Speedith: A Diagrammatic Reasoner for Spider Diagrams

Abstract

In this paper, we introduce Speedith which is a diagrammatic theorem prover for the language of spider diagrams. Spider diagrams are a well-known logic for which there is a sound and complete set of inference rules. Speedith provides a way to input diagrams, transform them via the diagrammatic inference rules, and prove diagrammatic theorems. It is designed as a program that plugs into existing general purpose theorem provers. This allows for seamless formal verification of diagrammatic proof steps within established proof assistants such as Isabelle. We describe the general structure of Speedith, the diagrammatic language, the automatic mechanism that draws the diagrams when inference rules are applied on them, and how formal diagrammatic proofs are constructed.

Logical and Geometrical Complementarities between Aristotelian Diagrams

“…The description of the internal structure of the RDH, and more in particular of the various types of complementarities, crucially relies on the idea that smaller diagrams occur inside bigger diagrams. These notions of subdiagram or diagram embedding/nesting have been studied for other types of diagrams as well, more in particular Euler diagrams [7], Venn diagrams [8], spider diagrams [9] or algebra diagrams [10]. The analysis proposed in the present paper is very much in line with the visual grammar or visual syntax approach developed by Engelhardt [11, p. 104] in that "various syntactic principles can be identified in graphics of different types, and the nature of visual representation allows for visual nesting and recursion [...] any object may contain a set of (sub-)objects within the space that it occupies.…”

Inspection and Selection of Representations

“…Various formalisations of specific reasoning systems using a single kind of representation have been implemented, including first order [16,15], higher order [11,21], diagrammatic [27,13,28], among many others. A few heterogeneous reasoning systems that integrate multiple representations have also been built [2,26], as well as some tools for re-representing problems and knowledge across and within systems [19,12,23].…”

Speedith: A Reasoner for Spider Diagrams

“…We enumerate the following compound inference rules (others can be found in Howse et al [12] and Urbas et al [33]):…”

“…Speedith is our stand-alone interactive diagrammatic theorem prover for the logic of spider diagrams [33]. Moreover, it was also designed to be easily pluggable into other proof-assisting software.…”

“…Thus, Speedith's diagrammatic proofs are representations of concepts from algebra, number theory, analysis, topology and category theory. 2 We first introduced Speedith in [33]. This paper gives a comprehensive account of Speedith, its design and theoretical properties, and also its further developments regarding the selection of inference rules, hand-drawing interface and pluggable infrastructure.…”