Interfaces in a Game-theoretic Setting for Controlling the Plasmodium

Motions

Andrew Schumann

and Krzysztof Pancerz

1,2

University of Information Technology and Management in Rzeszow, Sucharskiego 2, 35-225 Rzeszow, Poland

University of Management and Administration, Akademicka 4, 22-400 Zamo

c, Poland

Keywords:

Physarum polycephalum, Badhamia utricularis, Bio-inspired Game-theory, Timed Transition System, p-adic

Valued Probability, Knowledge State of Plasmodium, Strategy of Plasmodium, User Interface.

Abstract:

The plasmodium is the large one-cell organism containing a mass of multinucleate protoplasm. It is an active

feeding stage of Physarum polycephalum or Badhamia utricularis and it moves by protoplasmic streaming

which reverses every 30-60 s. In moving, the plasmodium switches its direction or even multiplies in ac-

cordance with different biosignals 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. Furthermore, we can deﬁne p-adic probabilities

on these transitions and, using them, we can deﬁne a knowledge state of plasmodium and its game strategy

in occupying attractants as payoffs for the plasmodium. As a result, we can regard the task of controlling

the plasmodium motions as a game and we can design different interfaces in a game-theoretic setting for the

controllers of plasmodium transitions.

1 INTRODUCTION

Conventionally, the intelligent behavior of animals is

explained by their nervous system that coordinates

voluntary and involuntary actions of animal’s body

and transmits signals between different parts of its

body, which allows animals to act intentionally and

efﬁciently. There is an approach in artiﬁcial intelli-

gence, consisting in building computational models

inspired by these nervous systems, that is called arti-

ﬁcial neural network.

Nevertheless, there are one-cell organisms like

Physarum polycephalum or Badhamia utricularis

(supergroup Amoebozoa, phylum Mycetozoa, class

Myxogastria) without any nervous system and they

are able at their plasmodial stage to build complex

networks for solving different tasks: maze-solving

(Nakagaki, Yamada, and Toth, 2000), minimum-risk

path ﬁnding (Nakagaki, Yamada, and Toth, 2001),

(Nakagaki et al., 2007), associative learning (Shi-

rakawa, Gunji, and Miyake, 2011), etc. In other

References on this new culture are contained in (Neu-

bert et al., 1995)

words, Physarum polycephalum and Badhamia utric-

ularis demonstrate an intelligent behavior with inten-

tionality and efﬁciency, although they do not have ner-

vous systems at all. In particular, they demonstrate

the ability to memorize and anticipate repeated events

(Saigusa et al., 2008). Furthermore, by means of plas-

modium behavior, it is possible to simulate the behav-

ior of some collectives such as collectives of parasites

(Schumann and Akimova, 2013). Thus, the complex

intelligent behavior of plasmodium is biologically un-

explained still and shows the limits of our understand-

ing what natural intelligence is.

Now, there are many attempts to involve the plas-

modium into semi-electrical devices to obtain a semi-

biological and semi-electrical chip in due course (Sun

et al., 2009), (Tsuda, Aono, and Gunji, 2004), (Tsuda

et al., 2011), (Adamatzky, 2010). The point is that

the plasmodium spread by networks can be pro-

grammable and thereby it may simulate different in-

telligent processes. We are working on this problem,

too (Adamatzky et al., 2012). In this paper, we are

going to present our results in modelling the plasmod-

ium networks as timed transition systems (Section

2). Propagations in these systems can be calculated

338

Schumann A. and Pancerz K..

Interfaces in a Game-theoretic Setting for Controlling the Plasmodium Motions.

DOI: 10.5220/0005285203380343

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2015), pages 338-343

ISBN: 978-989-758-069-7

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

by means of p-adic valued probabilities and fuzzi-

ness (Khrennikov and Schumann, 2006), (Schumann,

2008), (Schumann, 2010), see Section 3. In terms of

these probabilities we can deﬁne a knowledge state

of plasmodium and its game strategy in occupying at-

tractants as payoffs for the plasmodium (Section 4).

Hence, we can control the plasmodium motions as a

game (Section 5). As a consequence, user interfaces

for the controllers of plasmodium propagations can

have a natural form of game-theoretic setting.

2 TIMED TRANSITION SYSTEMS

FOR PROGRAMMING THE

PLASMODIUM MOTIONS

The plasmodium is an amorphous yellowish mass

with networks of protoplasmic tubes, programmed by

spatial conﬁgurations of attracting and repelling stim-

uli. Any motion of plasmodium proceeds from one

stimuli to others. As a result, we deal with a kind

of natural transition systems with states presented by

attractants and events presented by plasmodium tran-

sitions between attractants. We can distinguish sev-

eral operations (instructions) in the plasmodium net-

works like: add node, remove node, add edge, re-

move edge (Adamatzky, 2010). Adding and remov-

ing nodes can be implemented through activation and

deactivation of attractants, respectively. Adding and

removing edges can be implemented by means of re-

pellents put in proper places in the space. An activated

repellent can avoid a plasmodium transition between

attractants.

