Fredkin and Toffoli gates implemented in Oregonator modelof Belousov-Zhabotinsky medium

Andrew Adamatzky∗

University of the West of England, Bristol, UK

(Dated: December 28, 2016)

A thin-layer Belousov-Zhabotinsky (BZ) medium is a powerful computing device capable forimplementing logical circuits, memory, image processors, robot controllers, and neuromorphic archi-tectures. We design the reversible logical gates — Fredkin gate and Toffoli gate — in a BZ mediumnetwork of excitable channels with sub-excitable junctions. Local control of the BZ medium ex-citability is an important feature of the gates’ design. A excitable thin-layer BZ medium respondsto a localised perturbation with omnidirectional target or spiral excitation waves. A sub-excitableBZ medium responds to an asymmetric perturbation by producing travelling localised excitationwave-fragments similar to dissipative solitons. We employ interactions between excitation wave-fragments to perform computation. We interpret the wave-fragments as values of Boolean variables.A presence of a wave-fragment at a given site of a circuit represents logical truth, absence of thewave-fragment — logical false. Fredkin gate consists of ten excitable channels intersecting at elevenjunctions eight of which are sub-excitable. Toffoli gate consists of six excitable channels intersectingat six junctions four of which are sub-excitable. The designs of the gates are verified using numericalintegration of two-variable Oregonator equations.

I. INTRODUCTION

A thin-layer Belousov-Zhabotinsky (BZ) medium [16,119] exhibits target waves, spiral waves and localisedwave-fragments and their combinations. A numberof theoretical and experimental laboratory prototypesof BZ computing devices have been produced. Theyare image processes and memory devices [52, 59, 60],logical gates implemented in geometrically constrainedBZ medium [86, 88], approximation of shortest pathby excitation waves [6, 74, 89], memory in BZ micro-emulsion [52], information coding with frequency of os-cillations [39], onboard controllers for robots [8, 107, 114],chemical diodes [49], neuromorphic architectures [36,41, 42, 44, 91, 99] and associative memory [92, 93],wave-based counters [40], and other information proces-sors [28, 38, 43, 117]. First steps have been alreadymade towards prototyping arithmetical circuits with BZ:simulation and experimental laboratory realisation ofgates [1, 5, 7, 86, 88, 103], clocks [25] and evolvinglogical gates [102]. A one-bit half-adder, based on aballistic interaction of growing patterns [2], was imple-mented in a geometrically-constrained light-sensitive BZmedium [23]. Models of multi-bit binary adder, decoderand comparator in BZ are proposed in [45, 95, 96, 120].These architectures typically employ network of ‘conduc-tive’ channels made agar saturation with the reaction so-lution, crossover structures as T-shaped coincidence de-tectors [37] and chemical diodes [49]. Excitation pat-terns in the light-sensitive BZ medium [106] can be con-trolled with illumination, therefore it is possible to make‘conductive’ just by varying illumination of the medium.

∗ andrew.adamatzky@uwe.ac.uk

TABLE I. Truth table of Fredkin and Toffoli gates.

Input OutputFredkin Toffoli

z x y c a b a b

0 0 0 0 0 0 0 0

0 0 1 0 0 1 0 1

0 1 0 0 1 0 1 0

0 1 1 0 1 1 1 1

1 0 0 1 0 0 0 0

1 0 1 1 1 0 0 1

1 1 0 1 0 1 1 1

1 1 1 1 1 1 1 0

By controlling excitability [50] in different loci of themedium we can achieve substantial results, e.g. imple-ment analogs of dendritic trees [99], polymorphic logicalgates [4], integer square root circuits [90].

In our previous paper on a BZ fusion gate and an addermade from fusion gates [3] we developed a hybrid ap-proach to design of BZ-based logical circuits. We keepchannels excitable, to remove the need of monitoring thewhole medium, and junctions sub-excitable. In the chan-nels BZ medium behaves as a ‘classical’ excitable mediumwhile allows for localised, soliton-like, wave-fragments tointeract in the sub-excitable junctions. In present paperwe apply our approach to design and numerically simu-late logically reversible Boolean gates: Fredkin [35] andToffoli [100] gates.

We adopt the following symbolic notations. Booleanvariables x and y take values ‘0’, logical False, and ‘1’,logical True; xy is a conjunction (operation and), x +

arX

iv:1

612.

0825

4v1

[cs

.ET

] 2

5 D

ec 2

016

2

y is a disjunction (or), x ⊕ y is exclusive disjunction(operation xor), x is a negation of variable x (not).In all designs presented excitation waves are initiated attop ends of input channels and propagate down to outputchannels. The input channels are labelled by x, y, z andoutput channels by a, b, c.

The Fredkin and Toffoli gates have three inputs andthree outputs each. Their truth tables are shown inTab. I. Both gates transform input signals x and y con-trolled by signal z to output signals a, b and c. In bothgates c = z, that is a control signal leaves the gate ‘un-changed’ and a and b are calculated as follows:

• Fredkin gate: a = xz + yz and b = xz + yz

• Toffoli gate: a = x and b = y ⊕ (zx)

Input variables x and y are recombined with the con-trol signal. In Fredkin gate an output is a disjunctionof two conjunctions: one input variable and control sig-nal, another input variable and a negation of a controlsignal. In Toffoli gate one output is an identity of an in-put and second output is an exclusive disjunction of oneinput variable with conjunction of another input vari-able with a control signal. The gates are important be-cause they are key elements of low-power computing cir-cuits [17, 26] and they are amongst key components ofquantum and optical computing circuits. These gateswere implemented in theoretical and simulation design inoptical [20, 24, 46, 62, 65, 72, 78, 84], quantum [14, 18,87, 121], single electron [109, 118], and nano-mechanicalsystems [110]; membrane P-systems [61], DNA [82], mag-netic bubbles [21], enzymatic reactions [55, 112], andslime mould [81]. Experimental laboratory prototypeswere produced with optical systems [57], nuclear mag-netic resonance [22, 29, 67], and enzymatic reactions [31–34, 54, 66, 69].

The paper is structured as follows. The Oregonatormodel is described in Sect. II. Section III outlines designprinciples of the logical circuits. Functioning of the gatesis presented in full details in Sects. IV and V. Physical re-versibility of BZ circuits is discussed in Sect. VI. Furtherdevelopments are outlined in Sect. VII.

