175 research outputs found
Approximation Algorithms using Allegories and Coq
In this thesis, we implement several approximation algorithms for solving optimization problems on graphs. The result computed by the algorithm may or may not be optimal. The approximation factor of an algorithm indicates how close the computed result is to an optimal solution. We are going to verify two properties of each algorithm in this thesis.First, we show that the algorithm computes a solution to the problem, and, second, we show that the approximation factor is satisfied. To implement these algorithms, we use the algebraic theory of relations, i.e., the theory of allegories and various extension thereof. An implementation of various kinds of lattices and the theory of categories is required for the declaration of allegories. The programming language and interactive theorem prover Coq is used for the implementation purposes. This language is based on Higher-Order Logic (HOL) with dependent types which support both reasoning and program execution. In addition to the abstract theory, we provide the model of set-theoretic relations between finite sets. This model is executable and used in our examples. Finally, we provide an example for each of the approximation algorithm
Extensions of Functors From Set to V-cat
We show that for a commutative quantale V every functor from Set to V-cat has an enriched left-Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric
Extending Set Functors to Generalised Metric Spaces
For a commutative quantale V, the category V-cat can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor T (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor TV on V-cat. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones.
Conceptually, this allows us to to solve the same recursive domain equation X ≅ TX in different categories (such as sets and metric spaces) and we study how their solutions (that is, the final coalgebras) are related via change of base.
Mathematically, the heart of the matter is to show that, for any commutative quantale V, the “discrete functor Set → V-cat from sets to categories enriched over V is V-cat-dense and has a density presentation that allows us to compute left-Kan extensions along D
Axioms for the category of sets and relations
We provide axioms for the dagger category of sets and relations that recall
recent axioms for the dagger category of Hilbert spaces and bounded operators.Comment: 14 pages; corrected proof of Corollary 1.
Mysteriously Meant
In Mysteriously Meant, Professor Allen maps the intellectual landscape of the Renaissance as he explains the discovery of an allegorical interpretation of Greek, Latin, and finally Egyptian myths and the effect this discovery had on the development of modern attitudes toward myth. He believes that to understand Renaissance literature one must understand the interpretations of classical myth known to the sixteenth and seventeenth centuries. In unraveling the elusive strands of myth, allegory, and symbol from the fabric of Renaissance literature such as Milton's Paradise Lost, Allen is a helpful guide. His discussion of Renaissance authors is as authoritative as it is inclusive. His empathy with the scholars of the Renaissance keeps his discussion lively—a witty study of interpreters of mythography from the past
Role of an artefact of Dynamic algebra in the conceptualisation of the algebraic equality
In this contribution, we explore the impact of Alnuset, an artefact of dynamic algebra, on the conceptualisation of algebraic equality. Many research works report about obstacles to conceptualise this notion due to interference of the previous arithmetic knowledge. New meanings need to be assigned to the equal sign and to letters used in algebraic expressions. Based on the hypothesis that Alnuset can be effectively used to mediate the conceptual development necessary to master the algebraic equality notion, two experiments have been designed and implemented in Italy and in France. They are reported in the second part of this pape
Hermeneutical Outlines in and of Dante’s Legal Theory
Based upon the concept of Law qualified in Monarchia, II.50, Dante was not only a general philosopher (a lover of knowledge) as well as a political disputant in his times, but also his primary contribution (not always obvious) in legal speculation could be demonstrated. In fact, if his thought reflected the platonic ordo sapientiae through a deep intersection between téchne and episteme (phronesis) toward a linguistic koiné, could we say the same thing on his concept of justice as a rational ars boni et aequi? This essay aims to depict Dante as legal theorist of his times and theorist of Justice beyond them, adopting the hermeneutical point of view, not just as an interest into textual interpretation but referring his use of language as form of life and his works an inexhaustible sources of education for legal philosophy
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