Adding and removing edges (in fact, adding and

removing protoplasmic tubes) can change dynami-

cally over time. To model such behavior, we have

proposed to use timed transition systems as a high-

level model of behavior of plasmodium. Let N be a

set of nonnegative integers. Formally, a timed tran-

sition system T T S = hS,E,T,s

,l, ui consists of the

non-empty set of states S, the set of events E, the

transition relations T ⊆ S × E × S, the initial state s

as well as a minimal delay function (a lower bound)

l : E → N assigning a nonnegative integer to each

event and a maximal delay function (an upper bound)

u : E → N ∪ {∞} assigning a nonnegative integer or

inﬁnity to each event. Usually transition systems are

based on actions which may be viewed as labelled

events. If hs,e,s

i ∈ T , then the idea is that T S can

go from s to s

as a result of the event e occurring at

s. In timed transition systems, timing constraints re-

strict the times at which events may occur. The timing

constraints are classiﬁed into two categories: lower-

bound and upper-bound requirements. A transition

system can be presented as a graph structure with

nodes corresponding to states and edges correspond-

ing to transitions. In case of plasmodium, states repre-

sent attractants whereas edges represent protoplasmic

tubes (plasmodium transitions between attractants).

To program computation tasks for the plas-

modium propagations, we are developing a new

object-oriented programming language (Schumann

and Pancerz, 2013) called the Physarum language,

where the following three basic set descriptions are

deﬁned: (i) T S.State – setting states of plasmodium,

including initial states; (ii) T S.Event – setting events

transiting one states to others; (iii) T S.Transition –

setting transitions of plasmodium. The proposed lan-

guage can be used for developing programs for plas-

modium motions by the spatial conﬁguration of stim-

uli.

Let us consider a simple timed transition sys-

tem shown as a graph structure in Figure 1 with the

following timing constraints: l(e

) = 0, u(e

) = ∞,

l(e

) = 0, u(e

) = ∞, l(e

) = 5, u(e

) = 10.

Figure 1: An example of timed transition system.

The code in the Physarum language has the fol-

lowing form:

#TRANSITION_SYSTEM s1=new TS.State("s1");

s1.setAsInitial;

s2=new

TS.State("s2");

s3=new TS.State("s3");

e1=new TS.Event("e1");

t1=new TS.Transition(s1,e1,s2);

e2=new TS.Event("e2");

t2=new

TS.Transition(s1,e2,s3);

e3=new

TS.Event("e3");

e3.setTimingConstraints(5,10);

t3=new

TS.Transition(s2,e3,s3);

The default timing constraints are 0 as a lower

bound and ∞ as an upper bound.

As a result of programming the plasmodium tran-

sitions, we obtain spatial conﬁgurations of stimuli

InterfacesinaGame-theoreticSettingforControllingthePlasmodiumMotions

339

presented in Figure 2: (a) for the time instant t = 4,

(b) for the time instant t = 8, where P is plasmodium,

, A

are attractants, and R is a repellent. It is

easy to see that the event e

is allowed only if actual

time t ∈ {5, 6, . . . , 10}. Therefore, in the model in Fig-

ure 2(a), a repellent, avoiding the transition between

states s

and s

as a result of the event e

, is present,

i.e. it is activated.

(a)

(b)

Figure 2: Spatial conﬁgurations of stimuli for the plasmod-

ium motions.

3 P-ADIC VALUED

PROBABILITIES AND

FUZZINESS

We have supplemented our language with instructions

enabling us to determine (in the simulation stage) pos-

sible properties of experiments in terms of the proba-

bility space:

• setTimeStep – setting a time step from which the

experiment starts, t = 0,1,2,...,n,

• setTimeEnd – setting a time end when the exper-

iment stops, t = 0, 1, 2, . . . , ∞,

• getNeighCard – getting a cardinality number of

neighboring attractants for a given attractant at the

given setTimeStep and setTimeEnd,

• getAccessCard – getting a cardinality number of

attractants accessible for a given attractant by

protoplasmic tubes at the given setTimeStep and

setTimeEnd.

Instructions for the simulation stage are preceded

with $. Let us consider a simple timed transition sys-

tem given earlier.

If we add the following instructions to the code:

$setTimeStep(0);

$setTimeEnd(10);

$getNeighCard(s2);

$getAccessCard(s2);

we obtain the following 3-adic streams:

• 2 2 2 2 2222222 for getting a cardinality number

of neighboring attractants for A

• 2 2 2 2 2211111 for getting a cardinality number

of attractants accessible for A

Thus, if we have p−1 neighbor attractants for A

we deal with p-adic streams. If setTimeStep(0) and

setTimeEnd(∞) we deal with inﬁnite p-adic streams.