II. OREGONATOR MODEL OFSUB-EXCITABLE MEDIUM

We use two-variable Oregonator equations [30] adaptedto a light-sensitive Belousov-Zhabotinsky (BZ) reactionwith applied illumination [15]. The Oregonator equa-tions in chemistry bear the same importance as Hodgkin-Huxley and FitzHugh-Nagumo equations in neurophysi-ology, Brusselator in thermodynamics, Meinhardt-Giererin biology, Lotka-Volterra in ecology, and Fisher equa-tion in genetics. The Oregonator equations are used tomodel a wide range of phenomena in BZ, e.g. analysisof rotating waves [51], chaos in flow BZ [101], stochas-tic resonance in BZ [9], effect of macro mixing [48]. The

FIG. 1. Time lapsed snapshots of two excitation wave-fragments in sub-excitable medium. The grid of 500×500node is excited with a rectangular domains 3×20 nodes atthe centre of a grid. This excitation produces two excitationwave-fragments. One wave-fragment propagates to the leftanother to the right. The picture is the time lapsed snapshotof the medium recorded every 150th step of numerical inte-gration. See videos at https://drive.google.com/open?id=0BzPSgPF_2eyUVTMxbzFuYTZvVzg

Oregonator equations is the simplest continuous modelof the BZ medium yet showing very good agreementwith laboratory experiments. Let us provide few exam-ples. A stable three-dimensional organising centre thatperiodically emits trigger excitation waves found exper-imentally is reproduced in the Oregonator model [13].Studies of the BZ system with a global negative feedbackdemonstrate that the Oregonator model shows the samebifurcation scenario of bulk oscillations and wave pat-terns emerging when the global feedback exceed a criti-cal value as the bifurcation scenario observed in labora-tory experiments [105]. There is a good match betweenlab experiments on modifying excitation wave patterns inBZ using external DC field and the Oregonator model ofthe same phenomena [83]. The Oregonator model usedin [27] to evaluate the dispersion relation for periodicwave train solutions in BZ shows agrees with experi-mental results. Patterns produced by the Oregonatormodel of a three-dimensional scrolls waves are indistin-guishable from patterns produced in the laboratory ex-periments [111]. Excitation spiral breakup demonstratedin the Oregonator model is verified in experiments [97].The Oregonator model can be finely tuned, e.g. adjustedfor temperature dependence [73], scaled [47], modifiedfor oxygen sensitivity [58]. Author with colleagues per-sonally used the Oregonator model as a fast-prototypingtool and virtual testbed in designing BZ medium basedcomputing devices which were implemented experimen-tally [5, 7, 25, 90, 102, 103].

The Oregonator equations are following:

∂u

∂t=

1

ε(u− u2 − (fv + φ)

u− qu+ q

) +Du∇2u

∂v

∂t= u− v (1)

The variables u and v represent local concentrations of anactivator, or an excitatory component of BZ system, andan inhibitor, or a refractory component. Parameter ε setsup a ratio of time scale of variables u and v, q is a scalingparameter depending on rates of activation/propagationand inhibition, f is a stoichiometric coefficient. Constantφ is a rate of inhibitor production. In a light-sensitive

3

BZ φ represents the rate of inhibitor production propor-tional to intensity of illumination. We integrate the sys-tem using Euler method with five-node Laplace operator,time step ∆t = 0.001 and grid point spacing ∆x = 0.25,ε = 0.02, f = 1.4, q = 0.002, Du = 1.0, Dv = 0.0.To generate excitation waves of wave-fragments we per-turb the medium by a square solid domains of excitation,10 × 1 sites (unless otherwise stated) in state u = 1.0.The parameter φ characterises excitability of the simu-lated medium. The medium is excitable, it exhibits ‘clas-sical’ target waves when φ = 0.05. The medium is non-excitable when φ > 0.09. The medium is sub-excitablewith propagating localizations, or wave-fragments, whenφ = 0.0766 (Fig. 1).

We guide wave-fragment by constraining them to ex-citable channels and by colliding the wave-fragments witheach other at the junctions of the channels [3]. The chan-nels used in the designs presented have width 30 nodes.At most junctions channels intersect at a right angle tokeep ‘contact size’ between the channels minimal andthus to reduce chances of a single wave propagating alongone channel to ‘spread’ into another channel.

There are two type of junctions: excitable and sub-excitable. In the excitable junction, as in the channels,φ = 0.05. In the sub-excitable junction, each node beingat a distance less than 15 nodes from the centre of thejunction has φ = 0.0766, on the design schemes all nodesof the junctions inside the circles has φ = 0.0766.

There are three scenarios of a wave crossing the junc-tion. If the junction is excitable (φ = 0.05) the wavespreads in all channels. If the junction is sub-excitable(φ = 0.0766) and only one wave approaches the junction,this lonely wave is transformed to a wave-fragment whichcrosses the junction conserving its shape. If two wave en-ter the sub-excitable junction, they collide and fuse intoa single wave-fragment.

Time lapse snapshots provided in the paper wererecorded at every 150 time steps, we display sites withu > 0.04. Videos supplementing figures were producedby saving a frame of the simulation every 10th step ofnumerical integration and assembling them in the videowith play rate 30 fps.

III. DESIGN PRINCIPLES

A collision-based computation, inspired by Fredkin-Toffoli conservative logic [35], employs mobile compactfinite patterns which implement computation while in-teracting with each [6]. Information values (e.g. truthvalues of logical variables) are given by either absence orpresence of the localisations or other parameters of the lo-calisations. The localisations travel in space and performcomputation when they collide with each other. Almostany part of the medium space can be used for compu-tation. The localisations undergo transformations, theychange velocities, form bound states, annihilate or fusewhen they interact with other localisations. Information

X

Y

X Y

XYXY

XYXY

(a)

x y

a b

c d

j1

j2

(b) (c)

(d) (e)

FIG. 2. Time lapsed snapshots of wave-fragments propagat-ing in simulated BZ medium implementing Margolus gate.Bounding box of the gate is 234 × 334 nodes. (a) Scheme ofthe gate. (b) x = 0, y = 1, (c) x = 1, y = 0, (d) x = 1, y = 1.

values of localisations are transformed in the collisionsand thus a computation is implemented. The conceptof the collision-based logical gates with excitation wave-fragments is best illustrated using a gate based on colli-sion between two soft balls, the Margolus gate [64], shownin Fig. 2a. Logical value x = 1 is given by a ball presentedin input trajectory marked x, and x = 0 by absence ofthe ball in the input trajectory x; the same applies toy = 1 and y = 0. When two balls approaching the colli-sion gate along paths x and y collide, they compress butthen spring back and reflect. The balls come out alongthe paths marked xy. If only one ball approaches thegate, for inputs x = 1 and y = 0 or x = 0 and y = 1,the ball exits the gate via path xy (for input x = 1 andy = 0) or xy (for input x = 0 and y = 1).

