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## Über dieses Buch

This book shows that the plasmodium of Physarum polycephalum can be considered a natural labelled transition system, and based on this, it proposes high-level programming models for controlling the plasmodium behaviour. The presented programming is a form of pure behaviourism: the authors consider the possibility of simulating all basic stimulus–reaction relations. As plasmodium is a good experimental medium for behaviouristic models, the book applies the programming tools for modelling plasmodia as unconventional computers in different behavioural sciences based on studying the stimulus–reaction relations. The authors examine these relations within the framework of a bio-inspired game theory on plasmodia they have developed i.e. within an experimental game theory, where, on the one hand, all basic definitions are verified in experiments with Physarum polycephalum and Badhamia utricularis and, on the other hand, all basic algorithms are implemented in the object-oriented language for simulations of plasmodia. The results allow the authors to propose that the plasmodium can be a model for concurrent games and context-based games.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Physarum polycephalum, called also slime mould, belongs to the species of order Physarales, subclass Myxogastromycetidae, class Myxomycetes, division Myxostelida. Plasmodium is its vegetative phase represented as a single cell with a myriad of diploid nuclei.
Andrew Schumann, Krzysztof Pancerz

### Chapter 2. Natural Labelled Transition Systems and Physarum Spatial Logic

Usually, a labelled transition system is used for describing the behaviour and tempo-spatial structure of concurrent systems [148]. The latter were first introduced by Milner (Communicating and mobile systems: the p-calculus. Cambridge University Press, Cambridge, 1999 [54]).
Andrew Schumann, Krzysztof Pancerz

### Chapter 3. Decision Logics and Physarum Machines

The slime mould can simulate many intelligent processes connecting to transporting. So, we can try to explicate a decision mechanism of Physarum polycephalum in building transporting networks. Let us start with some basic definitions in decision theory.
Andrew Schumann, Krzysztof Pancerz

### Chapter 4. Petri Net Models of Plasmodium Propagation

The slime mould behaviour can be examined as rational from the standpoint of decision theory. So, we can try to implement some strong mathematical tools on the slime mould transitions, such as Petri nets, cf. [119, 121]. Let us remember that Petri nets were developed by C.A. Petri [67] as a graphical and mathematical tool, among others, for describing information processing systems. As a graphical tool, Petri nets can be used as a visual-communication aid. As a mathematical tool, Petri nets are represented, among others, by algebraic equations. For detailed information on Petri nets, we refer the readers to [69].
Andrew Schumann, Krzysztof Pancerz

### Chapter 5. Rough Set Based Descriptions of Plasmodium Propagation

The main problem with formalizing the slime mould behaviours is that Physarum polycephalum tries to occupy all achievable goals simultaneously. In other words, timed transition systems realized by the slime mould cannot be monogenic.
Andrew Schumann, Krzysztof Pancerz

### Chapter 6. Non-Well-Foundedness

A non-well-founded set theory belongs to axiomatic set theories that violate the rule of well-foundedness and, as an example, allow sets to contain themselves: $$X\in X$$. In non-well-founded set theories, the foundation axiom of Zermelo-Fraenkel set theory is replaced by axioms implying its negation.
Andrew Schumann, Krzysztof Pancerz

### Chapter 7. Physarum Language

To build high-level models of plasmodium propagation in Physarum machines and analyze their behaviours in the non-well-founded universe, an object-oriented programming language, called the Physarum language, has been developed by us.
Andrew Schumann, Krzysztof Pancerz

### Chapter 8. p-Adic Valued Logic

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.
Andrew Schumann, Krzysztof Pancerz

### Chapter 9. p-Adic Valued Arithmetic Gates

Let $$\mathscr {PM}=(Ph, Attr, Rep)$$ be a structure of the Physarum machine. In Sect. 3, we have described a dynamics of the Physarum machine $$\mathscr {PM}$$ by the family of the sets of protoplasmic veins formed by plasmodium during its action. In this chapter, the dynamics of the Physarum machine $$\mathscr {PM}$$ will be additionally considered by means of two sets of attractants (occupied by plasmodium and unoccupied) determined for each time instant t.
Andrew Schumann, Krzysztof Pancerz

### Chapter 10. The Rudiments of Physarum Games

In this chapter, we are going to propose a context-based game theory as an example of new mathematics which could be applied in programming different biological devices, not only the Physarum Chips.
Andrew Schumann, Krzysztof Pancerz

### Chapter 11. Physarum Go Games and Rough Sets of Payoffs

Go is a game, originated in ancient China, in which two persons play with a Go board and Go stones. In general, two players alternately place black and white stones, on the vacant intersections of a board with a $$19 \times 19$$ grid of lines, to surround the territory. Whoever has more territory at the end of the game is the winner.
Andrew Schumann, Krzysztof Pancerz

### Chapter 12. Interfaces in a Game-Theoretic Setting for Controlling the Physarum Motions

Physarum polycephalum and Badhamia utricularis demonstrate an intelligent behaviour with intentionality and efficiency, although they do not have nervous systems at all. In particular, they demonstrate the ability to memorize and anticipate repeated events.
Andrew Schumann, Krzysztof Pancerz

### Chapter 13. Conclusions

In moving, the plasmodium switches its direction or even multiplies in accordance with different bio-signals attracting or repelling its motions, e.g. in accordance with pheromones of bacterial food, which attract the plasmodium, and high salt concentrations, which repel it. So, the plasmodium motions can be controlled by different topologies of attractants and repellents so that the plasmodium can be considered a programmable biological device in the form of a timed transition system, where attractants and repellents determine the set of all plasmodium transitions.
Andrew Schumann, Krzysztof Pancerz

### Backmatter

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