Completeness and Universality of Arithmetical Numberings

Badaev, S. A.; Goncharov, S. S.; Sorbi, Andrea

doi:10.1007/978-1-4615-0755-0_2

Cited by 31 publications

(31 citation statements)

References 7 publications

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“…In what follows, we denote by the inverse Fourier transform of the Butterworth filter as a function in the variable (5) Convergence Criterion of the Adaptive ILC: In [19] and [20] a rigorous convergence analysis for a similar adaptive ILC has been carried out. Next, we prove briefly the convergence of the applied adaptive algorithm by use of an uniform version of the fixed point theorem [21]. As explained in [8], in the case when the adaptive filter is used, the update formula (2) remains the same while (1) becomes (6) known as the nonstationary convolutional integral [22], [23] which is an extension of the convolutional method to nonstationary processes.…”

Section: Ilc With Adaptive Robustness Filtermentioning

confidence: 99%

Adaptive Iterative Learning Control for High Precision Motion Systems

Rotariu

Steinbuch

Ellenbroek

2008

IEEE Trans. Contr. Syst. Technol.

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Abstract-Iterative learning control (ILC) is a very effectivetechnique to reduce systematic errors that occur in systems that repetitively perform the same motion or operation. However, several characteristics have prevented standard ILC from being widely used for high precision motion systems. Most importantly, the learned feedforward signal depends on the motion profile (setpoint trajectory) and if this is altered, the learning process has to be repeated. Secondly, ILC amplifies non-repetitive disturbances and noise. Finally, its performance may be limited due to position-dependent dynamics. This paper presents the design and implementation of a time-frequency adaptive ILC that is applicable for motion systems which executes the same motion or operation. It employs the same control system block diagram as standard ILC, but instead of a fixed robustness filter it uses a time-varying filter. By using the results of the time-frequency adaptive ILC, i.e., the shape of the learned feedforward signal, a "piecewise ILC" is proposed that leads to the design of a single learned feedforward signal suitable for different setpoints. The results are experimentally shown to work for a high precision motion system.

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Section: Ilc With Adaptive Robustness Filtermentioning

confidence: 99%

Adaptive Iterative Learning Control for High Precision Motion Systems

Rotariu

Steinbuch

Ellenbroek

2008

IEEE Trans. Contr. Syst. Technol.

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“…N → X is a numbering of X [1,18,19,21,22] def ⇔ ψ is onto. 4 We will often write ψ p as shorthand for ψ(p).…”

Section: Introduction: Krt and Fprtmentioning

confidence: 99%

Program Self-Reference in Constructive Scott Subdomains

Case

Moelius²

2011

Theory Comput Syst

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Intuitively, a recursion theorem asserts the existence of self-referential programs. Two well-known recursion theorems are Kleene's Recursion Theorem (krt) and Rogers' Fixpoint Recursion Theorem (fprt). Does one of these two theorems better capture the notion of program self-reference than the other? In the context of the partial computable functions over the natural numbers (PC), fprt is strictly weaker than krt, in that fprt holds in any effective numbering of PC in which krt holds, but not vice versa. It is shown that, in this context, the existence of self-reproducing programs (a.k.a. quines) is assured by krt, but not by fprt. Most would surely agree that a self-reproducing program is self-referential. Thus, this result suggests that krt is better than fprt at capturing the notion of program self-reference in PC.A generalization of krt to arbitrary constructive Scott subdomains is then given. (For fprt, a similar generalization was already known.) Surprisingly, for some such subdomains, the two theorems turn out to be equivalent. A precise characterization is given of those constructive Scott subdomains in which this occurs. For such subdomains, the two theorems capture the notion of program self-reference equally well. This is an expanded version of [7].

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“…Reducibility of numberings is a pre-order relation on Com 0 n+1 (A), which induces, in the usual way, a quotient structure R 0 n+1 (A), an upper semilattice, called the Rogers semilattice of Σ 0 n+1 -computable numberings of the family A. We refer the reader to [2,4] for details on Σ 0 n+1 -computable numberings and related topics.THEOREM. For every k ∈ N , there exist infinitely many infinite Σ 0 k+1 -computable families with pairwise elementarily different Rogers semilattices.…”

mentioning

confidence: 99%

“…For the classical case of computable families of computably enumerable sets, it was shown that there exist infinitely many families with pairwise elementarily different theories of Rogers semilattices (see [1]). Nevertheless, resent studies on Rogers semilattices of arithmetic numberings for infinite families of sets at levels greater than one in the Kleene-Mostowsky hierarchy have revealed the following: -significant differences as regards algebraic and elementary properties of the semilattices in question compared with the classical case;-homogeneity of the structure of their ideals and intervals (see [2][3][4][5]). Initially, this led to the conjecture that the Rogers semilattices for infinite families of sets of any fixed high level of the arithmetic hierarchy would have similar properties.…”

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confidence: 99%

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Elementary Theories for Rogers Semilattices

Badaev¹,

Гончаров

Sorbi³

2005

Algebr Logic

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It is proved that for every level of the arithmetic hierarchy, there exist infinitely many families of sets with pairwise non-elementarily equivalent Rogers semilattices.Research in elementary theories for Rogers semilattices is one of the main problems in the theory of numberings. For the classical case of computable families of computably enumerable sets, it was shown that there exist infinitely many families with pairwise elementarily different theories of Rogers semilattices (see [1]). Nevertheless, resent studies on Rogers semilattices of arithmetic numberings for infinite families of sets at levels greater than one in the Kleene-Mostowsky hierarchy have revealed the following: -significant differences as regards algebraic and elementary properties of the semilattices in question compared with the classical case;-homogeneity of the structure of their ideals and intervals (see [2][3][4][5]). Initially, this led to the conjecture that the Rogers semilattices for infinite families of sets of any fixed high level of the arithmetic hierarchy would have similar properties. Pretty soon, however, it was found out that for every level of the hierarchy, there are at least two families of that level such that their Rogers semilattices are not elementarily equivalent (see [6]). The objective of the present paper is to prove that for every level of the arithmetic hierarchy, there exist infinitely many families with pairwise non-elementarily equivalent Rogers semilattices. We recall some necessary notation and definitions. A numbering α of a family A ⊆ Σ 0 n+1 is said to beA numbering α is reducible to a numbering β if α = β • f for some computable function f . Reducibility of numberings is a pre-order relation on Com 0 n+1 (A), which induces, in the usual way, a quotient structure R 0 n+1 (A), an upper semilattice, called the Rogers semilattice of Σ 0 n+1 -computable numberings of the family A. We refer the reader to [2,4] for details on Σ 0 n+1 -computable numberings and related topics.THEOREM. For every k ∈ N , there exist infinitely many infinite Σ 0 k+1 -computable families with pairwise elementarily different Rogers semilattices.Proof. Let k be an arbitrary natural number. We will construct a sequence {B e } e 1 of infinite Σ 0 k+1 -computable families such that Th(R 0 k+1 (B e )) = Th(R 0 k+1 (B e )) *

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Completeness and Universality of Arithmetical Numberings

Cited by 31 publications

References 7 publications

Adaptive Iterative Learning Control for High Precision Motion Systems

Adaptive Iterative Learning Control for High Precision Motion Systems

Program Self-Reference in Constructive Scott Subdomains

Elementary Theories for Rogers Semilattices

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