On Complexity of Two Dimensional Languages Generated by Transducers

We consider two-dimensional languages, called here 2d transducer languages, generated by iterative applications of transducers (finite state automata with output). To each transducer a two-dimensional language consisting of blocks of symbols is associated: the bottom row of a block is an input string accepted by the transducer and, by iterative application of the transducer, each row of the block is an output of the transducer on the preceding row. We observe that this class of languages is a proper subclass of recognizable picture languages containing the class of all factorial local 2d languages. By taking the average growth rate of the number of blocks in the language as a measure of its complexity, also known as the entropy of the language, we show that every entropy value of a one-dimensional regular language can be obtained as an entropy value of a 2d transducer language.

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