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Transducer generated arrays of robotic nano-arms

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Abstract

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.

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Notes

  1. Often in literature the symbol used as the boundary symbol is # which we use to denote the cardinality of a set.

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Acknowledgements

This work has been supported in part by the NSF grants CCF #0523928 and CCF #0726396 to N.J. and by grants GM-29554 from NIGMS, grants DMI-0210844, EIA-0086015, CCF-0432009, CCF-0523290 and CTS-0548774, CTS-0608889 from the NSF, 48681-EL and W911NF-07-1-0439 from ARO, N000140910181 from the office of Naval Research and a grant from the W.M. Keck Foundation, to N.C.S.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620, USA

    Egor Dolzhenko & Nataša Jonoska

  2. Chemistry Department, New York University, New York, NY, 10003, USA

    Nadrian C. Seeman

Authors
  1. Egor Dolzhenko
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  2. Nataša Jonoska
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  3. Nadrian C. Seeman
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Correspondence to Egor Dolzhenko.

Additional information

This paper is written for the occasion of the 60th birthday of our dear friend Giancarlo Mauri. Happy birthday Giancarlo!

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Dolzhenko, E., Jonoska, N. & Seeman, N.C. Transducer generated arrays of robotic nano-arms. Nat Comput 9, 437–455 (2010). https://doi.org/10.1007/s11047-009-9157-5

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