Transducer generated arrays of robotic nano-arms

We consider sets of two-dimensional arrays, called here transducer generated languages, obtained by iterative applications of transducers (finite state automata with output). Each transducer generates a set of blocks of symbols such that 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 show how these arrays can be implemented through molecular assembly of triple crossover DNA molecules. Such assembly could serve as a scaffold for arranging molecular robotic arms capable of simultaneous movements. We observe that transducer generated languages define a class of languages which is a proper subclass of recognizable picture languages, but it contains the class of all factorial local two-dimensional languages. By taking the average growth rate of the number of blocks in the language as a measure of its complexity, we further observe that arrays with high complexity patterns can be generated in this way.