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We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP $_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP $_{1}$ theories. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and k -ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler's construction for groups, Chatzidakis' style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras. [ABSTRACT FROM AUTHOR]

We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analog of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories. [ABSTRACT FROM AUTHOR]

In a scenario of global conjuncture, resources generated by the internet have become relevant regarding relationships and cyber communication. The rising of virtual movements, which initiated a new social-historical perspective of communities and their own vocabularies in the digital environment, have reduced barriers and brought people and social networks closer, according to their ideas and similarities. To carry out this work, we brought as a sample a streaming platform, a means of use and distribution of digital content, without the need for data storage by the user. Considering Netflix as a platform strongly present into society, we proposed a comparative analysis on the classification and organization of information in the catalogs that are part of Netflix. For this purpose, we selected four samples present in the catalog, and defined that in order to meet the criteria for the study, they needed to integrate the culture of convergence (transmedia, fandoms and niche market): Productions in series format and Netflix Original Productions. It is believed that the objectives were achieved, but some points are worth mentioning: beside the lack of historical scientific bibliography on the subject, chain divisions were noted only on the terms visible on screen; the division did not follow a hierarchy, making the classification system flawed and poorly structured, without showing important aspects of extension and comprehension. [ABSTRACT FROM AUTHOR]

We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the "main gap" by reducing the dependence of certain versions on (highly independent) cardinal arithmetic. [ABSTRACT FROM AUTHOR]

We discuss dividing lines (in model theory) and some test questions, mainly the universality spectrum. So there is much on conjectures, problems and old results, mainly of the author and also on some recent results. [ABSTRACT FROM AUTHOR]

The classification may be created on a basis of the analysis and grouping of the soil data or some typical images (standards) of the soil, which were imagined or selected from the existing data. These will be the representative profiles. The expression "representative profile or soil pit" may serve as a summary (denotation) of concepts different in meaning or content. As reference points may be taken mean, modal, or median profiles of the marked taxons, holotype – profile, which is the most similar to the whole taxon profiles. However, the calculated representatives are not entirely convenient because of the dependence on the volumes of multidimensional samples. Two approaches are combined. However, this also causes the increase of the subjectivity share, which is contained in the procedures of representative processes identification in the groups of initial objects. Nevertheless, we found more objective approach of typical samples detection, which is presented by the idea of classification-specification. The purpose of classification (genetic, factorial, substantive, combined, etc.) is determined through a fixed system of soil indicators (specificities) corresponding to this goal, organized in the form of a classification-specification graph. The number of objects for such a classification is determined by the number of gra-dation combinations for the values of the selected specificities (Кj), where j = 1,2, ... m, and m is the number of specificities taken into account. The edges of the graph reflect descriptions of representative images (types) of classification taxa. The word “type” here is related to the representative profile, and not to a soil.

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory  T in which all partially recursive functions are representable, yet T does not interpret Robinson's theory  R. To this end, we borrow tools from model theory — specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of ∃ ∀ theories interpretable in existential theories in the process. [ABSTRACT FROM AUTHOR]

In this paper we give an additional characterizations of the first order complete superstable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned class of first-order theories can be regarded as "not very complicated". This was done for "stable" and for "À0-stable." Here we give a complete answer for "superstable". [ABSTRACT FROM AUTHOR]

In 1967 the author introduced a pre-ordering of all first order complete theories where T is lower than U if it is easier for an ultrapower of a model of T than an ultrapower of a model of U to be saturated. In a long series of recent papers, Malliaris and Shelah showed that this pre-ordering is very rich and gives a useful way of classifying simple theories. In this paper we investigate the analogous pre-ordering in continuous model theory. [ABSTRACT FROM AUTHOR]

Abstract We investigate the interpretability ordering ⊴ ⁎ using generalized Ehrenfeucht–Mostowski models. This gives a new approach to proving inequalities and investigating the structure of types. [ABSTRACT FROM AUTHOR]

A classification table for geometric programming is given in this paper. The table is exhaustive and exclusive with only one state in each row and each column. It proves that out of 49 possible duality states, only seven are possible. The proofs of theorems leading to the classification table are based on the new states, which are defined according to the newly defined homogenized programs for both the primal and dual geometric programming. [ABSTRACT FROM AUTHOR]

INFOSOIL is a new information-reference system aimed at elaborating purposeful classifications of soils by using a definite set of soil features. The stages and procedures of this system permit to determine the maximum number of objects taking into complete account their most informative features, the method and quality of objects grouping into classes as well as to recognize newly presented soils to be classified. This system should be named as a generator or engine of soil classifications developed for different purposes; it is based on the concept of enumeration classification, thus connecting a totality of soil features to meet requirements of the soil classification. The amount of features (classifier positions) determines the maximum number of objects and brings them in correspondence. It is shown how is realized the suggested approach in practice; the position of soils in the classification system is indicated. In this paper the first INFOSOIL version (www.infosoil) is presented to show the development of its real possibilities and accessibility for users and classification authors. In the system the interfaces are envisaged with the soil-geographical database, the State register of soil resources and the Soil map, 1: 2.5 M scale. In essence the system contains the soil classification system widely adopted now (2004), the automated guide to determine the soils (2008) and creates all the conditions for its further development. INFOSOIL has being elaborated as an intellectual information system that includes the MERON system, algorithm means to create new enumeration classifications as well as to use and study the available classifications.

