scholarly journals K-Knuth Equivalence for Increasing Tableaux

10.37236/4805 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Christian Gaetz ◽  
Michelle Mastrianni ◽  
Rebecca Patrias ◽  
Hailee Peck ◽  
Colleen Robichaux ◽  
...  
Keyword(s):  
Equivalence Classes ◽  
Equivalence Relations ◽  
Knuth Equivalence

A $K$-theoretic analogue of RSK insertion and the Knuth equivalence relations were introduced by Buch, Kresch, Shimozono, Tamvakis, and Yong (2006) and Buch and Samuel (2013), respectively. The resulting $K$-Knuth equivalence relations on words and increasing tableaux on $[n]$ has prompted investigation into the equivalence classes of tableaux arising from these relations. Of particular interest are the tableaux that are unique in their class, which we refer to as unique rectification targets (URTs). In this paper, we give several new families of URTs and a bound on the length of intermediate words connecting two $K$-Knuth equivalent words. In addition, we describe an algorithm to determine if two words are $K$-Knuth equivalent and to compute all $K$-Knuth equivalence classes of tableaux on $[n]$.

scholarly journals Equivalence Projective Simulation as a Framework for Modeling Formation of Stimulus Equivalence Classes

2020 ◽  
Vol 32 (5) ◽  
pp. 912-968 ◽  
Author(s):  
Asieh Abolpour Mofrad ◽  
Anis Yazidi ◽  
Hugo L. Hammer ◽  
Erik Arntzen
Keyword(s):  
Stimulus Equivalence ◽  
Human Subjects ◽  
Network Models ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Learning Context ◽  
Neural Network Models ◽  
Equivalence Theory ◽  
Hypothetical Experiment ◽  
Theory Research

Stimulus equivalence (SE) and projective simulation (PS) study complex behavior, the former in human subjects and the latter in artificial agents. We apply the PS learning framework for modeling the formation of equivalence classes. For this purpose, we first modify the PS model to accommodate imitating the emergence of equivalence relations. Later, we formulate the SE formation through the matching-to-sample (MTS) procedure. The proposed version of PS model, called the equivalence projective simulation (EPS) model, is able to act within a varying action set and derive new relations without receiving feedback from the environment. To the best of our knowledge, it is the first time that the field of equivalence theory in behavior analysis has been linked to an artificial agent in a machine learning context. This model has many advantages over existing neural network models. Briefly, our EPS model is not a black box model, but rather a model with the capability of easy interpretation and flexibility for further modifications. To validate the model, some experimental results performed by prominent behavior analysts are simulated. The results confirm that the EPS model is able to reliably simulate and replicate the same behavior as real experiments in various settings, including formation of equivalence relations in typical participants, nonformation of equivalence relations in language-disabled children, and nodal effect in a linear series with nodal distance five. Moreover, through a hypothetical experiment, we discuss the possibility of applying EPS in further equivalence theory research.

scholarly journals New Dichotomies for Borel Equivalence Relations

1997 ◽  
Vol 3 (3) ◽  
pp. 329-346 ◽  
Author(s):  
Greg Hjorth ◽  
Alexander S. Kechris
Keyword(s):  
Quotient Space ◽  
Complete Classification ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Recent Developments ◽  
Borel Map ◽  
Image Position ◽  
Completely Metrizable

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish (i.e., separable, completely metrizable) space and E a Borel equivalence relation on X, a (complete) classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). To be of any value we would expect I and c to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial (pre-)ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ f(x)Ff(y). Then if is an embedding of X/E into Y/F, which is “Borel” (in the sense that it has a Borel lifting).Intuitively, E ≤BF might be interpreted in any one of the following ways:(i) The classi.cation problem for E is simpler than (or can be reduced to) that of F: any invariants for F work as well for E (after composing by an f as above).(ii) One can classify E by using as invariants F-equivalence classes.(iii) The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F.

