Barroso Mancha, Marc

idUS. Depósito de Investigación Universidad de Sevilla

On the cellular automata P systems and chainreactionsMarc Barroso ManchaUniversitat de BarcelonaEmail: marc.barroso4@gmail.comThis paper has been written after attending the 14th Brainstorming Week onMembrane Computing as a physics student from University of Barcelona. Thework presented here tries to represent what I have learned during that period, whiletrying to apply some of the most innovative concepts shown in the presentationsattended during that week into some interesting situations.1 IntroductionThis brief paper aims to introduce the cellular automata P systems as an exampleof ESNP (extended spiking neural P systems) with transmittable states, and thenapply the available rules to simulate a simple model of a random walk in 2D. Let’sstart by formally introducing the system. We have the usual definition:Π = (O,Q, σ1, ..., σn, in, out) (1)· whereO = {a}, such that a is called the spike·Q is a finit alphabet of states· σi are neurons, defined by: σi = (αi, ni, fi, Ri) where→ αi ∈ Q, is the initial state→ ni is the initial number of spikes→ fi is the state combining function→ Ri is a finite set of rules· in is the input neuron· out is the output neuron132 Marc Barroso ManchaIt is also necessary to explain how the rules work. The most general form is:α/ac → (t1, ak1 , β1), ..., (tm, akm , βm) such that α, βj ∈ Q (2)that just means that when the state of the given neuron is α, and if the neuronhas exactly c objects ‘a’ inside, it should send kj spikes to the neuron tj , whilealso transmitting the state βj , for 1 ≤ j ≤ m. When multiple rules are appliedinto the same neuron in the same clock tick (i.e. different states are transmittedto the same neuron σi), the function fi dictates how they should be combined toobtain a unique final state.2 Cellular automataA cellular automata is defined as a grid of cells, such as every cell can be inone of a finite number of states. The set of cells next to each one it’s called itsneighborhood. Then, at every generation, some fixed rules determine the new stateof each cell in terms of the current state of the cell and the states of the cells in itsneighborhood. One of the most important applications is the known as “Conway’sGame of Life”, originally created by the mathematician John Conway in 1970.Its importance is due to the fact that it can be proven to be a universal Turingmachine (that is, anything that can be computed algorithmically can be computedwithin Conway’s Game of Life - even the Game of Life itself!). Rudolf Freund andSergiu Ivanov show in their presentation “Extended SNP Systems with States”that the ESNP with transmittable states is analogue to a cellular automata (withthe only change we are going to do is talk about neurons instead of cells). Morespecifically, when only two states are considered, there is an easy set of rules thatenables us to simulate the Game of Life. Thus, they showed that we can obtainuniversality with only two states in the ESNP paradigm. The description of suchsystem can be found on their presentation.3 Simple nuclear chain reaction modelIn order to implement these ideas into something more tangible, we can think ofthe following situation: we have an organized grid of atoms (represented by theneurons), where all of them are stable (in the sense that no rules could be appliedinitially, that is, the system would halt immediately). In the position (i, j) weintroduce an unstable atom that will explode in one clock tick, releasing n num-ber of particles (represented by the objects), that will go to any of its neighbors,making them unstable, and thus propagating some kind of state (generating thechain reaction). This is known as a two dimensional random walk. Some attractiveOn the cellular automata P systems and chain reactions 133studies can consist of varying the number of objects an explosion yields, observ-ing whether the reaction consume all the possible atoms or not (if an atom thathas already exploded is considered to be destroyed insted of being replaced), orconsidering some time of interaction in every step. Another interesting behavior(also more difficult to implement) would be trying to change the geometry of thesystem (so the number of neighbors could vary from atom to atom), and see if youcould get more efficiency some way or another.This is obviously the first iteration of the model one can think of: furthercomplications can be considered, as making the grid in 3D (just by adding layersupon layers of atoms), or even trying to add probabilities so further atoms thanthe more direct neighbors have a chance of becoming unstable. This model canbe used to simulate the path of a photon that emerges from the Sun’s core and istrying to reach the surface. A very simple model has been implemented at whataims to be an ESNP simulator, programmed in Python. It consists of a square gridof arbitrary dimensions, and an unstable atom in the middle. The initial reactionlasts two ticks, and send one spike in two random different directions. For the sakeof simplicity, we have used three different states: 0 - stable atom, 1 - unstable atom,2 - already exploded atom, even if two were already enough, as explained before.We understand that this model has some difficulties (we only used the most directneighbors; this can be extended to have probabilities for all 8 adjoin neurons), andcan even be too simplified (we know that atoms won’t be organized in a rectangulargrid, as shown here, but will have some kind of spatial distribution). But whilemodeling this problem, we found out that this could also be implemented as somekind of A∗ path-finding algorithm or some kind of path algorithm, only haltingthe computation when a certain point is reached or if no rules have been applied,and then letting a lot of systems run in parallel. With a comparison between thenumber of rules used, one can actually get a good representation of the optimalroute. Could we explot from the fact that we can acces more states and takeadvantage of it, working under the P-systems paradigm? Further research andmodeling must be done to answer this.

On the cellular automata P systems and chain reactions

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On the cellular automata P systems and chain reactions

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