Algebraic Properties of Qualitative Spatio-Temporal Calculi

Qualitative spatial and temporal reasoning is based on so-called qualitative calculi. Algebraic properties of these calculi have several implications on reasoning algorithms. But what exactly is a qualitative calculus? And to which extent do the qualitative calculi proposed meet these demands? The literature provides various answers to the first question but only few facts about the second. In this paper we identify the minimal requirements to binary spatio-temporal calculi and we discuss the relevance of the according axioms for representation and reasoning. We also analyze existing qualitative calculi and provide a classification involving different notions of a relation algebra.Comment: COSIT 2013 paper including supplementary materia

Zeno's Paradoxes. A Cardinal Problem 1. On Zenonian Plurality

In this paper the claim that Zeno's paradoxes have been solved is contested. Although no one has ever touched Zeno without refuting him (Whitehead), it will be our aim to show that, whatever it was that was refuted, it was certainly not Zeno. The paper is organised in two parts. In the first part we will demonstrate that upon direct analysis of the Greek sources, an underlying structure common to both the Paradoxes of Plurality and the Paradoxes of Motion can be exposed. This structure bears on a correct - Zenonian - interpretation of the concept of division through and through. The key feature, generally overlooked but essential to a correct understanding of all his arguments, is that they do not presuppose time. Division takes place simultaneously. This holds true for both PP and PM. In the second part a mathematical representation will be set up that catches this common structure, hence the essence of all Zeno's arguments, however without refuting them. Its central tenet is an aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some number theoretic and geometric implications will be shortly discussed. Furthermore, it will be shown how the Received View on the motion-arguments can easely be derived by the introduction of time as a (non-Zenonian) premiss, thus causing their collapse into arguments which can be approached and refuted by Aristotle's limit-like concept of the potentially infinite, which remained - though in different disguises - at the core of the refutational strategies that have been in use up to the present. Finally, an interesting link to Newtonian mechanics via Cremona geometry can be established.Comment: 41 pages, 7 figure

Explaining Gabriel-Zisman localization to the computer

This explains a computer formulation of Gabriel-Zisman localization of categories in the proof assistant Coq. It includes both the general localization construction with the proof of GZ's Lemma 1.2, as well as the construction using calculus of fractions. The proof files are bundled with the other preprint "Files for GZ localization" posted simultaneously