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Two-Dimensional Uniquely Parsable Isometric Array Grammars

We propose a new subclass of two-dimensional isometric array grammar, called Uniquely Parsable Array Grammar (UPAG). UPAG is a grammar class such that when we parse (i.e. apply rewriting rules reversely to) a given two-dimensional word, the rewritable portions of the word do not overlap each other and the rule that is applicable to each portion is uniquely determined. Thus parsing can be done in a deterministic manner. We study basic properties of UPAG such as unique parsability and parallel parsability. Then we define Monotone Terminating UPAG (MTUPAG), a subclass of UPAG, and show that any two-dimensional word generated by MTUPAG can be parsed in linear time. We also show several examples of MTUPAGs which generate interesting sets of geometrical patterns.