Further, we will take into consideration types of rules 1 and 3, only. Unconventional computing becomes an interdisciplinary field of science, where computer scientists, physicists and mathematicians apply principles of information processing in natural systems to design novel computing devices and architectures. In
Physarum Chip Project: Growing Computers from Slime Mould [
2
] supported by FP7,
we are going to implement programmable amorphous biological computers in
plasmodium of Physarum polycephalum. Physarum polycephalum is a one-cell organism
manifesting some primitive intelligence in its propagating and foraging behavior (cf.
[
9
]). A biological computing device implemented in the plasmodium of Physarum
polycephalum is said to be a Physarum machine. A comprehensive information on
Physarum machines can be found in [
1
]. The Physarum machine comprises an
amorphous yellowish mass with networks of protoplasmic veins, programmed by spatial
configurations of attracting and repelling stimuli.
To program Physarum machines, i.e., to set the spatial distribution of stimuli, we
are designing a new object-oriented programming language [
10
], [
11
], [
13
], called the
Physarum language. Moreover, to support research on programming Physarum
machines, we are developing a specialized software tool, called the Physarum software
system, shortly PhysarumSoft (see [
16
]). Our language is based on the prototype-based
approach (cf. [
6
]). According to this approach, there are inbuilt sets of prototypes,
implemented in the language, that correspond to both the high-level models used for
describing behaviour of Physarum polycephalum (e.g., ladder diagrams, transition
systems, timed transition systems, Petri nets) and the low-level model (distribution of
stimuli). In [
15
], we proposed to use Petri nets with inhibitor arcs (cf. [
3
]) as one of
the high-level models to describe behaviour of Physarum polycephalum. The inhibitor
arcs test the absence of tokens in a place and they can be used to disable transitions.
This fact can model repellents in Physarum machines. A transition can only fire if all
its places connected through inhibitor arcs are empty (cf. [
20
]). Each high-level model
(including a Petri net one) is translated into the low-level language, i.e., spatial
distribution of stimuli (attractants and/or repellents). Such distribution can be treated as a
program for the Physarum machine.
In the literature, one can find a lot of approaches using Petri nets as models of
rule-based systems (e.g., [
5
], [
7
], [
18
], [
19
]). First of all, structures of Petri nets reflect
structures of rule-based systems. Various options of structures have been considered,
according to respective approaches. Moreover, the proposed approaches differ in the
dynamics that models reasoning processes. In our research, we propose another approach
in order to reflect dynamics of Physarum machines, i.e., propagation of protoplasmic
veins of the plasmodium according to activation/deactivation of stimuli. In the proposed
Petri net models of Physarum machines, we can distinguish several kinds of places:
– Places representing Physarum polycephalum.
– Places representing control stimuli (attractants or repellents) corresponding to
propositions in antecedent parts of rules.
– Places representing auxiliary stimuli (attractants) corresponding to partial results of
evaluation of logical expressions in antecedent parts of composite production rules.
– Places representing output stimuli (attractants) corresponding to propositions in
consequence parts of rules.
For each kind of places, we adopt different meaning (interpretation) of tokens (see,
for example, Tables 1, 2 and 3 for places representing control stimuli and places
representing output stimuli, respectively). Each token corresponds to proper evaluation
of the proposition according to the role played by a given stimulus. In our models
of simple rule-based systems, we have implemented an idea of flowing power used
in ladder diagrams to model digital circuits. The same idea was used by us to
construct logic gates through the proper geometrical distribution of stimuli in Physarum
machines (see [
17
]). Flowing power is replaced with propagation of plasmodium of
Physarum polycephalum. Therefore, in each Petri net model of a rule, a place
representing Physarum polycephalum is present. Petri net models are a useful tool to
reflect dynamics of Physarum machines, i.e., propagation of protoplasmic veins of the
plasmodium in consecutive time instants. Tokens present in places representing output
stimuli show which attractants of Physarum machines are occupied by the plasmodium
at given time instants.
In general, we can distinguish two techniques to control behavior of Physarum
polycephalum: repellent-based and attractant-based [
1
]. Attractants are sources of nutrients
or pheromones, on which the plasmodium feeds. In case of repellents, the fact that
plasmodium of Physarum avoids light and some thermo- and salt-based conditions is used.
These possibilities are reflected in the cretaed Petri net models. Technically, the second
approach is easier to implement. In case of repellent-based control approach, Petri net
models of production rules of type 1 and 3 have the form as in Figures 1 and 2,
respectively. In these models, the places Rdi1, Rdi2, ..., Rdi3 correspond to propositions in
the antecedent parts of the rules. The relationship between meaning of tokens and
evaluation of propositions is shown in Table 1. These places are translated into repellents in
the low-level model (distribution of stimuli). The places A1, A2, ..., Ak−1 correspond
to auxiliary stimuli. The relationship between the meaning of tokens and the evaluation
of propositions is shown in Table 2. The place Adj correspond to the output stimulus.
The relationship between meaning of tokens and evaluation of propositions is shown in
Table 3. It is easy to see that, in case of type 1 of production rules, the token is present
in Adj (the proposition in the consequence part is true), if all places Rdi1, Rdi2, ..., Rdi3
do not hold tokens (the propositions in the antecedent part are true). In case of type
3 of production rules, the token is present in Adj (the proposition in the consequence
part is true), if at least one of the places Rdi1, Rdi2, ..., Rdi3 does not hold a token (at
least one of the propositions in the antecedent part is true). The structures of Physarum
machines for production rules of type 1 and 3 are shown in Figure 3 (a) and (b),
respectively. Distributions of stimuli can be treated as programs for these machines. In
the further research, we will consider more complex rule-based systems. However, we
are aware of the topological constraints if the Physarum machine is implemented in the
two-dimensional space (e.g., on the Petri dish). In this case, propagation of
protoplasmic veins forming a planar graph is admissible only.
Another challenging problem is to use Physarum machines in the process of
optimization of rule-based systems. Physarum polycephalum is originally famous as a
computing biological substrate due to its alleged ability to approximate shortest path
from its inoculation site to a source of nutrients.
Acknowledgments