Counting the Number of Equivalence Classes of Borel and Coanalytic Equivalence Relations.

1987 ◽  
Vol 52 (3) ◽  
pp. 869
Author(s):  
Alain Louveau ◽  
Jack H. Silver ◽  
John P. Burgess ◽  
L. Harrington ◽  
R. Sami ◽  
...  
Keyword(s):  
Equivalence Classes ◽  
Equivalence Relations

scholarly journals EQUIVALENCE RELATIONS WHICH ARE BOREL SOMEWHERE

2017 ◽  
Vol 82 (3) ◽  
pp. 893-930 ◽  
Author(s):  
WILLIAM CHAN
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Proper Forcing ◽  
Image Position

AbstractThe following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$. If for all $z \in {H_{{{\left( {{2^{{\aleph _0}}}} \right)}^ + }}}$, z♯ exists, then there exists an I+${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence relation.

Investigation of properties of equivalence classes of permutations by inverse Robinson — Schensted — Knuth transformation

2019 ◽  
pp. 11-22
Author(s):  
N. N. Vassiliev ◽  
V. S. Duzhin ◽  
A. D. Kuzmin
Keyword(s):  
Software Package ◽  
Numerical Experiments ◽  
Young Tableau ◽  
Infinite Sequence ◽  
Equivalence Classes ◽  
Young Tableaux ◽  
Young Diagrams ◽  
C Programming ◽  
Set Of Permutations ◽  
Knuth Equivalence

Introduction:All information about a permutation, i.e. about an element of a symmetric groupS(n), is contained in a pair of Young tableaux mapped to it by RSK transformation. However, when considering an infinite sequence of natural or real numbers instead of a permutation, all information about it is contained only in an insertion infinite Young tableau. The connection between the first element of an infinite sequence of uniformly distributed random values and the limit angle of the recording tableau nerve was found in a recent work by D. Romik and P. Śniady. However, so far there were no massive numerical experiments devoted to the reconstruction of the beginning of such a sequence by the beginning of an insertion Young tableau. The reconstruction accuracy is very important, because even the value of the first element of a sequence can be determined only by an infinite tableau.Purpose:Developing a software package for operations on Young diagrams and Young tableaux, and its application for numerical experiments with large Young tableaux. Studying the properties of Knuth equivalence classes and dual Knuth equivalence classes on a set of permutations by numerical experiments using direct and inverse RSK transformation.Results:A software package is developed using the C ++ programming language. It includes functions for dealing with Young diagrams and tableaux. The dependence of values of the first element of a permutation obtained by inverse RSK transformation on the recording tableau nerve end coordinates was investigated by conducting massive numerical experiments. Standard deviations of these values were calculated for permutations of different sizes. We determined possible positions of 1 in permutations of the same Knuth equivalence class. It has been found out that the number of these positions does not exceed the number of corner boxes of the corresponding Young diagram. Experiments showed that for a fixed insertion tableau, the value of the first element of a permutation depends only on the recording tableau nerve end coordinates.

scholarly journals Strong shift equivalence and algebraic K-theory

2019 ◽  
Vol 2019 (752) ◽  
pp. 63-104 ◽  
Author(s):  
Mike Boyle ◽  
Scott Schmieding
Keyword(s):  
Symbolic Dynamics ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Bijective Correspondence ◽  
Strong Shift ◽  
Shift Equivalence ◽  
K Theory ◽  
Strong Shift Equivalence ◽  
Stabilizer Group ◽  
General Ring

