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"Recursive functions"
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An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert’s 17th problem
We prove an elementary recursive bound on the degrees for Hilbert’s 17th problem. More precisely we express a nonnegative polynomial
as a sum of squares of rational functions, and we obtain as degree estimates for the numerators and denominators the following tower of
five exponentials
eBook
Minimal Weak Truth Table Degrees and Computably Enumerable Turing Degrees
Two of the central concepts for the study of degree structures in computability theory are computably enumerable degrees and minimal degrees. For strong notions of reducibility, such as m-deducibility or truth table reducibility, it is possible for computably enumerable degrees to be minimal. For weaker notions of reducibility, such as weak truth table reducibility or Turing reducibility, it is not possible to combine these properties in a single degree. We consider how minimal weak truth table degrees interact with computably enumerable Turing degrees and obtain three main results. First, there are sets with minimal weak truth table degree which bound noncomputable computably enumerable sets under Turing reducibility. Second, no set with computable enumerable Turing degree can have minimal weak truth table degree. Third, no \\Delta^0_2 set which Turing bounds a promptly simple set can have minimal weak truth table degree.
eBook
Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem
Goncharov and Peretyat’kin independently gave necessary and sufficient conditions for when a set of types of a complete theory
Along the way, we analyze a number of related principles. Some of these turn out to fall into well-known reverse
mathematical classes, such as ACA
eBook
On the Computability of Primitive Recursive Functions by Feedforward Artificial Neural Networks
We show that, for a primitive recursive function h(x,t), where x is a n-tuple of natural numbers and t is a natural number, there exists a feedforward artificial neural network N(x,t), such that for any n-tuple of natural numbers z and a positive natural number m, the first m+1 terms of the sequence h(z,t) are the same as the terms of the tuple (N(z,0),…,N(z,m)).
Journal Article
On Proofs of Properties of Semirecursive Sets
In this paper, we present proofs of properties of semirecursive sets based directly on the definition of these sets and on the recursiveness of Kleene predicates. These proofs are shorter and clearer than traditional proofs of similar statements for recursively enumerable sets.
Journal Article
On Correspondences between Feedforward Artificial Neural Networks on Finite Memory Automata and Classes of Primitive Recursive Functions
When realized on computational devices with finite quantities of memory, feedforward artificial neural networks and the functions they compute cease being abstract mathematical objects and turn into executable programs generating concrete computations. To differentiate between feedforward artificial neural networks and their functions as abstract mathematical objects and the realizations of these networks and functions on finite memory devices, we introduce the categories of general and actual computabilities and show that there exist correspondences, i.e., bijections, between functions computable by trained feedforward artificial neural networks on finite memory automata and classes of primitive recursive functions.
Journal Article
Tracking and Data Association Based on Reinforcement Learning
Currently, most multi-target data association methods require the assumption that the target motion model is known, but this assumption is clearly not valid in a real environment. In the case of an unknown system model, the influence of environmental clutter and sensor detection errors on the association results should be considered, as well as the occurrence of strong target maneuvers and the sudden appearance of new targets during the association process. To address these problems, this paper designs a target tracking and data association algorithm based on reinforcement learning. First, this algorithm combines the dynamic exploration capability of reinforcement learning and the long-time memory function of LSTM network to design a policy network that predicts the probability of associating a point with its various possible source targets. Then, the Bayesian network and the multi-order least squares curve fitting method are combined to predict the location of target, and the results are fed into the Bayesian recursive function to obtain the reward. Simultaneously, some corresponding mechanisms are proposed for possible problems that interfere with the association process. Finally, the simulation experimental results show that this algorithm associates the results with higher accuracy compared to other algorithms when faced with the above problem.
Journal Article
The recursive Green’s function method for graphene
We describe how to apply the recursive Green’s function method to the computation of electronic transport properties of graphene sheets and nanoribbons in the linear response regime. This method allows for an amenable inclusion of several disorder mechanisms at the microscopic level, as well as inhomogeneous gating, finite temperature, and, to some extend, dephasing. We present algorithms for computing the conductance, density of states, and current densities for armchair and zigzag atomic edge alignments. Several numerical results are presented to illustrate the usefulness of the method.
Journal Article