RD Semantics

8/3/2019 RD Semantics

1/14

Toward semantical modelof reaction-diffusion computing

Andrew SchumannDepartment of Philosophy and Science Methodology,

Belarusian State University, Minsk, Belarus, and

Andrew AdamatzkyDepartment of Computer Science, University of the West of England, Bristol, UK

Abstract

Purpose The purpose of this paper is to fill a gap between experimental and abstract-theoreticmodels of reaction-diffusion computing. Chemical reaction-diffusion computers are amongst leadingexperimental prototypes in the field of unconventional and nature-inspired computing. In the

reaction-diffusion computers, the data are represented by concentration profiles of reagents,information is transferred by propagating diffusive and phase waves, computation is implemented ininteraction of the traveling patterns, and results of the computation are recorded as a finalconcentration profile.

Design/methodology/approach The paper analyzes a possibility of co-algebraic representationof the computation in reaction-diffusion systems using reaction-diffusion cellular-automata models.

Findings Using notions of space-time trajectories of local domains of a reaction-diffusion mediumthe logic of trajectories is built, where well-formed formulas and their truth-values are defined byco-induction. These formulas are non-well-founded set-theoretic objects. It is demonstrated that thelogic of trajectories is a co-algebra.

Research limitations/implications The paper uses the logic defined to establish a semanticalmodel of the computation in reaction-diffusion media.

Originality/value The work presents the first ever attempt toward mathematical formalization of

reaction-diffusion processes and is built building up semantics of reaction-diffusion computing. It isenvisaged that the formalism produced will be used in developing programming techniques ofreaction-diffusion chemical media.

Keywords Cybernetics, Logic, Semantics, Computer applications

Paper type Research paper

1. IntroductionReaction-diffusion computers (Adamatzky, 2001; Adamatzky et al., 2005) are spatiallyextended chemical systems, which process information using interacting growingpatterns, of excitable and diffusive waves. In reaction-diffusion processors, both thedata and the results of the computation are encoded as concentration profiles of thereagents. The computation is performed via the spreading and interaction of

wave fronts. As specified in Adamatzky (2001) and Adamatzky et al. (2005), animplementation of reaction-diffusion computers is based on three principles of thephysics of computation (Margolus, 1984). First principle states that the physical actionmeasures the amount of information, i.e. a dynamics of the reaction-diffusion system isinterpreted as computation. Second principle is that physical information travels only afinite distance, i.e. the natural computation is always local. Third principle says thatthe nature is governed by propagating patterns and traveling waves, a computation isspatial.

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0368-492X.htm

K38,9

1518

Kybernetes

Vol. 38 No. 9, 2009

pp. 1518-1531

q Emerald Group Publishing Limited

0368-492X

DOI 10.1108/03684920910991504


2/14

It was proved theoretically and demonstrated in laboratory experiments thatreaction-diffusion computers are capable for solving advanced computational tasks,including image processing and computational geometry, logics and arithmetics, androbot control; see (Adamatzky et al., 2005) for detailed references, and overview of

theoretical and experimental results.There is a particular feature of reaction-diffusion chemical computers: the media are

fully conductive for chemical or excitation waves. Every point of a medium can beinvolved in the propagation of chemical waves and reactions between diffusingchemical species. Once a reaction is initiated in a point, it spreads all over thecomputing space by target and spiral waves. Such phenomena of wave-propagation,analogous to one-to-all broadcasting in massive-parallel systems, are employed tosolve problems ranging from the Voronoi diagram construction to robot navigation(Adamatzky, 2001; Adamatzky et al., 2005).

Despite extensive experimental research and a series of successful implementationsof working prototypes of chemical reaction-diffusion computers the field ofreaction-diffusion computing is yet far from being accepted by old-school computerscientists, educated in 1950-1970s. This is because the reaction-diffusion computinglacks formalisms, so common for any other classical models of computation. In presentwe are laying the road toward overcoming the deficiency, and filling a gap betweenexperimental and abstract-theoretic models of reaction-diffusion computing.

First step for formal representation of reaction-diffusion computers is proposed inSection 2 in terms of reaction-diffusion cellular automata. In Section 4, we introduce alogic of trajectories to formalize a computation as spatio-temporal behavior of anyparticular element of a reaction-diffusion medium. We then combine our findings inreaction-diffusion cellular automata and co-algebras to outline a semantical model ofthe computation in Section 5.

