of this work is a framework allowing to model systems with
dynamic process creation, equipped with a procedure to detect symmetric
executions (ie., which differ only by the identities of processes). This allows
to reduce the state space, potentially to an exponentially smaller size, and,
because process identifiers are never reused, this also allows to reduce to
finite size some infinite state spaces. However, in this approach, the
procedure to detect symmetries does not allow for computationally efficient
algorithms, mainly because each newly computed state has to be compared with
every already reached state.
In this paper, we propose a new approach to detect symmetries in this
framework that will solve this problem, thus enabling for efficient algorithms.
We formalise a canonical representation of states and identify a sufficient
condition on the analysed model that guarantees that every symmetry can be
detected. For the models that do not fall into this category, our approach is
still correct but does not guarantee a maximal reduction of state space.Comment: In Proceedings Infinity 2012, arXiv:1302.310
An Etude on Recursion Relations and Triangulations
Following~\cite{Arkani-Hamed:2017thz}, we derive a recursion relation by
applying a one-parameter deformation of kinematic variables for tree-level
scattering amplitudes in bi-adjoint theory. The recursion relies on
properties of the amplitude that can be made manifest in the underlying
kinematic associahedron, and it provides triangulations for the latter.
Furthermore, we solve the recursion relation and present all-multiplicity
results for the amplitude: by reformulating the associahedron in terms of its
vertices, it is given explicitly as a sum of "volume" of simplicies for any
triangulation, which is an analogy of BCFW representation/triangulation of
amplituhedron for SYM.Comment: 26 pages, 3 figure
Learning by stochastic serializations
Complex structures are typical in machine learning. Tailoring learning
algorithms for every structure requires an effort that may be saved by defining
a generic learning procedure adaptive to any complex structure. In this paper,
we propose to map any complex structure onto a generic form, called
serialization, over which we can apply any sequence-based density estimator. We
then show how to transfer the learned density back onto the space of original
structures. To expose the learning procedure to the structural particularities
of the original structures, we take care that the serializations reflect
accurately the structures' properties. Enumerating all serializations is
infeasible. We propose an effective way to sample representative serializations
from the complete set of serializations which preserves the statistics of the
complete set. Our method is competitive or better than state of the art
learning algorithms that have been specifically designed for given structures.
In addition, since the serialization involves sampling from a combinatorial
process it provides considerable protection from overfitting, which we clearly
demonstrate on a number of experiments.Comment: Submission to NeurIPS 201
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