Abstract We describe all binary simple homogeneous structures M in terms of ∅-definable equivalence relations on M , which "coordinatize" M and control dividing, and extension properties that respect these equivalence relations. [ABSTRACT FROM AUTHOR]

Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, ( K 3 , ≼ ) , is ω -stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in ( K 3 , ≼ ) the Smoothness Axiom fails, and so ( K 3 , ≼ ) is not an AEC. [ABSTRACT FROM AUTHOR]

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP 1 -like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence. [ABSTRACT FROM AUTHOR]

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:(1)For all high-enough λ,Khas no long splitting chains.(2)For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.(3)For all high-enough λ,Khas a unique limit model of cardinality λ.(4)For all high-enough λ,Khas a superlimit model of cardinality λ.(5)For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.(6)There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model. [ABSTRACT FROM AUTHOR]

We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we consider Definition 1 Let M 0 ≺ N be models from K and A be a set. We say that the Galois-type of A over N does not fork over M 0 , written A ⫝ M 0 N , iff for all small a ∈ A and all small N − ≺ N , we have that Galois-type of a over N − is realized in M 0 . Assuming property ( E ) (Existence and Extension, see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a “big cardinal”. Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful. In [10] , it is established that, if this notion is an independence notion, then it is the only one. [ABSTRACT FROM AUTHOR]

As technology transforms knowledge and information management systems, statistical data is becoming more accessible, available in bigger and more complex datasets and is able to be analysed and interpreted in so many different ways. Traditional approaches to the development, maintenance and revision of statistical classifications no longer support or enable description of data in ways that are as useful to users as they could be. The ability to search and discover information in ways that were previously not possible means that new methodologies for managing and describing the data, and its associated metadata, are required. The development of structured lists of categories, often hierarchic in nature, based on a single concept, limited by the constraints of the printed page, statistical survey processing system needs, sequential code structures or narrow user defined scopes results in statistical classifications neither dynamically reflecting the real world of official statistics nor maintaining relevance in a fast changing information society. Opportunities exist for modernising the developmental processes for statistical classifications by using, for example, semantic web technology, Simple Knowledge Organisation Systems (SKOS), and Resource Description Frameworks (RDF), and for better describing metadata and information within and across multiple, interconnected information and knowledge management systems. These opportunities highlight the difficulties that come with using traditional approaches to statistical classification development and management, and encourage new thinking for different and more flexible options for developers and users. This paper explores the need to dispense with traditional practices for developing statistical classifications as cornerstones of metadata, knowledge and information management, and comments on the need to change the underlying methodology within statistical classification theory, best practice principles and how they can be used in associated information management systems. [ABSTRACT FROM AUTHOR]

A theory is said to have exact saturation at a singular cardinal if it has a -saturated model which is not -saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality. [ABSTRACT FROM AUTHOR]

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $$\lambda {:}$$ Our proofs use independence calculus and a generalization of averages to this non first-order context. [ABSTRACT FROM AUTHOR]

We study the $$\kappa $$ -Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and $$T^{\prime }$$ , if T is classifiable and $$T^{\prime }$$ is not, then the isomorphism of models of $$T^{\prime }$$ is strictly above the isomorphism of models of T with respect to $$\kappa $$ -Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions. [ABSTRACT FROM AUTHOR]

In the setting of abstract elementary classes (AECs) with amalgamation, Shelah has proven a downward categoricity transfer from categoricity in a successor and Grossberg and VanDieren have established an upward transfer assuming in addition a locality property for Galois types that they called tameness. We further investigate categoricity transfers in tame AECs. We use orthogonality calculus to prove a downward transfer from categoricity in a successor in AECs that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: Theorem 0.1 Let K be an AEC and let LS ( K ) ≤ λ < θ be cardinals. If K has a type-full good [ λ , θ ] -frame and K is categorical in both λ and θ + , then K is categorical in all μ ∈ [ λ , θ ] . We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from ℶ ℶ ( 2 LS ( K ) ) + to ℶ ( 2 LS ( K ) ) + assuming that the AEC is LS ( K ) -tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form M ∪ { a } or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds. [ABSTRACT FROM AUTHOR]

