Chapter 15 The Second-Order Theories Of Linear Orderings

AbstractWe develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.

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INDECOMPOSABLE LINEAR ORDERINGS AND HYPERARITHMETIC ANALYSIS

Journal of Mathematical Logic ◽

10.1142/s0219061306000517 ◽

2006 ◽

Vol 06 (01) ◽

pp. 89-120 ◽

Cited By ~ 8

Author(s):

ANTONIO MONTALBÁN

Keyword(s):

Second Order ◽

The Other ◽

Mathematical Statement ◽

Second Order Arithmetic ◽

Linear Orderings ◽

Mathematical Theorem ◽

Order Arithmetic ◽

Analysis And Study

A statement of hyperarithmetic analysis is a sentence of second order arithmetic S such that for every Y⊆ω, the minimum ω-model containing Y of RCA0 + S is HYP(Y), the ω-model consisting of the sets hyperarithmetic in Y. We provide an example of a mathematical theorem which is a statement of hyperarithmetic analysis. This statement, that we call INDEC, is due to Jullien [13]. To the author's knowledge, no other already published, purely mathematical statement has been found with this property until now. We also prove that, over RCA0, INDEC is implied by [Formula: see text] and implies ACA0, but of course, neither ACA0, nor ACA 0+ imply it. We introduce five other statements of hyperarithmetic analysis and study the relations among them. Four of them are related to finitely-terminating games. The fifth one, related to iterations of the Turing jump, is strictly weaker than all the other statements that we study in this paper, as we prove using Steel's method of forcing with tagged trees.

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Monadic second-order logic and linear orderings of finite structures

Computer Science Logic - Lecture Notes in Computer Science ◽

10.1007/bfb0022254 ◽

1995 ◽

pp. 163-176

Author(s):

Bruno Courcelle

Keyword(s):

Second Order ◽

Order Logic ◽

Linear Orderings ◽

Finite Structures ◽

Second Order Logic ◽

Monadic Second Order Logic

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Disappearance Voltage Determinations Using a Convergent Beam

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100064979 ◽

1971 ◽

Vol 29 ◽

pp. 184-185

Author(s):

W. L. Bell

Keyword(s):

Second Order ◽

Objective Method ◽

Uniform Thickness ◽

Rocking Curve ◽

Crystal Perfection ◽

Kikuchi Line ◽

Central Maximum ◽

Diffraction Patterns ◽

Convergent Beam ◽

Beam Technique

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.

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