The Margolus gate is implemented by straightforwardmapping of balls trajectories (Fig. 2a) to a configura-tion of excitable channels (Fig. 2b). Junction j1 issub-excitable (φ = 0.0766). Junction j2 is excitable(φ = 0.05). Functioning of the gate is shown in Fig. 2c–iand the representation of logical values on each segmentof the gate in Tab IIa. If inputs are x = 0 and y = 1 theinput channel y excited (Fig. 2c). The wave-fragmentpropagates across junction j1 into output a; the wavedoes not spread into b because the junction j1 is sub-excitable. If inputs are x = 1 and y = 0 the input x isexcited (Fig. 2d). The wave-fragment propagates acrossjunction j1 into output b; the wave does not spread into

4

TABLE II. Representation of the logical values in segmentsof the BZ implementations of (a) Margolus gate, (b) Fredkingate, (c) Toffoli gate.

(a)

Segment Value

xj1 x

yj1 y

j1a xy

j1b xy

j1j2 xy

j2c xy

j2d xy

(b)

Segment Value

xj4 x

zj1 z

j1j3 z

j1j2 z

j2c z

j3j4 z

j3j5 z

yj2 y

j2j5 y

j4j7 xz

j4j6 xz

j5j6 yz

j5j9 yz

j6j7 xyz

j6j9 xyz

j7j10 xz + xyz

j9j11 yz + xyz

j6j8 xyz

j8j10 xyz

j8j11 xyz

j9j11 yz + xyz

j10a xz + yz

j11b xz + yz

(c)

Segment Value

xj1 z

j1c z

j1j3 z

yj2 x

j2j3 x

j2j4 x

j4a x

j3j4 zx

j4j5 zx

j5j7 zxy

j5j6 zxy

j6b y ⊕ (zx)

b because the junction j1 is sub-excitable. If inputs arex = 1 and y = 1 both input channels are excited (Fig. 2e).The wave-fragments propagating along channels x andy collide at junction j1: they fuse into a single wave-fragment which enters segment j1j2. On reaching theexcitable junction j2 the wave-fragment splits into twowave-fragments. One wave propagates into segment c,another wave into segment d. Excitation wave appearsat output a only if channel x was not excited and channely was excited. Thus we have a = xy. Excitation waveappears at output b only if channel x was excited andchannel y was not excited. Thus we have b = xy. Exci-tation wave propagating along segment j1j2 imitates twosoft balls propagating as a single body for some time aftertheir collision (Fig. 2a). Splitting of the wave-fragmentinto segments c and d (Fig. 2e) imitates soft balls spring-ing back and reflecting (Fig. 2a): c = xy and d = xy.

IV. IMPLEMENTATION OF FREDKIN GATE

Design of excitable implementation of Fredkin gateconsists of ten excitable channels intersecting at elevenjunctions (Fig. 3). All junctions but j1, j3, j8 are sub-

j1 j

2

j3

j4

j5

j6

j7

j8 j

9

j10

j11

x z y

a b c

FIG. 3. Implementation of Fredkin gate. Black channels areexcitable; white space is non-excitable. Junctions are labelledj1 to j1; sub-excitable junctions are encircled. Bounding boxof the taste is 532 × 532 nodes.

excitable. The exact domains of sub-excitable ares shownin circles in Fig. 3. The gate implement transformationof the Boolean triple (x, y, z) to (a, b, c). Inputs x = 1,y = 1 or z = 1 are represented by excitations initiatedat the entrances of the input channels; if an input vari-able is ‘0’ the corresponding input channel is not excited.Outputs a, b, c are assumed to have value ‘1’ when anexcitation wave appears at the corresponding output.

Input x = 0, y = 0, z = 1. Excitation wave is initiatedat entrance of channel z (Fig. 4a). The wave propagatesalong segment zj1. Junction j1 is excitable therefore thewave enters segments j1j3 and j1j2. The wave enteredj1j2 propagates across sub-excitable junction j2 into seg-ment j2c without spreading into segment j2j5; this wavereaches the output c. The wave propagating in segmentj1j3 splits into two waves at the excitable junction j3.One wave propagates along j3j4 and another along j3j5.Junctions j4 and j5 are sub-excitable therefore the wavesrun across the junctions and get extinguished in the cul-de-sacs.

Input x = 0, y = 1, z = 0. Excitation wave is initiatedin channel y. The wave crosses sub-excitable junction j2without spreading to neighbouring segments j1j2 and j2cbut propagates only into j2j5. Junctions j5 and j6 aresub-excitable. The excitation wave runs across these twojunctions without spreading to neighbouring segments.When the wave reaches junction j10 it collides with thewall of the segment j10a, slightly reflect and proceedstowards output a (Fig. 4b).

Input x = 0, y = 1, z = 1. Input channels y and z areexcited. Excitation wave-fronts propagate in zj1 and yj2

5

(a) (b) (c) (d)

(e) (f) (g)

FIG. 4. Time lapsed snapshots of wave-fragments propagating in simulated BZ medium implementing Fredkin gate. (a) x =0, y = 0, z = 1, (b) x = 0, y = 1, z = 0, (c) x = 0, y = 1, z = 1, (d) x = 1, y = 0, z = 0, (e) x = 1, y = 0, z = 1, (f) x = 1, y =1, z = 0, (g) x = 1, y = 1, z = 1. See videos at https://drive.google.com/open?id=0BzPSgPF_2eyUSjM2aWZlU21fM3c

(Fig. 4c). The wave-front originated in y travels acrosssub-excitable junction j2 into segment j2j5 without ex-panding into segments j1j2 and j2c. The wave-front orig-inated in z spreads into j1j2, j1j3, j3j4 (where the exci-tation dies) and j3j5. The wave-fronts propagating alongj3j5 and j2j5 collides at sub-excitable junction j5. Thesetwo wave-fronts collide because distance from z to j5equals to distance from y to j5. The wave-fronts merge on‘impact’ and fuse into a single excitation wave-front prop-agating into j5j9. Junctions j9 and j10 are sub-excitabletherefore the wave-front cross them without expanding,moving directly to output b. The wave-front propagat-ing along j1j2 crossed junction j2 without expanding andreaches output c. Note that on the time-lapsed snapshots(Fig. 4c) traces of wave-fronts travelling from j1 to c andform y to j5 intersect: wave-fronts per se do not collidebecause distance from z to j2 is large than distance from yto j2. See videos at https://drive.google.com/open?id=0BzPSgPF_2eyUSjM2aWZlU21fM3c.

Input x = 1, y = 0, z = 0. The excitation dynamics isanalogous to the dynamics for inputs x = 0, y = 0, z = 1.The wave-fragment initiated at x propagates along alongpath x to j4 to j6 to j9 to j11 to b (Fig. 4d).

Input x = 1, y = 1, z = 0. Excitation wave-front orig-inated in x collides with excitation wave-front originatedin y at junction j6. The wave-front fuse into a singlewave-front. This front travels along j6j8. Junction j8 isexcitable therefore the wave-front spreads into segments

j8j10 and j8j11 and to outputs a and b (Fig. 4e).