2. Reaction-diffusion automataEvery chemical non-stirred reaction-diffusion medium (micro-volume) can berepresented as a finite state machine or an automaton. Its states correspond toreagents which prevails in theirs concentration at any discrete time. For example, ifthere are two reagents, a and b, in a medium, then each micro-volume x can berepresented by a finite automaton axsuch that if at the time step tconcentration ofa inx exceed concentration ofbin x, then ax takes the state a, otherwise the stat b. Cellularautomata (Boccara, 2003; Chopard and Droz, 2005; Ilachinski, 2001) are bestcomputational structures to represent space-time dynamics of spatially extendednon-linear systems, including reaction-diffusion media. This is because a cellularautomaton is a regular network, or a lattice, array, of locally connected finite automataupdating their states in parallel. Recall that a cellular automaton is a four-tuple

A kZd;S; u;fl, where:(1) d[ N is a number of dimensions and the members of Zd are referred as cells;

(2) S is a finite set of elements called the states of an automaton A, the members ofZd take their values in S;

(3) u , Zd \{0}d is a finite ordered set of n elements, U(x ) is said to be aneighborhood for the cell x; and

(4) f : Sn1 ! S that is f is the local transition function (or local rule).

Reaction-diffusion

computing

1519


3/14

As we see an automaton is considered on the endless d-dimensional space of integers,i.e. on Zd. Discrete time is introduced for t 0, 1, 2, . . . For instance, the sell x at time tis denoted by x t. Each automaton calculates its next state depending on states of itsclosest neighbors. The cellular automata thus represent locality of physics of

information and massive-parallelism in space-time dynamics of natural systems.To represent a chemical reaction-diffusion system in a cellular automaton

(Adamatzky, 1994) with local transition function f and cell-x neighborhoodux ky1; . . . ;ynl, one needs to select a substrate state, let us call it s, such that:

fs; . . .s s

a set of reactants Q {r1, . . . rm}, and then determine a diffusion and reactionsequations. In most primitive form the diffusion can be specified as follows:

x t1

ri if xt s and Dtx {s; ri}

s if x t s and jDtx={s}j . 1

x t otherwise;

8>>>:

where Dtx {yt[ Q : y t[ ux} is a set of states observed in a neighborhood of x at

time t.Reactions between reactants ofQcan be represented in cell-state transition rules by

many different ways, the more generalized totalistic coding suggested inAdamatzky et al. (2006), Wuensche and Adamatzky (2006) and Adamatzky andWuensche (2007): a cells update depends on the number of different cell-states in itsneighborhood irrespective of the cell-states positions. Thus, if there are p m 1components in the chemical system then the update (transition) rule can be written asfollows:

x t1 fdpxt; dp21xt; . . . ; d0xt;

where da(x)t is the number of cell xs neighbors with cell-state a [ {s} < Q at time

step t.

3. Formalization of reaction-diffusion computingLet us consider most famous task approximation of Voronoi diagram inreaction-diffusion systems, see overview in Adamatzky et al. (2005). Let P be anonempty finite set of planar points and jPj n. For points p (p1,p2) andx (x1,x2)

let dp;x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p1 2x12 p2 2x2

2p

denote their Euclidean distance. A planarVoronoi diagram of the set P is a partition of the plane into such regions, that for any

element of P, a region corresponding to a unique point p contains all those points ofthe plane which are closer to p in respect to the distance d than to any other nodeof P. A unique region:

vorp m[P;mp> {z [ R2 : dp;z , dm;z}

assigned to point p is called a Voronoi cell of the point p.Voronoi cells of a planar set represent the natural or geographical neighborhood

of the sets elements. Therefore, the computation of a Voronoi diagram based

K38,9

1520


4/14

on the spreading of some substance from the data points is usually the first approachof those trying to design massively parallel algorithms, see overview of experimentaltechniques in de Lacy Costello et al. (2004). Let us consider a simplistic implementationof the Voronoi diagram construction in a reaction-diffusion medium with one diffusing

reagent a and a substrate s.The reagent a diffuses from sites introduced to correspond to the elements of a given

planar set P. When two diffusing wave fronts meet a super-threshold concentration ofreagents prevents waves from spreading further. A cellular-automaton model representsthis as follows.

Every cell has two possible states: s (resting state or a substrate) and a (reagent). Ifthe cell is in state a it remains in this state forever. If the cell is in state s and betweenone and three of its neighbors are in state a, then the cell takes the state a; otherwise,the cell remains in the state s (this reflects the super-threshold inhibition, or aself-inhibition idea). A cell state transition rule is as follows:

x t1 a; if x t s and 1 # dxt# a

x t; otherwise

(1

where dxt j{y [ ux : y t a}j, and a 3 for a rectangular, eight-cellneighborhood, and a 2 for a hexagonal, six-cell neighborhood, cellular automaton.

