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Turing completeness of water computing

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Abstract

We further develop water computing as a variant of P systems. We propose an improved modular design, which duplicates the main water flows by associated control flows. We first solve the three open problems of the previous design by demonstrating: how functions can be stacked without a combinatorial explosion of valves; how termination of the system can be detected; and how to reset the system. We then prove that the system is Turing complete by modelling the construction of \(\mu\)-recursive functions. The new system is based on directed acyclic graphs, where tanks are nodes and pipes are arcs; there are no loops anymore, waterfalls strictly in a ‘top down’ direction. Finally, we demonstrate how our water tank system can be viewed as a restricted version of cP systems. We conclude with a list of further challenging problems.

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Author information

Authors and Affiliations

  1. School of Computer Science, University of Auckland, Auckland, New Zealand

    Alec Henderson, Radu Nicolescu & Michael J. Dinneen

  2. Compucon New Zealand, Auckland, New Zealand

    T. N. Chan

  3. Department of Bioinformatics, Friedrich Schiller University Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Hendrik Happe & Thomas Hinze

Authors
  1. Alec Henderson
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  2. Radu Nicolescu
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  3. Michael J. Dinneen
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  4. T. N. Chan
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  5. Hendrik Happe
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  6. Thomas Hinze
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Correspondence to Alec Henderson.

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Example trace of controlled subtraction

Example trace of controlled subtraction

Here we present a trace of saturation subtraction for both the tank system presented in Figure 4, and cP rules in Table 9 (see Figs. 15, 16, 17, 18, 19, 20).

Fig. 15
figure 15

The initial state of controlled subtraction of \(3\ominus 2\)

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Fig. 16
figure 16

The first step of controlled subtraction of \(3-2\)

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Fig. 17
figure 17

The second step of controlled subtraction of \(3\ominus 2\)

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Fig. 18
figure 18

The third step of controlled subtraction of \(3\ominus 2\)

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Fig. 19
figure 19

The fourth step of controlled subtraction of \(3\ominus 2\)

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Fig. 20
figure 20

The last step of controlled subtraction of \(3\ominus 2\)

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Henderson, A., Nicolescu, R., Dinneen, M.J. et al. Turing completeness of water computing. J Membr Comput 3, 182–193 (2021). https://doi.org/10.1007/s41965-021-00081-3

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