Input x = 1, y = 0, z = 1. Excitation initiated inx propagates towards j4 (Fig. 4f). Excitation initiatedin z propagates along path zj1j2c to output c and alsoalong path j1j3j4 to junction j4. Distance from x to j4 isthe same as distance from z to j4. Therefore wave-frontstravelling along paths xj4 and zj4 collide with each other.They fuse into a single excitation wave-front. This wave-front propagates to output a.

Input x = 1, y = 1, z = 1. Excitations are initiatedin all three inputs (Fig. 4g). Wave-fronts originated in xand z collide with each other at junction j4. They fuseinto a wave-fragment travelling towards output a. Wave-fronts originated in z and y collide with each other atjunction j5. They fuse into a wave-fragment travellingtowards output b. The excitation wave-front from z alsotravels to output c.

Thus, we have c = z, a = xz + yz, b = xzyz. Ex-act Boolean functions represented by a wave-fragmenton each of the segments are shown in Tab. IIb.

In design Figs. 3 and 4 the path from z to c is longerthan paths from x, y or z to a and b therefore a signalarrives at output c later than signals arrived at outputs aand b. This can be amended by making output segmentsa and b longer by transforming them in zig-zag segments.We did not show this compensation on the scheme orvideo not to clutter the designs.

6

xz y

a bc

j1

j2

j3

j4

j5

j6 j

6

j7

FIG. 5. Excitable medium implementation of Toffoli gate.Black channels are excitable and sub-excitable; white spaceis non-excitable. Junction are labelled j1 to j7; sub-excitablejunctions are encircled. Bounding box of the taste is 720 ×580 nodes.

V. IMPLEMENTATION OF TOFFOLI GATE

An excitable medium device implementing Toffoli gate(Fig. 5) consists of six excitable channels intersectingat six junctions four of which — j3, j4, j6 — are sub-excitable.

Input z = 0, x = 0, y = 1. Excitation wave-fragmentpropagates from y to j5 (Fig. 6a). Junction j5 is sub-excitable therefore the wave-fragment travels across thejunction without spreading into lateral openings. Theexcitation front reaches output b.

Input z = 0, x = 1, y = 0. Wave-front initiatedin x splits into two wave-front at excitable junction j2.The wave travelling along j2j3 enters the cul-de-sac anddies. The wave running along j2j4 reaches the output a(Fig. 6b).

Input z = 0, x = 1, y = 1. Paths leading from x to aand from y to b do not intersect, therefore wave-fragmentsoriginated in x and y do not interact. Dynamics of ex-citation is an super-position of the excitation scenariosz = 0, x = 0, y = 1 and z = 0, x = 1, y = 0 (Fig. 6c).

Input z = 1, x = 0, y = 0. Excitation wave-fronttravels from z to j1 (Fig. 6d). At j1 the wave-front splitsinto segment j1c, where it reaches output c, and segmentj1j3, where the wave-front dies in the cul-de-sac aftercrossing the junction.

Input z = 1, x = 0, y = 1. Wave-fronts originatingin z and y do not interact. The wave-front started inz reaches output c, the wave-front initiated in y reachesoutput b (Fig. 6e).

Input z = 1, x = 1, y = 0. Wave-fronts from z andx propagate to outputs c and a, respectively. They alsopropagate along segments j1j3 and j2j3 and collide atjunction j3. They fuse and travel as a single wave-front

along j3j4 and j4j5; this wave-front appears at output b(Fig. 6f). Not that traces of wave-fragments along pathsj3 to b and j2 to a intersect on the time-lapsed snap-shot (Fig. 6f) however wave-fragments do not collide atjunction j4 because they enter the junction at differenttimes.

Input z = 1, x = 1, y = 1. Wave-fronts from z andx propagate to outputs c and a, respectively (Fig. 6g).They also propagate along segments j1j3 and j2j3 andcollide at junction j3. They fuse and travel as a singlewave-front along j3j4 and j4j5. Wave-front from y trav-els towards j5. Distance from x (or z) to j5 is the samedistance from y to j5. Therefore wave-front travellingfrom j3 to j5 collides at junction j5 with wave-front trav-elling from y to j5. These two wave-fronts merge intoa single wave-front, which collides into a separation be-tween segments j5j7 and j5j6, the wave-front annihilates(Fig. 6g). Not that traces of wave-fragments along pathsj3 to b and j2 to a intersect on the time-lapsed snap-shot (Fig. 6g) however wave-fragments do not collide atjunction j4 because they enter the junction at differenttimes.

Thus, we have c = z, a = z, b = y ⊕ (zx). ExactBoolean represented by a wave-fragment on each of thesegments is shown in Tab. IIc.

VI. ON PHYSICAL REVERSIBILITY

The designs of BZ implementation of collision-basedgates are logically reversible but not physically reversible(under ’physical reversibility’ we mean only that swap-ping inputs with outputs not reversibility of chemical re-actions or excitation processes). Take for example imple-mentation of Margolus gate (Fig. 2ab). Assume outputchannels in the original Margolus gate are inputs, andinputs in the original Margolus gate are outputs. If allinputs but d are ’0’ excitation is initiated only in channeld (Fig. 7b). The wave-fragment propagates into junc-tion j2 where it collides to a protruding non-excitableseparator of segments j1j2 and j2c. The wave-fragmentchanges its direction of movement in the result of colli-sion and therefore propagates into segment j1y (Fig. 7b).Analogically, if only c is excited the excitation wave-frontexits the gate through output y (Fig. 7c). If a is excitedthe wave-fragment propagates straight into output y; ifb is excited the wave-fragment propagates into x. Whenboth channels a and b are excited two excitation wave-fragments fuse into a single wave-fragments; this wavecollides into a protruding separators between segmentsj1x and j1y, and this wave annihilates (Fig. 7d). Whenc and d are excited the wave-front is formed in segmentj2j1. This wave-fragment splits into two wave-fragmentspropagating into segments j1x and j1y. Thus, we havex = ab+ c and y = ab+ d.

If we apply input signals to channels a, b, c of Fredkinand Toffoli gates we get the following output signals onchannels x, y, z.

7

(a) (b) (c)

(d) (e) (f)

(g)

FIG. 6. Time lapsed snapshots of wave-fragments propagating in simulated BZ medium implementing Toffoli gate. (a) z = 0, x =0, y = 1, (b) z = 0, x = 1, y = 0, (c) z = 0, x = 1, y = 1, (d) z = 1, x = 0, y = 0, (e) z = 1, x = 0, y = 1, (f) z = 1, x = 1, y = 0,(g) z = 1, x = 1, y = 1. See videos at https://drive.google.com/open?id=0BzPSgPF_2eyUa2JwZ2FDTXBSdDQ

If inputs are swapped with output in the Fredkin gate(Fig. 3) then z = a + b + c, x = a, y = b. Wave-frontsrepresenting signals generated at a, b and c do not in-teract with each other. If at least one of the channels a,b or c is excited the wave-front always propagates intochannel z. Wave-front starting at a splits at j4 and goesinto x and z. Wave from b splits at j5 and propagates toz and y. Wave initiated at c propagates along path c toj2 to j1 to z. Thus we have z = a + b + c. Excitationwave-front gets into x only if channel a is excited, thusx = a. Channel y gets excited only if excitation wave isinitiated in channel b, thus y = b.