Abstract For a semiring \mathcal{R} , the relations of shift equivalence over \mathcal{R} ( \textup{SE-}\mathcal{R} ) and strong shift equivalence over \mathcal{R} ( \textup{SSE-}\mathcal{R} ) are natural equivalence relations on square matrices over \mathcal{R} , important for symbolic dynamics. When \mathcal{R} is a ring, we prove that the refinement of \textup{SE-}\mathcal{R} by \textup{SSE-}\mathcal{R} , in the \textup{SE-}\mathcal{R} class of a matrix A, is classified by the quotient NK_{1}(\mathcal{R})/E(A,\mathcal{R}) of the algebraic K-theory group NK_{1}(\mathcal{R}) . Here, E(A,\mathcal{R}) is a certain stabilizer group, which we prove must vanish if A is nilpotent or invertible. For this, we first show for any square matrix A over \mathcal{R} that the refinement of its \textup{SE-}\mathcal{R} class into \textup{SSE-}\mathcal{R} classes corresponds precisely to the refinement of the \mathrm{GL}(\mathcal{R}[t]) equivalence class of I-tA into \mathrm{El}(\mathcal{R}[t]) equivalence classes. We then show this refinement is in bijective correspondence with NK_{1}(\mathcal{R})/E(A,\mathcal{R}) . For a general ring \mathcal{R} and A invertible, the proof that E(A,\mathcal{R}) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For \mathcal{R} commutative, we show \cup_{A}E(A,\mathcal{R})=NSK_{1}(\mathcal{R}) ; the proof rests on Nenashev’s presentation of K_{1} of an exact category.

Classifying positive equivalence relations

1983 ◽  
Vol 48 (3) ◽  
pp. 529-538 ◽  
Author(s):  
Claudio Bernardi ◽  
Andrea Sorbi
Keyword(s):  
Topological Space ◽  
Equivalence Relation ◽  
Recursive Function ◽  
Equivalence Classes ◽  
Point Of View ◽  
Classification Theorem ◽  
Equivalence Relations ◽  
Natural Numbers ◽  
Uniformity Property

AbstractGiven two (positive) equivalence relations ~1, ~2 on the set ω of natural numbers, we say that ~1 is m-reducible to ~2 if there exists a total recursive function h such that for every x, y ∈ ω, we have x ~1y iff hx ~2hy. We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a “uniformity property” holds). This result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration.From this fact we deduce that an equivalence relation on ω can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.

Countable models of nonmultidimensional ℵ0-stable theories

1983 ◽  
Vol 48 (1) ◽  
pp. 197-205 ◽  
Author(s):  
Elisabeth Bouscaren ◽  
Daniel Lascar
Keyword(s):  
Finite Number ◽  
Homogeneous Model ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Stable Theory ◽  
Strong Type ◽  
First Case ◽  
Skolem Functions ◽  
Finite Sequences ◽  
Homogeneous Models

In this paper T will always be a countable ℵ0-stable theory, and in this introduction a model of T will mean a countable model.One of the main notions we introduce is that of almost homogeneous model: we say that a model M of T is almost homogeneous if for all ā and finite sequences of elements in M, if the strong type of ā is the same as the strong type of (i.e. for all equivalence relations E, definable over the empty set and with a finite number of equivalence classes, ā and are in the same equivalence class), then there is an automorphism of M taking ā to . Although this is a weaker notion than homogeneity, these models have strong properties, and it can be seen easily that the Scott formula of any almost homogeneous model is in L1. In fact, Pillay [Pi.] has shown that almost homogeneous models are characterized by the set of types they realize.The motivation of this research is to distinguish two classes of ℵ0-Stable theories:(1) theories such that all models are almost homogeneous;(2) theories with 2ℵ0 nonalmost homogeneous models.The example of theories with Skolem functions [L. 1] (almost homogeneous is then equivalent to homogeneous) seems to indicate a link between these properties and the notion of multidimensionality, and that nonmultidimensional theories are in the first case.

Thin equivalence relations and effective decompositions

1993 ◽  
Vol 58 (4) ◽  
pp. 1153-1164 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Perfect Set

AbstractLet E be a equivalence relation for which there does not exist a perfect set of inequivalent reals. If 0* exists or if V is a forcing extension of L, then there is a good well-ordering of the equivalence classes.

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