Continuous Domains in Formal Concept Analysis*

Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.

Stable recovery of planar regions with algebraic boundaries in Bernstein form

We present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations associated to a set of image moments. However, the sensitivity of the moments to noise makes the numerical solution highly unstable. To derive a robust image recovery algorithm, we represent algebraic polynomials and the corresponding image moments in terms of bivariate Bernstein polynomials and apply polynomial-generating, refinable sampling kernels. This approach is robust to noise, computationally fast and simple to implement. We illustrate the performance of our reconstruction algorithm from noisy samples through extensive numerical experiments. Our code is released open source and freely available.

A semi-canonical reduction for periods of Kontsevich–Zagier

The [Formula: see text]-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of [Formula: see text]-rational functions over [Formula: see text]-semi-algebraic domains in [Formula: see text]. The Kontsevich–Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stokes formula. In this paper, we prove that every non-zero real period can be represented as the volume of a compact [Formula: see text]-semi-algebraic set obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich–Zagier period conjecture.

All Liouville Numbers are Transcendental

Summary In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation.

On naturally continuous non-dcpo domains

After works of Normann and the author on sequentiality (Normann 2006Mathematical Structures in Computer Science16 (2) 279–289; Normann and Sazonov 2012Annals of Pure and Applied Logic163 (5) 575–603; Sazonov 2007Logical Methods in Computer Science3 (3:7) 1–50), the necessity and possibility of a non-dcpo domain theory became evident. In this paper, the category of continuous dcpo domains is generalized to a category of ‘naturally’ continuous non-dcpo domains with ‘naturally’ continuous maps as arrows. A full subcategory of the latter, assuming a kind of bounded-completeness requirement of domains and presence of ⊥ in each, proves to be Cartesian closed and equivalent to a subclass of Ershov's general A-spaces (Ershov 1974Algebra and Logics12 (4) 369–416). This extends a non-dcpo generalization of Scott (algebraic) domains introduced and proved to be equivalent to Ershov's general f-spaces (Ershov 1972Algebra and Logic11 (4) 367–437) in Sazonov (2007 op. cit.; 2009 Annals of Pure and Applied Logic159 (3) 341–355).The current approach to natural domains (v-domains) is different from f-spaces and A-spaces in that it has arisen in Sazonov (2007 op. cit.) in a different way from defining fully abstract models for some versions of the language PCF over Integers, whereas the Ershov's approach was not initially related with full abstraction, and non-dcpo version of f-spaces and A-spaces were originally considered in an abstract (mainly topological) style. In this paper devoted to naturally continuous natural domains (v-continuous v-domains), we also work in an abstract (mainly order-theoretic) style but with the hope to relate it in the future with the ideas of PCF over Reals by exploring and adapting the ideas in Escardó (1996Theoretical Computer Science162 (1) 79–115), Escardó et al. (2004Mathematical Structures in Computer Science, 14 (6), Cambridge University Press 803–814), Marcial-Romero and Escardó (2007Theoretical Computer Science379 (1-2) 120–141), Sazonov (2007 op. cit.).

Representation of algebraic domains by formal association rule systems

In this paper, we introduce the notion of consistent F-augmented contexts by adding a special family of finite subsets into the structure of a formal context, which essentially establishes the basis of the representation of general algebraic domains. In particular, we investigate the association rule systems which are derived from the consistent F-augmented contexts and propose the notion of formal association rule systems. By the notion of antecedent connections, we obtain the equivalence between the category of formal association rule systems and that of algebraic domains, which demonstrates that the proposed notion of formal association rule systems provides a concrete approach to representing algebraic domains.

A Topological Approach in the Extended Fraenkel-Mostowski Model of Set Theory

Lattices of subgroups are presented as algebraic domains. Given an arbitrary group, we define the Scott topology over the subgroups lattice of that group. A basis for this topology is expressed in terms of finitely generated subgroups. Several properties of the continuous functions with respect the Scott topology are obtained; they provide new order properties of groups. Finally there are expressed several properties of the group of permutations of atoms in a permutative model of set theory. We provide new properties of the extended interchange function by presenting some topological properties of its domain. Several order and topological properties of the sets in the Fraenkel-Mostowski model remains also valid in the Extended Fraenkel-Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel-Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel-Mostowski model.