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Turing completeness of water computing

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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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
figure15

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

Fig. 16
figure16

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

Fig. 17
figure17

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

Fig. 18
figure18

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

Fig. 19
figure19

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

Fig. 20
figure20

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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Keywords

  • Water-based computing
  • Membrane systems
  • \(\mu~{\textrm{recursion}}\)