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Computational completeness of complete, star-like, and linear hybrid networks of evolutionary processors with a small number of processors

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Abstract

A hybrid network of evolutionary processors (HNEP) is a graph where each node is associated with a special rewriting system called an evolutionary processor, an input filter, and an output filter. Each evolutionary processor is given a finite set of one type of point mutations (insertion, deletion or a substitution of a symbol) which can be applied to certain positions in a string. An HNEP rewrites the strings in the nodes and then re-distributes them according to a filter-based communication protocol; the filters are defined by certain variants of random-context conditions. HNEPs can be considered both as languages generating devices (GHNEPs) and language accepting devices (AHNEPs); most previous approaches treated the accepting and generating cases separately. For both cases, in this paper we show that five nodes are sufficient to accept (AHNEPs) or generate (GHNEPs) any recursively enumerable language by showing the more general result that any partial recursive relation can be computed by an HNEP with (at most) five nodes with the underlying graph structure for the communication between the evolutionary processors being the complete or the linear graph with five nodes, whereas with a star-like communication graph we need six nodes. If the final results are defined by only taking the terminal strings out of the designated output node, then for these extended HNEPs we can prove that only four nodes are needed in all cases—for computing any partial recursive relation as well as for generating and accepting any recursively enumerable language—and the underlying communication structure can be a complete or a linear graph, but now even a star-like graph, too.

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Notes

  1. These results were already claimed in the original paper (Alhazov et al. 2014b) presented at UCNC 2014; in this paper we not only establish a new extended proof of these results, but also consider the variants with terminal extraction and with linear and star-like communication structures.

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Acknowledgments

The work of the first author and the fourth author was supported by Project STCU-5384 awarded by the Science and Technology Center in the Ukraine. The authors gratefully acknowledge the suggestions and comments of the two anonymous referees.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Academiei 5, 2028, Chişinău, Moldova

    Artiom Alhazov & Yurii Rogozhin

  2. Faculty of Informatics, TU Wien, Favoritenstraße 9-11, 1040, Wien, Austria

    Rudolf Freund

  3. Department of Information Technologies, Åbo Akademi University, Joukahainengatan 3-5, 20520, Turku, Finland

    Vladimir Rogozhin

Authors
  1. Artiom Alhazov
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  2. Rudolf Freund
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  3. Vladimir Rogozhin
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  4. Yurii Rogozhin
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Corresponding author

Correspondence to Rudolf Freund.

Additional information

Yurii Rogozhin: deceased on March 10th 2014.

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Alhazov, A., Freund, R., Rogozhin, V. et al. Computational completeness of complete, star-like, and linear hybrid networks of evolutionary processors with a small number of processors. Nat Comput 15, 51–68 (2016). https://doi.org/10.1007/s11047-015-9534-1

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