Higher dimensional trees, algebraically

• Authors: Neil Ghani;Alexander Kurz
• Venue: CALCO'07 Proceedings of the 2nd international conference on Algebra and coalgebra in computer science
• Year: 2007

In formal language theory, James Rogers published a series of innovative papers generalising strings and trees to higher dimensions. Motivated by applications in linguistics, his goal was to smoothly extend the core theory of the formal languages of strings and trees to higher dimensions. Rogers' definitions rely on a specific representation of higher dimensional trees. This paper presents an alternative approach which focusses more on their universal properties and is based upon category theory, algebras, coalgebras and containers. Our approach reveals that Rogers' trees are canonical constructions which are also particularly beautiful. We also provide new theoretical results concerning higher dimensional trees. Finally, we provide evidence for our devout conviction that clean mathematical theories provide the basis for clean implementations by indicating how our abstract presentation will make computing with higher dimensional trees easier.