If inputs are swapped with outputs in Toffoli gate(Fig. 5) then z = b + c, x = a + b, y = b. Only exci-tation wave-front initiated at channel b can excite x, y

and z. This is because excitation wave splits at the junc-tion j6 and thus excitation propagates along paths b toj6 to j7 to j4 (splits in j3) to z and x. At the same time,the wave propagates from b to j6 to j5 to y. Channel yis only excited from b, thus we have y = b. Channel xcan be excited by wave-fronts initiated at a or at b, thusx = a+b. Channel z is excited from b or c, thus z = b+c.

To make physically — at a macro-level not at the levelof chemical reactions — reversible gates we can take twocopies of the original gate (Fig. 5) flip one copy aroundan axis perpendicular to inputs and link these two sub-circuits in a single-circuit as shown in Fig. 8. Originalcopy of Toffoli gate is labelled T in Fig. 8, and flippedversion T ∗. A routing of input signals between T and T ∗

is implemented using sub-excitable junctions encircled in

8

x y

a b

c d

j1

j2

(a) (b) (c)

(d) (e)

FIG. 7. Time lapsed snapshots of wave-fragments propa-gating in simulated BZ medium implementing reversed (in-puts are swapped with outputs) Margolus gate: x = ab + c,y = ab + d. (a) Scheme of the original gate. (b) a = 0, b = 0,c = 0, d = 1, (c) a = 0, b = 0, c = 1, d = 0, (d ) a = 1, b = 1,c = 0, d = 0, (e) a = 0, b = 0, c = 1, d = 1.

Fig. 8. The junction functioning is illustrated in Fig. 9.When a wave-front is initiated either at port p1 (Fig. 9b)or p2 (Fig. 9c) the wave-front propagates towards portp3 without back-spreading into port p2 (p1). When awave-front is initiated at port p3 it propagates towardsthe port p2 only (Fig. 9d) but does not enter the channel

leading to port p1. In terms of electrical circuits, this isanalogous of having a diode at p1p3 connection (Fig. 9e).

Two examples of the circuit responding to xyz andabc inputs are shown in Fig. 8c. When signals enter thecircuit via a, b or c the excitation waves propagate intothe sub-circuit T∗, the signals entered x, y or z are routedinto the sub-circuit T . An example of traces of excitationpropagation when signals are applied to inputs x, y, z isshown in Fig. 8b; and, when signals are applied to inputsa, b, c in Fig. 8c.

VII. DISCUSSION

The architectures of Fredkin and Toffoli gates pro-posed are evaluated in numerical model of a light sen-sitive Belousov-Zhabotinsky (BZ) reaction. Advantageof the light-sensitive BZ is that there is no need for al-tering a homogeneous substrate: an architecture of com-puting circuits is projected with light onto the mediumsuch that conductive channels are non-illuminated andnon-conductive domains are illuminated. Further stud-ies might also focus on mapping the designs in molec-ular arrays [56, 75, 80, 104]; arrays of single-electronoscillators,, locally coupled with capacitors [70], solidstate reaction-diffusion devices with minority-carrier dif-fusion [12, 98]; CNN chips, where pairs of layers rep-resent activator and inhibitor concentrations, and dif-fusion and reaction are controlled via external biasvoltages [10, 11, 19, 53, 71, 76, 85]. Unconventionalrobotics is another application domain. The BZ com-puting circuits can be integrated with self-propulsive BZdroplets [94]. The BZ circuits can play a key role in em-bedded parallel controllers for soft robots made of pHsensitive polymers and gels [63, 113, 115, 116], especiallyto implement decision-making circuits for crawling robotsmade of BZ gels [77] as processors complimentary to afluidic logic [68, 79, 108].

[1] Andrew Adamatzky. Collision-based computing inBelousov–Zhabotinsky medium. Chaos, Solitons &Fractals, 21(5):1259–1264, 2004.

[2] Andrew Adamatzky. Slime mould logical gates: explor-ing ballistic approach. arXiv preprint arXiv:1005.2301,2010.

[3] Andrew Adamatzky. Binary full adder, made of fusiongates, in a subexcitable Belousov-Zhabotinsky system.Physical Review E, 92(3):032811, 2015.

[4] Andrew Adamatzky, Ben de Lacy Costello, and LarryBull. On polymorphic logical gates in subexcitablechemical medium. International Journal of Bifurcationand Chaos, 21(07):1977–1986, 2011.

[5] Andrew Adamatzky, Ben De Lacy Costello, Larry Bull,and Julian Holley. Towards arithmetic circuits in sub-excitable chemical media. Israel Journal of Chemistry,51(1):56–66, 2011.

[6] Andrew Adamatzky and Benjamin de Lacy Costello.Collision-free path planning in the Belousov-Zhabotinsky medium assisted by a cellular automaton.Naturwissenschaften, 89(10):474–478, 2002.

[7] Andrew Adamatzky and Benjamin de Lacy Costello.Binary collisions between wave-fragments in a sub-excitable Belousov–Zhabotinsky medium. Chaos, Soli-tons & Fractals, 34(2):307–315, 2007.

[8] Andrew Adamatzky, Benjamin de Lacy Costello, ChrisMelhuish, and Norman Ratcliffe. Experimental imple-mentation of mobile robot taxis with onboard Belousov–Zhabotinsky chemical medium. Materials Science andEngineering: C, 24(4):541–548, 2004.

[9] Takashi Amemiya, Takao Ohmori, Masaru Nakaiwa,and Tomohiko Yamaguchi. Two-parameter stochasticresonance in a model of the photosensitive Belousov-Zhabotinsky reaction in a flow system. The Journal ofPhysical Chemistry A, 102(24):4537–4542, 1998.

9

x y

c a b

z

T

T*

(a)

x y

c a b

z

T

T*

(b)

x y

c a b

z

T

T*

(c)

FIG. 8. (a) A scheme of physically reversible Toffoli gate made of two copies of original Toffoli gate Fig. 5. Sub-excitablejunctions, in addition to junctions shown in Fig. 5, are encircled. (b) Trajectory of excitation wave-fronts for inputs x = 0,y = 1, z = 1. (c) Trajectory of excitation wave-fronts for inputs a = 0, b = 1, c = 1.

[10] Paolo Arena, Adriano Basile, Luigi Fortuna, and Mat-tia Frasca. Cnn wave based computation for robot nav-igation planning. In Circuits and Systems, 2004. IS-CAS’04. Proceedings of the 2004 International Sympo-sium on, volume 5, pages V–500. IEEE, 2004.

