Related papers: Difference hierarchies and duality with an applica…
We investigate in a method for proving separation results for abstract classes of languages. A well established method to characterize varieties of regular languages are identities. We use a recently established generalization of these…
Hierarchy is a common and effective way of organizing data and representing their relationships at different levels of abstraction. However, hierarchical data dependencies cause difficulties in the estimation of "separable" models that can…
Quasi-set theory was proposed as a mathematical context to investigate collections of indistinguishable objects. After presenting an outline of this theory, we define an algebra that has most of the standard properties of an orthocomplete…
We classify all apartness relations definable in propositional logics extending intuitionistic logic using Heyting algebra semantics. We show that every Heyting algebra which contains a non-trivial apartness term satisfies the weak law of…
In this article, we present a fresh perspective on language, combining ideas from various sources, but mixed in a new synthesis. As in the minimalist program, the question is whether we can formulate an elegant formalism, a universal…
Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N.D. Belnap in a 1977 paper entitled 'How a computer should think'. Prioritised default bilattices include not only Belnap's…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
For every fixed class of regular languages, there is a natural hierarchy of increasingly more general problems: Firstly, the membership problem asks whether a given language belongs to the fixed class of languages. Secondly, the separation…
The symmetric difference in Boolean lattices can be defined in two different but equivalent forms. However, it can be introduced also in every bounded lattice with complementation where these two forms need not coincide. We study lattices…
We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic…
A discriminative structured analysis dictionary is proposed for the classification task. A structure of the union of subspaces (UoS) is integrated into the conventional analysis dictionary learning to enhance the capability of…
Distributed First Order Logic (DFOL) has been introduced more than ten years ago with the purpose of formalising distributed knowledge-based systems, where knowledge about heterogeneous domains is scattered into a set of interconnected…
We generalise the concept of duality to systems of ordinary difference equations (or maps). We propose a procedure to construct a chain of systems of equations which are dual, with respect to an integral $H$, to the given system, by…
A regular language is $k$-lookahead deterministic (resp. $k$-block deterministic) if it is specified by a $k$-lookahead deterministic (resp. $k$-block deterministic) regular expression. These two subclasses of regular languages have been…
This report is mostly written for educational purposes. It is meant as a self contained introduction to regular languages, regular expressions, and regular expression matching by using Brzozowski derivatives. As such it is mostly based on…
The scientific and practical needs of the twenty-first century lead humankind to convergence of the specialized and diverse branches of science and technology. This convergence reveals the need for new mathematical theories capable of…
The impressive performance of neural networks on natural language processing tasks attributes to their ability to model complicated word and phrase compositions. To explain how the model handles semantic compositions, we study hierarchical…
Cartesian differential categories were introduced to provide an abstract axiomatization of categories of differentiable functions. The fundamental example is the category whose objects are Euclidean spaces and whose arrows are smooth maps.…
A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the…
We address the classical problem of hierarchical clustering, but in a framework where one does not have access to a representation of the objects or their pairwise similarities. Instead, we assume that only a set of comparisons between…