An example of a Voronoi diagram computed in an automaton model of areaction-diffusion medium with one reagent and one substrate is shown in Figure 1.The diagram constructed is just an approximation of Voronoi diagram metric L

1, see

discussion in Adamatzky (2001).Let X be the two-dimensional Euclidean space R2. Let T denote the discrete time

and Qbe a finite set of states of the following three sorts: substrate s, activator a, and

inhibitor i. At each point on the metric space X, we allocate an infinite sequence of statetransitions. Let sdenote a function from Xto (Q T)w, i.e. for each point x [ X, sx is anonempty infinite sequence of pairs from Q T. Further, we will use some basicnotions of stream calculus such as co-induction and bisimulation, for more details see(Pavlovic and Escardo, 1998).

The function sx is a kind of stream and will be said to be a trajectory ofx. The set ofall trajectories is denoted by Tr(X, Q, T).

For a trajectory sx, we call sx (0) the initial value ofsx. We define the derivative of atrajectory sx, for all n $ 0, by sx

0n sxn 1. For any n $ 0, sx(n) is called then-th element of sx. It can also be expressed in terms of higher-order trajectoryderivatives, defined, for all k $ 0, by s0x sx; s

k1x s

kx

0. In this case, the nthelement of a trajectory sx is given by sxn s

nx 0. Also, the trajectory is

understood as an infinite sequence of derivatives. It will be denoted by aninfinite sequence of values or by an infinite tuple: sx sx0


5/14

If there exists a bisimulation relation R with ksx; tyl [ R then we write sx , ty andsay that sx and ty are bisimilar. In other words, the bisimilarity relation , is the union

of all bisimulations: ,:


6/14

computes the greatest bisimulation relation R that contains the pair ksx; tyl. Bycoinduction, it follows that sx ty for all pairs ksx; tyl [ R.

Notice that, sx ty means that the points x and y of X have the same trajectory.Meanwhile, one point x can have different trajectories sx tx.

Let us assume that our metric space Xhas a partition on Voronoi cells of the formvor(p ) in accordance with planar points p [ P. Each Voronoi cell vor(p) has just onestate. This means that it contains either a substrate (resting state), or only one reagent.Initial values of trajectories are substrate or reagents. In this case, it is natural that ifx,y [ vor(p ), then sx0 sy0 and if x [ vor(q), y [ vor(p) (vorp vorq ), thensx0 sy0. In other words, if points ofXbelong to the same Voronoi cell, then theirtrajectories have the same initial value and if points of X belong to different Voronoicells, then their trajectories have different initial values. Trajectories depend onreactions among Voronoi cells (i.e. among substrate and reagents).

Let us distinguish two kinds of neighborhood: for points ofX(p-neighborhood) andfor cells of {vorp : p [ P} (c-neighborhood). While the p-neighborhood for openVoronoi cells consists of an infinite number of members that have the same initial state,

the c-neighborhood consists of a finite number of members (they are other Voronoicells) which are necessarily in different initial states. Thus, the p-neighborhood doesnot play a significant role in the transition of the whole system (it is important only fora transition within the framework of one Voronoi cell). Therefore, we will considerc-neighborhood more often.

A trajectory sx for every x [ Xdepends on reactions among Voronoi cells, e.g. on atransition rule fcharacteristic for an appropriate cellular automaton of an appropriateVoronoi diagram.

Consider now a three-adic state automaton, where every cell takes one of thecell-states of the following three sorts: substrate S {s1, . . . , sm}, activatorA {a1, . . . , al} or inhibitor I {i1, . . . , ik}. The cardinality of the set of statesQ S A Iis no less than the number of members of the set P: jQj m l k $jPj (we suppose that some states are superposition of basic states and the number ofbasic states are equal to jPj, moreover jQj $ jPj, because some reagents may beobtained as result of reactions of basic reagents of P). The state of the first sort S is adedicated substrate state: a point in state sj [ S, whose c-neighborhood is filled onlywith states sj [ S, does not change its state (sj are analogous of quiescent state incellular automaton models). The states of two other sorts, A and I, are assigned to bereactants. The cell-state transition rule f can be written as follows:

sxn 1 fCi1sxn; . . . ;Ciksxn; Ca1sxn; . . . ;Calsxn;

Cs1sxn; . . . ;Csmsxn;2

where k jIj, l jAj, m jSj, Cpsxn is the number of point xs c-neighbors withstate p [ {i1; . . . ; ik; a1; . . . ; al;s1; . . . ;sm} at time step tn.