[11] Paolo Arena, Luigi Fortuna, and Marco Branciforte. Re-alization of a reaction-diffusion cnn algorithm for loco-motion control in an hexapode robot. Journal of VLSIsignal processing systems for signal, image and video

technology, 23(2-3):267–280, 1999.[12] Tetsuya Asai, Yuusaku Nishimiya, and Yoshihito

Amemiya. A novel reaction-diffusion system based onminority-carrier transport in solid-state cmos devices.In Semiconductor Device Research Symposium, 2001 In-ternational, pages 141–144. IEEE, 2001.

[13] Arash Azhand, Jan Frederik Totz, and Harald Engel.Three-dimensional autonomous pacemaker in the pho-tosensitive Belousov-Zhabotinsky medium. EPL (Euro-

10j4

p1

p2

p3

(a) (b) (c)

(d)

10j4

p1

p2

p3

(a) (b) (c)

(d)

p1

p2

p3

(e)

FIG. 9. Junctions used to route input signals between sub-circuits in Fig. 8. (a) Scheme of a junction with ports p1, p2,p3, domain of the junction inside circle is sub-excitable, otherparts are excitable. (bcd) Time-lapsed snapshots of excitationfronts: (b) p1 = 1, (c) p2 = 1, (d) p3 = 1, (e) Equivalentelectrical scheme. See videos at https://drive.google.com/open?id=0BzPSgPF_2eyUb3lLSThwNVYzN3M

Tycho Sleator, John A Smolin, and Harald Weinfurter.Elementary gates for quantum computation. Physicalreview A, 52(5):3457, 1995.

[15] Valentina Beato and Harald Engel. Pulse propagation ina model for the photosensitive Belousov-Zhabotinsky re-action with external noise. In SPIE’s First InternationalSymposium on Fluctuations and Noise, pages 353–362.International Society for Optics and Photonics, 2003.

[16] Boris P Belousov. A periodic reaction and its mecha-nism. Compilation of Abstracts on Radiation Medicine,147(145):1, 1959.

[17] Charles H Bennett. Notes on the history of reversiblecomputation. IBMs Journal of Research and Develop-ment, 32(1):16–23, 1988.

[18] Gennady P Berman, Gary D Doolen, Darryl D Holm,and Vladimir I Tsifrinovich. Quantum computer on aclass of one-dimensional Ising systems. Physics LettersA, 193(5-6):444–450, 1994.

[19] V Bonaiuto, A Maffucci, G Miano, M Salerno, F Sar-geni, and C Visone. Design of a cellular nonlinearnetwork for analogue simulation of reaction-diffusionpdes. In Circuits and Systems, 2000. Proceedings. IS-CAS 2000 Geneva. The 2000 IEEE International Sym-posium on, volume 3, pages 431–434. IEEE, 2000.

[20] NJ Cerf, C Adami, and PG Kwiat. Optical simulation ofquantum logic. Physical Review A, 57(3):R1477, 1998.

[21] Hsu Chang. Magnetic-bubble conservative logic. Inter-national Journal of Theoretical Physics, 21(12):955–960,1982.

[22] David G Cory, Mark D Price, and Timothy F Havel.Nuclear magnetic resonance spectroscopy: An experi-mentally accessible paradigm for quantum computing.Physica D: Nonlinear Phenomena, 120(1):82–101, 1998.

[23] Ben De Lacy Costello, Andy Adamatzky, Ishrat Jahan,and Liang Zhang. Towards constructing one-bit binaryadder in excitable chemical medium. Chemical Physics,381(1):88–99, 2011.

[24] Robert Cuykendall and Debra McMillin. Control-specific optical Fredkin circuits. Applied optics,26(10):1959–1963, 1987.

[25] Ben de Lacy Costello, Rita Toth, Christopher Stone,Andrew Adamatzky, and Larry Bull. Implementation ofglider guns in the light-sensitive Belousov-Zhabotinskymedium. Physical Review E, 79(2):026114, 2009.

[26] Alexis De Vos. Reversible computing: fundamentals,quantum computing, and applications. John Wiley &Sons, 2011.

[27] JD Dockery, James P Keener, and JJ Tyson. Dispersionof traveling waves in the Belousov-Zhabotinskii reac-tion. Physica D: Nonlinear Phenomena, 30(1):177–191,1988.

[28] Gabi Escuela, Gerd Gruenert, and Peter Dittrich. Sym-bol representations and signal dynamics in evolvingdroplet computers. Natural Computing, 13(2):247–256,2014.

[29] Xue Fei, Du Jiang-Feng, Shi Ming-Jun, Zhou Xian-Yi,Han Rong-Dian, and Wu Ji-Hui. Realization of theFredkin gate by three transition pulses in a nuclear mag-netic resonance quantum information processor. Chi-nese physics letters, 19(8):1048, 2002.

[30] Richard J Field and Richard M Noyes. Oscillations inchemical systems. iv. limit cycle behavior in a modelof a real chemical reaction. The Journal of ChemicalPhysics, 60(5):1877–1884, 1974.

[31] Brian E Fratto, Nataliia Guz, and Evgeny Katz.Biomolecular computing realized in parallel flow sys-tems: Enzyme-based double feynman logic gate. Paral-lel Processing Letters, 25(01):1540001, 2015.

[32] Brian E Fratto and Evgeny Katz. Reversible logic gatesbased on enzyme-biocatalyzed reactions and realizedin flow cells: A modular approach. ChemPhysChem,16(7):1405–1415, 2015.

[33] Brian E Fratto and Evgeny Katz. Controlledlogic gatesswitch gate and fredkin gate based onenzyme-biocatalyzed reactions realized in flow cells.ChemPhysChem, 2016.

[34] Brian E Fratto and Evgeny Katz. Utilization of a flu-idic infrastructure for the realization of enzyme-basedBoolean logic operations. International Journal of Par-allel, Emergent and Distributed Systems, pages 1–18,2016.

[35] Edward Fredkin and Tommaso Toffoli. Conserva-tive logic. In Collision-based computing, pages 47–81.Springer, 2002.

[36] Pier Luigi Gentili, Viktor Horvath, Vladimir K Vanag,and Irving R Epstein. Belousov-Zhabotinsky “chemicalneuron” as a binary and fuzzy logic processor. IJUC,8(2):177–192, 2012.

[37] J Gorecka and J Gorecki. T-shaped coincidence detectoras a band filter of chemical signal frequency. PhysicalReview E, 67(6):067203, 2003.