All these streams including both ﬁnite and inﬁnite can

be identiﬁed with p-adic integers. Let us recall that

each p-adic integer has a unique expansion

n = α

+ α

· p + . . . + α

· p

+ ... =

∞

∑

t=0

· p

where α

∈ {0,1,..., p − 1}, ∀t ∈ N. This num-

ber sometimes has the following notation: n =

...α

, where α

can be interpreted as a value

of α at time step t = 0,1,2,...,∞. We have used

the latter notation in our example. The set of p-adic

integers is denoted by Z

. For more details about

p-adic numbers, please see (Koblitz, 1984). Now,

p-adic analysis is used in many applications includ-

ing quantum mechanics (Vladimirov and Volovich,

1989), (Volovich, 1987).

The set Z

cannot be linearly ordered, but there

are many possibilities to deﬁne a partial ordering re-

lation. For example, we can assume that (i) for any

ﬁnite p-adic integers σ,τ ∈ N, we have σ ≤ τ in N

iff σ ≤ τ in Z

; (ii) each ﬁnite p-adic integer n =

...α

(i.e. such that α

= 0 for any i > j) is

less than any inﬁnite number τ, i.e. σ < τ for any

σ ∈ N and τ ∈ Z

\ N; (iii) each inﬁnite p-adic in-

teger σ is less, than p-adic integer τ iff σ

≤ τ

for

all t = 0, 1, 2, . . . Let us denote this ordering rela-

tion by O

. We can see that there exist p-adic inte-

gers, which are incompatible by O

. For example,

let p = 2 and let σ represents the p-adic integer −1/3

= . . . 10101. . . 101 and τ the p-adic integer −2/3 =

. . . 01010. . . 010. Then the p-adic streams σ and τ

are incompatible. Now we can deﬁne sup and inf

digit by digit. Then if σ ≤ τ, so in f (σ,τ) = σ and

sup(σ,τ) = τ. The greatest p-adic integer according

to our deﬁnition is −1 = ...xxxxxx, where x = p − 1,

and the smallest is 0 = . . . 00000.

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Let us deﬁne the Boolean operations on attractants

, A

, . . . so that

getNeighCard(si ∩ s j) :=

in f (getNeighCard(si),getNeighCard(s j);

getAccessCard(si ∩ s j) :=

in f (getAccessCard(si),getAccessCard(s j);

getNeighCard(si ∪ s j) :=

sup(getNeighCard(si),getNeighCard(s j);

getAccessCard(si ∪ s j)

:= in f (getAccessCard(si),getAccessCard(s j);

getNeighCard(¬si) := −1 − getNeighCard(si);

getAccessCard(¬si) := −1 − getAccessCard(si).

Let Ω

∗

denote all attractants both activated and de-

activated at each t = 0,1,2,...,∞. It is a union of all

attractants A

, A

, . . . at each time step. Its subsets

will be denoted by A

∗

⊆ Ω

∗

Let us deﬁne p-adic fuzziness as follows: a p-adic

fuzzy measure is a set function F

(·) deﬁned for sets

∗

⊆ Ω

∗

, it runs over the set Z

and satisﬁes the

following properties:

• F

(Ω

∗

) = −1 and F

(

∗

) = 0.

• If A

∗

⊆ Ω

∗

and B

∗

⊆ Ω

∗

are disjoint, i.e. inf(F

∗

), F

∗

)) = 0, then F

∗

∪ B

∗

)= F

∗

)

+ F

∗

). Otherwise, F

∗

∪ B

∗

)= F

∗

) +

∗

) − inf(F

∗

), F

∗

)) = sup(F

∗

)).

• If A

∗

⊆ Ω

∗

, then F

∗

∩ B

∗

) = in f (F

∗

)).

• F

(¬A

∗

) = −1 − F

∗

) for all A

∗

⊆ Ω

∗

, where

¬A

∗

= Ω

∗

\ A

∗

A p-adic probability measure is a set function

(·) deﬁned for sets A

∗

⊆ Ω

∗

thus:

• P

∗

) = −F

∗

) ∈ Z

• P

∗

) ∈ Q

is characterized by the follow-

ing constraint:

∗

) =

∗

∩B

∗

)

∗

)

∗

∩B

∗

)

∗

)

where P

∗

) 6= 0, P

∗

∩ B

∗

) = inf(P

∗

)).

The measure P

(·) runs over the set Q

of all p-

adic numbers (not only integers). Notice that while

is the ring of p-adic integers, Q

is the ﬁeld of

p-adic numbers.

