Abstract
To implement arithmetic circuits on plasmodia we face the problem that the plasmodium is propagated in many directions simultaneously in accordance with stimuli and their topology. So, to manage this behaviour we need to limit possible ways of propagation by a number \(p-1\) of attractants for each original point of the plasmodium and for each next step of its transitions.
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Notes
- 1.
Logical operations over formulas \(\forall \varphi (\varphi )\), \(\exists \varphi (\varphi )\), \( \exists \varphi (\lnot \varphi )\), \(\forall \varphi (\lnot \varphi )\) of the second-order logic are closed for the set of truth values 0, 1, 2, 3, therefore any logical operation over the expressions \(\ulcorner \forall \varphi (\varphi )\urcorner \), \(\ulcorner \exists \varphi (\varphi )\urcorner \), \(\ulcorner \exists \varphi (\lnot \varphi )\urcorner \), \(\ulcorner \forall \varphi (\lnot \varphi )\urcorner \) is equivalent to one of them.
- 2.
In the case of 2-adic valued logic we have only two degrees: 0 and 1. The expression “\(v_1(\varphi _1)\) holds for all valuations with the degree 0” means that \(\varphi _1\) is a contradiction. The expression “\(v_1(\varphi _1)\) holds for all valuations with the degree 1” means that \(\varphi _1\) is a tautology. The expression “\(v_{i-1}(\varphi _{i-1})\) holds for all valuations with the degree 0” means that \(\varphi _{i-1}\) is a contradiction. The expression “\(v_{i-1}(\varphi _{i-1})\) holds for all valuations with the degree 1” means that \(\varphi _{i-1}\) is a tautology.
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Schumann, A., Pancerz, K. (2019). p-Adic Valued Logic. In: High-Level Models of Unconventional Computations. Studies in Systems, Decision and Control, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-91773-3_8
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DOI: https://doi.org/10.1007/978-3-319-91773-3_8
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