[38] J Gorecki, K Gizynski, J Guzowski, JN Gorecka,P Garstecki, G Gruenert, and P Dittrich. Chemi-

(e)

FIG. 9. Junctions used to route input signals between sub-circuits in Fig. 8. (a) Scheme of a junction with ports p1, p2,p3, domain of the junction inside circle is sub-excitable, otherparts are excitable. (bcd) Time-lapsed snapshots of excitationfronts: (b) p1 = 1, (c) p2 = 1, (d) p3 = 1, (e) Equivalentelectrical scheme. See videos at https://drive.google.com/open?id=0BzPSgPF_2eyUb3lLSThwNVYzN3M

physics Letters), 108(1):10004, 2014.[14] Adriano Barenco, Charles H Bennett, Richard Cleve,

David P DiVincenzo, Norman Margolus, Peter Shor,Tycho Sleator, John A Smolin, and Harald Weinfurter.Elementary gates for quantum computation. Physicalreview A, 52(5):3457, 1995.

[15] Valentina Beato and Harald Engel. Pulse propagation ina model for the photosensitive Belousov-Zhabotinsky re-action with external noise. In SPIE’s First InternationalSymposium on Fluctuations and Noise, pages 353–362.International Society for Optics and Photonics, 2003.

[16] Boris P Belousov. A periodic reaction and its mecha-nism. Compilation of Abstracts on Radiation Medicine,147(145):1, 1959.

[17] Charles H Bennett. Notes on the history of reversiblecomputation. IBMs Journal of Research and Develop-ment, 32(1):16–23, 1988.

[18] Gennady P Berman, Gary D Doolen, Darryl D Holm,and Vladimir I Tsifrinovich. Quantum computer on aclass of one-dimensional Ising systems. Physics LettersA, 193(5-6):444–450, 1994.

[19] V Bonaiuto, A Maffucci, G Miano, M Salerno, F Sar-geni, and C Visone. Design of a cellular nonlinearnetwork for analogue simulation of reaction-diffusionpdes. In Circuits and Systems, 2000. Proceedings. IS-CAS 2000 Geneva. The 2000 IEEE International Sym-posium on, volume 3, pages 431–434. IEEE, 2000.

[20] NJ Cerf, C Adami, and PG Kwiat. Optical simulation ofquantum logic. Physical Review A, 57(3):R1477, 1998.

[21] Hsu Chang. Magnetic-bubble conservative logic. Inter-national Journal of Theoretical Physics, 21(12):955–960,1982.

[22] David G Cory, Mark D Price, and Timothy F Havel.Nuclear magnetic resonance spectroscopy: An experi-mentally accessible paradigm for quantum computing.Physica D: Nonlinear Phenomena, 120(1):82–101, 1998.

[23] Ben De Lacy Costello, Andy Adamatzky, Ishrat Jahan,and Liang Zhang. Towards constructing one-bit binaryadder in excitable chemical medium. Chemical Physics,381(1):88–99, 2011.

[24] Robert Cuykendall and Debra McMillin. Control-specific optical Fredkin circuits. Applied optics,26(10):1959–1963, 1987.

[25] Ben de Lacy Costello, Rita Toth, Christopher Stone,Andrew Adamatzky, and Larry Bull. Implementation ofglider guns in the light-sensitive Belousov-Zhabotinskymedium. Physical Review E, 79(2):026114, 2009.

[26] Alexis De Vos. Reversible computing: fundamentals,quantum computing, and applications. John Wiley &Sons, 2011.

[27] JD Dockery, James P Keener, and JJ Tyson. Dispersionof traveling waves in the Belousov-Zhabotinskii reac-tion. Physica D: Nonlinear Phenomena, 30(1):177–191,1988.

[28] Gabi Escuela, Gerd Gruenert, and Peter Dittrich. Sym-bol representations and signal dynamics in evolvingdroplet computers. Natural Computing, 13(2):247–256,2014.

[29] Xue Fei, Du Jiang-Feng, Shi Ming-Jun, Zhou Xian-Yi,Han Rong-Dian, and Wu Ji-Hui. Realization of theFredkin gate by three transition pulses in a nuclear mag-netic resonance quantum information processor. Chi-nese physics letters, 19(8):1048, 2002.

[30] Richard J Field and Richard M Noyes. Oscillations inchemical systems. iv. limit cycle behavior in a modelof a real chemical reaction. The Journal of ChemicalPhysics, 60(5):1877–1884, 1974.

[31] Brian E Fratto, Nataliia Guz, and Evgeny Katz.Biomolecular computing realized in parallel flow sys-tems: Enzyme-based double feynman logic gate. Paral-lel Processing Letters, 25(01):1540001, 2015.

[32] Brian E Fratto and Evgeny Katz. Reversible logic gatesbased on enzyme-biocatalyzed reactions and realizedin flow cells: A modular approach. ChemPhysChem,16(7):1405–1415, 2015.

[33] Brian E Fratto and Evgeny Katz. Controlledlogic gatesswitch gate and fredkin gate based onenzyme-biocatalyzed reactions realized in flow cells.ChemPhysChem, 2016.

[34] Brian E Fratto and Evgeny Katz. Utilization of a flu-idic infrastructure for the realization of enzyme-basedBoolean logic operations. International Journal of Par-allel, Emergent and Distributed Systems, pages 1–18,2016.

[35] Edward Fredkin and Tommaso Toffoli. Conserva-tive logic. In Collision-based computing, pages 47–81.Springer, 2002.

[36] Pier Luigi Gentili, Viktor Horvath, Vladimir K Vanag,and Irving R Epstein. Belousov-Zhabotinsky “chemicalneuron” as a binary and fuzzy logic processor. IJUC,8(2):177–192, 2012.

[37] J Gorecka and J Gorecki. T-shaped coincidence detectoras a band filter of chemical signal frequency. Physical

11

Review E, 67(6):067203, 2003.[38] J Gorecki, K Gizynski, J Guzowski, JN Gorecka,

P Garstecki, G Gruenert, and P Dittrich. Chemi-cal computing with reaction–diffusion processes. Phil.Trans. R. Soc. A, 373(2046):20140219, 2015.

[39] J Gorecki, JN Gorecka, and Andrew Adamatzky. In-formation coding with frequency of oscillations inBelousov-Zhabotinsky encapsulated disks. Physical Re-view E, 89(4):042910, 2014.

[40] J Gorecki, K Yoshikawa, and Y Igarashi. On chemi-cal reactors that can count. The Journal of PhysicalChemistry A, 107(10):1664–1669, 2003.

[41] Jerzy Gorecki and Joanna Natalia Gorecka. Informa-tion processing with chemical excitations–from instantmachines to an artificial chemical brain. InternationalJournal of Unconventional Computing, 2(4), 2006.

[42] Jerzy Gorecki, Joanna Natalia Gorecka, and YasuhiroIgarashi. Information processing with structured ex-citable medium. Natural Computing, 8(3):473–492,2009.

[43] Gerd Gruenert, Konrad Gizynski, Gabi Escuela, BasharIbrahim, Jerzy Gorecki, and Peter Dittrich. Under-standing networks of computing chemical droplet neu-rons based on information flow. International journalof neural systems, page 1450032, 2014.