Consider a ternary state automaton based on a two-dimensional lattice withhexagonal tiling. The automaton imitates reaction-diffusion medium in a sub-excitablemode. In such mode propagation of activator is limited and therefore not classicaltarget or spiral waves formed by travelling self-localizations emerge. See detaileddescription and particularly analysis of the automatons computational potential

Reaction-diffusion

computing

1523


7/14

in Adamatzky et al. (2006), Wuensche and Adamatzky (2006) and Adamatzky andWuensche (2007).

The c-neighborhood size is seven: the central cell and its six closest c-neighbors.To give a compact representation of the cell-state transition rule, we represent the

cell-state transition rule as a matrix M (mij ), where 0# i# j # 7, 0 # i j # 7,and mkl[ {s, a, i}. The output state of each c-neighborhood is given by the row-index k(the number of c-neighbors in cell-state i ) and column-index l (the number ofc-neighbors in cell-state a ). We do not have to count the number of c-neighbors incell-state s, because it is given by 7 2 (k l ). A point with a c-neighborhoodrepresented by indexes k and l will update to cell-state mkl which can be read off thematrix. In terms of the cell-state transition function this can be presented as follows:sxn 1 MCasx nCisxn.

Here, is the exact matrix structure, which corresponds to matrix M3 (i.e. ifCasxnCisxn 3). This is a so-called spiral rule (Adamatzky and Wuensche,2007), derived from the beehive rule (Wuensche and Adamatzky, 2006):

s a i a i i i i

s i i a i i i

s s i a i i

s i i a i

s s i a

s s i

s s

s

3

This matrix represents an example of transition rule (2) for the ternary state automatonbased on a two-dimensional lattice with hexagonal tiling. See example of theautomatons configuration in Figure 2.

Thus, as previously discussed in Adamatzky et al. (2006), Wuensche andAdamatzky (2006) and Adamatzky and Wuensche (2007), m01 a symbolizes thediffusion of activator a, m11 i represents the suppression of activator a by theinhibitor i, and mz2 i (z 0, . . . , 5) can be interpreted as self-inhibition of the activatorin particular concentrations. mz3a (z 0, . . . , 4) means a sustained excitation underparticular concentrations of the activator. mz0s (z 0, . . . , 7) means that the inhibitoris dissociated in absence of the activator, and that the activator does not diffuse in

subthreshold concentrations. And, finally, mzp i, p $ 4 is an upper-thresholdself-inhibition.

As we demonstrated in Wuensche and Adamatzky (2006) and Adamatzky andWuensche (2007), the cell-state transition rule (3) reflects the nonlinearity ofactivator-inhibitor interactions for sub-threshold concentrations of the activator.Namely, for a small concentration of the inhibitor and for threshold concentrations(values 1 and 3), the activator is suppressed by the inhibitor, while for criticalconcentrations of the inhibitor (value 2) both inhibitor and activator dissociate

K38,9

1524


8/14

producing the substrate, as symbolized in the following set of quasi-chemical reactions:a 6sa a, a ia i, a 2ia s, 3aa a, a 3ia i, 2aa i, ia s, etc.

Transition rule (2) may be represented as a superposition of logical functions of aspecial kind. As a result, we can obtain a semantical model of cellular automata ofsub-excitable reaction-diffusion media.

4. Logic of trajectories

Now, consider a propositional logic Lv for the set of all trajectories Tr(X, Q, T); itssyntax and semantics are defined by coinduction and their objects are trajectories(i.e. streams of pairs of cell-states and time-steps). The syntax of L

v is as follows:Variables. p< pjqjr. . .,where p, q, r are members of the product Q T of the set of states Q for an

appropriate reaction-diffusion cellular automaton and of the time line T.Constants. c< `j where ` means the truth and means the falsity.

Figure 2.Example of spatial

activity in automaton rule

(2), circles are cell in stateof activator, solid discs are

inhibitorsSource: See details in Adamatzky and Wuensche (2007)

Reaction-diffusion

computing

1525


9/14

Formulas. w;c< pjcj : cjw_ cjw^ cjw. cThese definitions are coinductive. For instance:

. a variable p is of the form of a stream p p0< p1< p2

Documents

RD Semantics