4 STATES OF KNOWLEDGE AND

STRATEGIES OF

PLASMODIUM

Using p-adic valued fuzziness and probabilities, we

can deﬁne games of plasmodia. So, in the given topol-

ogy of attractants, active zones of plasmodia (initial

states) can be considered players. Suppose, we have

a set of N players, call them i = 1, . . . , N. Agent i’s

knowledge structure is a function P

which assigns to

each attractant ω ∈ Ω

∗

a non-empty subset of Ω

∗

, so

that each thing ω belongs to one or more elements

of each P

, i.e. Ω

∗

is contained in a union of P

, but

are not mutually disjoint. Then P

(ω) is called i’s

knowledge state at the attractant ω. This means that

if the actual state is ω, the individual only knows that

the actual state is in P

(ω).

We can interpret P

(ω) probabilistically as fol-

lows: P

(ω) = {ω

: P

(ω

|ω) > 0}. Evidently that

(ω|ω) > 0 for all ω ∈ Ω

∗

, therefore for all ω ∈ Ω

∗

ω ∈ P

(ω).

Now we consider the relation A

∗

⊆ P

(ω), where

∗

⊆ Ω

∗

, as the statement that at ω agent i accepts the

performance A

∗

= {ω : A

∗

⊆ P

(ω)}.

Let B

∗

mean ‘Attractants, which can be occupied

by agent i’. After several steps, we expect fusions of

all protoplasmic tubes so that all attractants are occu-

pying by all agents. Does it mean that we observe a

union of B

∗

? No, it does not. We face just the situ-

ation that since a time step t = k the sets B

∗

are in-

tersected. Let C

∗

mean ‘Attractants accessible for the

attractant N

by protoplasmic tubes’. Assume, ω ∈ B

∗

and ω

∈ C

∗

. Evidently, P

(ω

|ω) > 0. As a conse-

quence, we assume according to our deﬁnitions that

each agent i knows ω at ω

and knows ω

at ω, i.e.

agent i accepts the performance B

∗

at ω

and i accepts

the performance C

∗

at ω.

Let getAccessSet(i,k) be a set of all at-

tractants such that i knows about them at the

given setTimeStep(t

) and setTimeEnd(t

). A

strategy of a player i is a mapping strat

i,k

getAccessSet(i,k) → Ω

∗

such that for any history

knowledge getAccessSet(i,k) it is true that strat

i,k

be-

longs to the set of attractants accessible at k.

InterfacesinaGame-theoreticSettingforControllingthePlasmodiumMotions

341

5 GAME-THEORETIC

INTERFACES FOR

PLASMODIUM

It is known due to the experiments performed by An-

drew Adamatsky and Martin Grube that if there are

only two agents of the plasmodium game, where the

ﬁrst agent is presented by a usual Physarum poly-

cephalum plasmodium and the second agent by its

modiﬁcation called a Badhamia utricularis plasmod-

ium, then both start to compete with each other.

In particular, the Physarum polycephalum plasmod-

ium grows faster and could grow into branches of

Badhamia utricularis, while the Badhamia utricu-

laris plasmodium could grow over Physarum poly-

cephalum veins. So, we face an interesting form of

zero-sum games.

The user interface for this game is designed on the

basis of the following game steps:

• ﬁrst, the system of Physarum language generates

locations of attractants and repellents;

• second, we can chose n plasmodia/agents of

Physarum polycephalum and m plasmodia/agents

of Badhamia utricularis;

• third, we obtain the task, for example to reach as

many as possible attractants or to construct the

longest path consisting of occupied attractants,

etc.;

• fourth, we can chose initial points for Physarum

polycephalum transitions and initial points for

Badhamia utricularis transitions;

• ﬁfth, we start to move step by step;

• sixth, we deﬁne who wins, either Physarum poly-

cephalum or Badhamia utricularis.

Thus, the plasmodium game has the form of cycle

of Figure 3.

Figure 3: The operative cycle of game-theoretic controller

of plasmodium motions.

In this game, we have two players (the ﬁrst plays

for the Physarum polycephalum plasmodia, the sec-

ond for the Badhamia utricularis plasmodia). The

system places attractants and repellents automatically.

Then the players choose which attractants are occu-

pied before the game and which rules of the game

hold (to reach as many as possible attractants or to

construct the longest path consisting of occupied at-

tractants, etc.). Then the system shows who wins and

who loses.

6 CONCLUSION

The plasmodium motion is an intelligent way of con-

structing expanding networks for solving complex

tasks. This motion has the form of transitions deter-

mined by locations of attractants and repellents. On

these transitions, it is possible to deﬁne p-adic proba-

bilities which are used for deﬁning a knowledge state

of plasmodium and its game strategy in occupying

attractants as payoffs for the plasmodium. Conse-

quently, the task of controlling the plasmodium mo-

tions is considered a game.

ACKNOWLEDGMENT

This research is supported by FP7-ICT-2011-8.

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