3,393 research outputs found
A verified algorithm enumerating event structures
An event structure is a mathematical abstraction modeling concepts as causality, conflict and concurrency between events. While many other mathematical structures, including groups, topological spaces, rings, abound with algorithms and formulas to generate, enumerate and count particular sets of their members, no algorithm or formulas are known to generate or count all the possible event structures over af inite set of events. We present an algorithm to generate such a family, along with a functional implementation verified using Isabelle/HOL. As byproducts, we obtain a verified enumeration of all possible preorders and partial orders. While the integer sequences counting preorders and partial orders are already listed on OEIS (On-line Encyclopedia of Integer Sequences), the one counting event structures is not. We therefore used our algorithm to submit a formally verified addition, which has been successfully reviewed and is now part of the OEIS.Postprin
Extracting few representative reconciliations with Host-Switches (Extended Abstract)
Phylogenetic tree reconciliation is the approach commonly used to in- vestigate the coevolution of sets of organisms such as hosts and symbionts. Given a phylogenetic tree for each such set, respectively denoted by H and S, together with a mapping φ of the leaves of S to the leaves of H, a reconciliation is a mapping ρ of the internal vertices of S to the vertices of H which extends φ with some constraints.
Given a cost for each reconciliation, a huge number of most parsimonious ones are possible, even exponential in the dimension of the trees. Without further information, any biological interpretation of the underlying coevolution would require that all optimal solutions are enumerated and examined. The latter is however impossible without pro- viding some sort of high level view of the situation. One approach would be to extract a small number of representatives, based on some notion of similarity or of equivalence between the reconciliations.
In this paper, we define two equivalence relations that allow one to identify many reconciliations with a single one, thereby reducing their number. Extensive experiments indicate that the number of output solutions greatly decreases in general. By how much clearly depends on the constraints that are given as input
Uniform decision problems in automatic semigroups
We consider various decision problems for automatic semigroups, which involve
the provision of an automatic structure as part of the problem instance. With
mild restrictions on the automatic structure, which seem to be necessary to
make the problem well-defined, the uniform word problem for semigroups
described by automatic structures is decidable. Under the same conditions, we
show that one can also decide whether the semigroup is completely simple or
completely zero-simple; in the case that it is, one can compute a Rees matrix
representation for the semigroup, in the form of a Rees matrix together with an
automatic structure for its maximal subgroup. On the other hand, we show that
it is undecidable in general whether a given element of a given automatic
monoid has a right inverse.Comment: 19 page
The Origins of Computational Mechanics: A Brief Intellectual History and Several Clarifications
The principle goal of computational mechanics is to define pattern and
structure so that the organization of complex systems can be detected and
quantified. Computational mechanics developed from efforts in the 1970s and
early 1980s to identify strange attractors as the mechanism driving weak fluid
turbulence via the method of reconstructing attractor geometry from measurement
time series and in the mid-1980s to estimate equations of motion directly from
complex time series. In providing a mathematical and operational definition of
structure it addressed weaknesses of these early approaches to discovering
patterns in natural systems.
Since then, computational mechanics has led to a range of results from
theoretical physics and nonlinear mathematics to diverse applications---from
closed-form analysis of Markov and non-Markov stochastic processes that are
ergodic or nonergodic and their measures of information and intrinsic
computation to complex materials and deterministic chaos and intelligence in
Maxwellian demons to quantum compression of classical processes and the
evolution of computation and language.
This brief review clarifies several misunderstandings and addresses concerns
recently raised regarding early works in the field (1980s). We show that
misguided evaluations of the contributions of computational mechanics are
groundless and stem from a lack of familiarity with its basic goals and from a
failure to consider its historical context. For all practical purposes, its
modern methods and results largely supersede the early works. This not only
renders recent criticism moot and shows the solid ground on which computational
mechanics stands but, most importantly, shows the significant progress achieved
over three decades and points to the many intriguing and outstanding challenges
in understanding the computational nature of complex dynamic systems.Comment: 11 pages, 123 citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/cmr.ht
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