Related papers: Aspects of an internal logic for persistence

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology,…

This chapter presents a state-of-the-art survey of relationships, traditionally referred to as `bridges', between interpolation properties for propositional logics -- including superintuitionistic, modal, and substructural logics -- and…

In this report, we introduce observation algebras, constructed by considering the downclosed subsets of a coherence space ordered by reverse inclusion. These may be interpreted as specifications of sets of events via some predicates with…

In this paper, we introduce the concept of a (lattice) skew Hilbert algebra as a natural generalization of Hilbert algebras. This notion allows a unified treatment of several structures of prominent importance for mathematical logic, e.g.…

We explore various semantic understandings of dual intuitionistic logic by exploring the relationship between co-Heyting algebras and topological spaces. First, we discuss the relevant ideas in the setting of Heyting algebras and…

This paper investigates the foundations of deep learning through insight of geometry, algebra and differential calculus. At is core, artificial intelligence relies on assumption that data and its intrinsic structure can be embedded into…

We give an algebraic proof of the criterion for hereditary structural completeness of an intermediate logic, or, equivalently, of the primitiveness of a variety of Heyting algebras.

Language theory, symbolic dynamics, modelisation of viral insertion into the genetic code of a host cell motivate the introduction of new types of bialgebras whose coalgebra parts are not necessarily coassociative. One of the aim of this…

The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…

In this paper we investigate two logics from an algebraic point of view. The two logics are: MALL (multiplicative-additive Linear Logic) and LL (classical Linear Logic). Both logics turn out to be strongly algebraizable in the sense of Blok…

The aim of this paper is to show that even if the natural algebraic semantic for modal (normal) logic is modal algebra, the more general class of subordination algebras (roughly speaking, the non symmetric contact algebras) is adequate too…

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…

Looking at some monoids and (semi)rings (natural numbers, integers and p-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function on such an algebra to behave…

This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…

We survey the notion and history of error-correcting codes and the algorithms needed to make them effective in information transmission. We then give some basic as well as more modern constructions of, and algorithms for, error-correcting…

We study several different notions of algebraicity in use in stable homotopy theory and prove implications between them. The relationships between the different meanings of algebraic are unexpectedly subtle, and we illustrate this with…

In this paper we introduce a new kind of topological space, called 'structured space', which locally resembles various kinds of algebraic structures. This can be useful, for instance, to locally study a space that cannot be globally endowed…

Algebraic operations are understood as topologiztion of algebra. They become an example of simplest convergence space. In our article the convergence is a arbitrary multivalued appointment. The continuity of some mapping between two…

The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions…

We present the algebra of assume-guarantee (AG) contracts. We define contracts, provide new as well as known operations, and show how these operations are related. Contracts are functorial: any Boolean algebra has an associated contract…