A Quest for Kleene Algebra in 2 Dimensions

Abstract

The term Kleene algebra refers to a certain algebraic structure, built up using sequential composition, its iterated version, and union. The key operation is composition: on strings, it connects the final point of the first string to the initial point of the second string. The quest for Kleene algebra in 2 dimensions starts with the clarification of the notions of word and composition in 2 dimensions. A 2-dimensional word is an arbitrary shape area, consisting of unit square cells, and filled with letters. Word composition puts two words together, without overlapping, and controls the contact elements of the contours of these words. This method actually defines a family of composition operations, indexed by the restrictions used to control the words’ contact parts. Finite automata and regular expressions are extended to 2 dimensions. The former is relatively easy and it reduces to tiling. For the latter, a few recently introduced classes of regular expressions n2RE and x2RE are presented. The formalism is completed with a mechanism to specify and solve recursive systems o equations for generating languages in 2 dimensions. Finite automata and regular expressions are equivalent and Kleene algebra provides a beautiful algebraic setting to formalize this result. A section on the limits of our current understanding on lifting this result to 2 dimensions is included. Finally, we briefly show that, enriched with spatial and temporal data attached to tiles, the formalism leads to a natural model for interactive, distributed programs.