scholarly journals AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

2018 ◽  
Vol 83 (3) ◽  
pp. 1147-1189 ◽  
Author(s):  
OLIVIER CARTON ◽  
THOMAS COLCOMBET ◽  
GABRIELE PUPPIS
Keyword(s):  
Open Problem ◽  
Algebraic Approach ◽  
Rational Numbers ◽  
Second Order ◽  
Linear Orderings ◽  
Definable Set ◽  
Mso Logic ◽  
Quantifier Alternation

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.

scholarly journals Perceived number is not abstract

2021 ◽  
Author(s):  
Lauren S Aulet ◽  
Stella F. Lourenco
Keyword(s):  
Rational Numbers ◽  
Second Order ◽  
Visual Illusions ◽  
Number Perception

To support the claim that the ANS represents rational numbers, C&B argue that number perception is abstract and characterized by a second-order character. However, converging evidence from visual illusions and psychophysics suggests that perceived number is not abstract, but rather, is perceptually interdependent with other magnitudes. Moreover, number, as a concept, is second-order, but number, as a percept, is not.

On Π1-automorphisms of recursive linear orders

1987 ◽  
Vol 52 (3) ◽  
pp. 681-688
Author(s):  
Henry A. Kierstead
Keyword(s):  
Linear Order ◽  
Rational Numbers ◽  
Order Type ◽  
Linear Orders ◽  
Arithmetical Hierarchy ◽  
Linear Orderings ◽  
Nontrivial Automorphism ◽  
Order Types ◽  
Element Chain

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

scholarly journals First order quantifiers in monadic second order logic

2004 ◽  
Vol 69 (1) ◽  
pp. 118-136 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wafik Boulos Lotfallah
Keyword(s):  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Existential Quantifier ◽  
Reachability Problem ◽  
Open Problems ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Quantifier Alternation

AbstractThis paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn (S) on properties S that says “there are n components having S”. We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Πn ⊈ FO(Σn), Σn ⊈ FO(∆n). and ∆n+1 ⊈ FOB(Σn), solving some open problems raised in [Mat98].

scholarly journals Finite-Time Trajectory Tracking of Second-Order Systems Using Acceleration Feedback Only

Automation ◽  
2021 ◽  
Vol 2 (4) ◽  
pp. 266-277
Author(s):  
Romain Delpoux ◽  
Thierry Floquet ◽  
Hebertt Sira-Ramírez
Keyword(s):  
Finite Time ◽  
Sliding Mode ◽  
Algebraic Approach ◽  
Second Order ◽  
Asymptotic State ◽  
Control Input ◽  
Positioning Systems ◽  
Acceleration Feedback ◽  
On Line ◽  
Second Order Systems

In this paper, an algebraic approach for the finite-time feedback control problem is provided for second-order systems where only the second-order derivative of the controlled variable is measured. In practice, it means that the acceleration is the only variable that can be used for feedback purposes. This problem appears in many mechanical systems such as positioning systems and force-position controllers in robotic systems and aerospace applications. Based on an algebraic approach, an on-line algebraic estimator is developed in order to estimate in finite time the unmeasured position and velocity variables. The obtained expressions depend solely on iterated integrals of the measured acceleration output and of the control input. The approach is shown to be robust to noisy measurements and it has the advantage to provide on-line finite-time (or non-asymptotic) state estimations. Based on these estimations, a quasi-homogeneous second-order sliding mode tracking control law including estimated position error integrals is designed illustrating the possibilities of finite-time acceleration feedback via algebraic state estimation.

scholarly journals Algebraic Construction of a Strongly Consistent, Permutationally Symmetric and Conservative Difference Scheme for 3D Steady Stokes Flow

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 269 ◽  
Author(s):  
Xiaojing Zhang ◽  
Vladimir Gerdt ◽  
Yury Blinkov
Keyword(s):  
Difference Scheme ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Integrability Condition ◽  
Three Dimensional ◽  
Algebraic Approach ◽  
Second Order ◽  
Algebraic Construction ◽  
Pressure Poisson Equation ◽  
Strongly Consistent

By using symbolic algebraic computation, we construct a strongly-consistent second-order finite difference scheme for steady three-dimensional Stokes flow and a Cartesian solution grid. The scheme has the second order of accuracy and incorporates the pressure Poisson equation. This equation is the integrability condition for the discrete momentum and continuity equations. Our algebraic approach to the construction of difference schemes suggested by the second and the third authors combines the finite volume method, numerical integration, and difference elimination. We make use of the techniques of the differential and difference Janet/Gröbner bases for performing related computations. To prove the strong consistency of the generated scheme, we use these bases to correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. As this takes place, our difference scheme is conservative and inherits permutation symmetry of the differential Stokes flow. For the obtained scheme, we compute the modified differential system and use it to analyze the scheme’s accuracy.

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