Boney and Grossberg [7] proved that every nice AEC has an independence relation. We prove that this relation is unique: in any given AEC, there can exist at most one independence relation that satisfies existence, extension, uniqueness and local character. While doing this, we study more generally the properties of independence relations for AECs and also prove a canonicity result for Shelah's good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theories) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework. [ABSTRACT FROM AUTHOR]

For many classes of models, there are universal members in any cardinal λ which 'essentially satisfies GCH, i.e., λ = 2<λ,' in particular for the class of models of a complete first-order T (well, if at least λ > | T|). But if the class is 'complicated enough', e.g., the class of linear orders, we know that if λ is 'regular and not so close to satisfying GCH', then there is no universal member. Here, we find new sufficient conditions (which we call the olive property), not covered by earlier cases (i.e., fail the so-called SOP4). The advantage of those conditions is witnessed by proving that the class of groups satisfies one of those conditions. [ABSTRACT FROM AUTHOR]

We prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models. [ABSTRACT FROM PUBLISHER]

We characterize model theoretic properties of the Urysohn sphere as a metric structure in continuous logic. In particular, our first main result shows that the theory of the Urysohn sphere is SOP n for all n ≥ 3 , but does not have the fully finite strong order property. Our second main result is a geometric characterization of dividing independence in the theory of the Urysohn sphere. We further show that this characterization satisfies the extension axiom, and so forking and dividing are the same for complete types. Our results require continuous analogs of several tools and notions in classification theory. While many of these results are undoubtedly known to researchers in the field, they have not previously appeared in publication. Therefore, we include a full exposition of these results for general continuous theories. [ABSTRACT FROM AUTHOR]

Deep neural networks are currently among the most commonly used classifiers. Despite easily achieving very good performance, one of the best selling points of these models is their modular design – one can conveniently adapt their architecture to specific needs, change connectivity patterns, attach specialised layers, experiment with a large amount of activation functions, normalisation schemes and many others. While one can find impressively wide spread of various configurations of almost every aspect of the deep nets, one element is, in authors’ opinion, underrepresented – while solving classification problems, vast majority of papers and applications simply use log loss. In this paper we try to investigate how particular choices of loss functions affect deep models and their learning dynamics, as well as resulting classifiers robustness to various effects. We perform experiments on classical datasets, as well as provide some additional, theoretical insights into the problem. In particular we show that L1 and L2 losses are, quite surprisingly, justified classification objectives for deep nets, by providing probabilistic interpretation in terms of expected misclassification. We also introduce two losses which are not typically used as deep nets objectives and show that they are viable alternatives to the existing ones. [ABSTRACT FROM AUTHOR]

Both basic and applied research on the construction, implementation, maintenance, and evaluation of classification schemes is called classification theory. If we employ Ritzer's metatheoretical method of analysis on the over one-hundred year-old body of literature, we can see categories of theory emerge. This paper looks at one particular part of knowledge organization work, namely classification theory, and asks 1) what are the contours of this intellectual space, and, 2) what have we produced in the theoretical reflection on constructing, implementing, and evaluating classification schemes? The preliminary findings from this work are that classification theory can be separated into three kinds: foundational classification theory, first-order classification theory, and second-order classification theory, each with its own concerns and objects of study. [ABSTRACT FROM AUTHOR]

We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first, we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular, we try to prove the generic pair conjecture and do it for measurable cardinals. [ABSTRACT FROM AUTHOR]

Multithreshold Entropy Linear Classifier (MELC) is a recent classifier idea which employs information theoretic concept in order to create a multithreshold maximum margin model. In this paper we analyze its consistency over multithreshold linear models and show that its objective function upper bounds the amount of misclassified points in a similar manner like hinge loss does in support vector machines. For further confirmation we also conduct some numerical experiments on five datasets. [ABSTRACT FROM AUTHOR]

We provide several crucial technical extensions of the theory of stable independence notions in accessible categories. In particular, we describe circumstances under which a stable independence notion can be transferred from a subcategory to a category as a whole, and examine a number of applications to categories of groups and modules, extending results of [16]. We prove, too, that under the hypotheses of [11] , a stable independence notion immediately yields higher-dimensional independence as in [26]. [ABSTRACT FROM AUTHOR]

We study the Borel-reducibility of isomorphism relations in the generalised Baire space κ κ. In the main result we show for inaccessible κ , that if T is a classifiable theory and T ′ is superstable with the strong dimensional order property (S-DOP), then the isomorphism of models of T is Borel reducible to the isomorphism of models of T ′. In fact we show the consistency of the following: If κ is inaccessible and T is a superstable theory with S-DOP, then the isomorphism of models of T is Σ 1 1 -complete. [ABSTRACT FROM AUTHOR]