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The universality of polynomial time Turing equivalence
We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
We develop a framework to give upper bounds on the "practical" computational
complexity of stability problems for a wide range of nonlinear continuous and
hybrid systems. To do so, we describe stability properties of dynamical systems
using first-order formulas over the real numbers, and reduce stability problems
to the delta-decision problems of these formulas. The framework allows us to
obtain a precise characterization of the complexity of different notions of
stability for nonlinear continuous and hybrid systems. We prove that bounded
versions of the stability problems are generally decidable, and give upper
bounds on their complexity. The unbounded versions are generally undecidable,
for which we give upper bounds on their degrees of unsolvability
Computational models of hemostasis: Degrees of complexity
The history of studies on blood clotting goes back to the emergence of civilized society itself. The foundations of the modern scientific study of hemostasis are based on the discovery of erythrocytes in blood in 1674 and, later, that of platelets in 1842. The causes of thrombosis are encapsulated in the Virchow Triad (dated to 1856), which refers, in modern terms, to hypercoagulability, alterations of hemodynamics (stasis), and endothelial injury. The understanding of coagulation, the network of reactions that underlies hemostasis and thrombosis, has evolved from a cascade (in 1964) into spatially distinct sets of reactions dependent on co-factors occurring on different cells in different tissues and linked together by diffusion and flow (as of 2015). Correspondingly, mathematical/computational models for hemostasis and thrombosis (which involve coagulation along with platelet aggregation in the presence of flow) have evolved in design complexity from Continuum temporal (or “homogeneous”) models to Continuum spatio-temporal models (with or without the flow) and lately into Discrete-Continuum spatio-temporal models with the flow. After a brief listing of the discoveries and historical personae that contributed to the understanding of hemostasis up to the present, the development of mathematical/computational models is traced from the late 1980s when they started gaining importance. Influential models are then highlighted. The models are reviewed in increasing order of design complexity (one of four possible methods of classification). The physiological significance of each and the insights they offer into hemostasis regulation are explained. © 2022 The Author
Physical portrayal of computational complexity
Computational complexity is examined using the principle of increasing
entropy. To consider computation as a physical process from an initial instance
to the final acceptance is motivated because many natural processes have been
recognized to complete in non-polynomial time (NP). The irreversible process
with three or more degrees of freedom is found intractable because, in terms of
physics, flows of energy are inseparable from their driving forces. In
computational terms, when solving problems in the class NP, decisions will
affect subsequently available sets of decisions. The state space of a
non-deterministic finite automaton is evolving due to the computation itself
hence it cannot be efficiently contracted using a deterministic finite
automaton that will arrive at a solution in super-polynomial time. The solution
of the NP problem itself is verifiable in polynomial time (P) because the
corresponding state is stationary. Likewise the class P set of states does not
depend on computational history hence it can be efficiently contracted to the
accepting state by a deterministic sequence of dissipative transformations.
Thus it is concluded that the class P set of states is inherently smaller than
the set of class NP. Since the computational time to contract a given set is
proportional to dissipation, the computational complexity class P is a subset
of NP.Comment: 16, pages, 7 figure
Efficient regularized isotonic regression with application to gene--gene interaction search
Isotonic regression is a nonparametric approach for fitting monotonic models
to data that has been widely studied from both theoretical and practical
perspectives. However, this approach encounters computational and statistical
overfitting issues in higher dimensions. To address both concerns, we present
an algorithm, which we term Isotonic Recursive Partitioning (IRP), for isotonic
regression based on recursively partitioning the covariate space through
solution of progressively smaller "best cut" subproblems. This creates a
regularized sequence of isotonic models of increasing model complexity that
converges to the global isotonic regression solution. The models along the
sequence are often more accurate than the unregularized isotonic regression
model because of the complexity control they offer. We quantify this complexity
control through estimation of degrees of freedom along the path. Success of the
regularized models in prediction and IRPs favorable computational properties
are demonstrated through a series of simulated and real data experiments. We
discuss application of IRP to the problem of searching for gene--gene
interactions and epistasis, and demonstrate it on data from genome-wide
association studies of three common diseases.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS504 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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