[44] Gerd Gruenert, Konrad Gizynski, Gabi Escuela, BasharIbrahim, Jerzy Gorecki, and Peter Dittrich. Under-standing networks of computing chemical droplet neu-rons based on information flow. International journalof neural systems, 25(07):1450032, 2015.

[45] Shan Guo, Ming-Zhu Sun, and Xin Han. Digital com-parator in excitable chemical media. International Jour-nal Unconventional Computing, 2015.

[46] James Hardy and Joseph Shamir. Optics inspired logicarchitecture. Optics Express, 15(1):150–165, 2007.

[47] Harold M Hastings, Richard Jeffrey Field, Sabrina God-frey Sobel, and David Guralnick. Oregonator scalingmotivated by showalter-noyes limit. The Journal ofPhysical Chemistry A, 2016.

[48] TJ Hsu, CY Mou, and DJ Lee. Effects of macromix-ing on the oregonator model of the belousovzhabotin-sky reaction in a stirred reactor. Chemical engineeringscience, 49(24):5291–5305, 1994.

[49] Yasuhiro Igarashi and Jerzy Gorecki. Chemical diodesbuilt with controlled excitable media. IJUC, 7(3):141–158, 2011.

[50] Yasuhiro Igarashi, Jerzy Gorecki, and Joanna NataliaGorecka. Chemical information processing devices con-structed using a nonlinear medium with controlled ex-citability. In Unconventional Computation, pages 130–138. Springer, 2006.

[51] W Jahnke, WE Skaggs, and Arthur T Winfree. Chem-ical vortex dynamics in the Belousov-Zhabotinskii re-action and in the two-variable oregonator model. TheJournal of Physical Chemistry, 93(2):740–749, 1989.

[52] Akiko Kaminaga, Vladimir K Vanag, and Irving R Ep-stein. A reaction–diffusion memory device. AngewandteChemie International Edition, 45(19):3087–3089, 2006.

[53] Koray Karahaliloglu and Sina Balkir. An mos cell circuitfor compact implementation of reaction-diffusion mod-els. In Neural Networks, 2004. Proceedings. 2004 IEEEInternational Joint Conference on, volume 1. IEEE,2004.

[54] Evgeny Katz and Brian E Fratto. Enzyme-based re-versible logic gates operated in flow cells. In Advancesin Unconventional Computing, pages 29–59. Springer,2017.

[55] Joshua P Klein, Thomas H Leete, and Harvey Rubin. Abiomolecular implementation of logically reversible com-putation with minimal energy dissipation. Biosystems,52(1):15–23, 1999.

[56] Zoran Konkoli and Goran Wendin. On information pro-cessing with networks of nano-scale switching elements.IJUC, 10(5-6):405–428, 2014.

[57] Natalie Kostinski, Mable P Fok, and Paul R Pruc-nal. Experimental demonstration of an all-optical fiber-based Fredkin gate. Optics letters, 34(18):2766–2768,2009.

[58] Hans Juergen Krug, Ludwig Pohlmann, and LotharKuhnert. Analysis of the modified complete oregonatoraccounting for oxygen sensitivity and photosensitivityof Belousov-Zhabotinskii systems. Journal of PhysicalChemistry, 94(12):4862–4866, 1990.

[59] L Kuhnert. A new optical photochemical memory devicein a light-sensitive chemical active medium. 1986.

[60] Lothar Kuhnert, KI Agladze, and VI Krinsky. Imageprocessing using light-sensitive chemical waves. 1989.

[61] Alberto Leporati, Claudio Zandron, and GiancarloMauri. Simulating the Fredkin gate with energy-basedp systems. J. UCS, 10(5):600–619, 2004.

[62] Seth Lloyd et al. A potentially realizable quantum com-puter. Science, 261(5128):1569–1571, 1993.

[63] Shingo Maeda, Yusuke Hara, Ryo Yoshida, and ShujiHashimoto. Control of the dynamic motion of a gelactuator driven by the Belousov-Zhabotinsky reaction.Macromolecular Rapid Communications, 29(5):401–405,2008.

[64] Norman Margolus. Universal cellular automatabased on the collisions of soft spheres. In AndrewAdamatzky, editor, Collision-based computing, pages107–134. Springer, 2002.

[65] GJ Milburn. Quantum optical Fredkin gate. PhysicalReview Letters, 62(18):2124, 1989.

[66] Fiona Moseley, Jan Halamek, Friederike Kramer, Ar-shak Poghossian, Michael J Schoning, and Evgeny Katz.An enzyme-based reversible cnot logic gate realized ina flow system. Analyst, 139(8):1839–1842, 2014.

[67] KVRM Murali, Neeraj Sinha, TS Mahesh, Malcolm HLevitt, KV Ramanathan, and Anil Kumar. Quantum-information processing by nuclear magnetic resonance:Experimental implementation of half-adder and sub-tractor operations using an oriented spin-7/2 system.Physical Review A, 66(2):022313, 2002.

[68] Ahmad Ahsan Nawaz, Xiaole Mao, Zackary S Stratton,and Tony Jun Huang. Unconventional microfluidics: ex-panding the discipline. Lab on a Chip, 13(8):1457–1463,2013.

[69] Ron Orbach, Francoise Remacle, RD Levine, and Ita-mar Willner. Dnazyme-based 2: 1 and 4: 1 multiplexersand 1: 2 demultiplexer. Chemical Science, 5(3):1074–1081, 2014.

[70] Takahide Oya, Tetsuya Asai, Takashi Fukui, and Yoshi-hito Amemiya. Reaction-diffusion systems consisting ofsingle-electron oscillators. International Journal of Un-conventional Computing, 1(2):179, 2005.

[71] Istvan Petras, Csaba Rekeczky, Tamas Roska, Ri-cardo Carmona, Francisco Jimenez-Garrido, and An-

12

gel Rodrıguez-Vazquez. Exploration of spatial-temporaldynamic phenomena in a 32× 32-cell stored programtwo-layer cnn universal machine chip prototype. Jour-nal of Circuits, Systems, and Computers, 12(06):691–710, 2003.

[72] AJ Poustie and KJ Blow. Demonstration of anall-optical Fredkin gate. Optics Communications,174(1):317–320, 2000.

[73] Srinivasa R Pullela, Diego Cristancho, Peng He, DaweiLuo, Kenneth R Hall, and Zhengdong Cheng. Tem-perature dependence of the oregonator model for theBelousov-Zhabotinsky reaction. Physical ChemistryChemical Physics, 11(21):4236–4243, 2009.

[74] NG Rambidi and D Yakovenchuk. Chemical reaction-diffusion implementation of finding the shortest pathsin a labyrinth. Physical Review E, 63(2):026607, 2001.

[75] John H. Reif. Local parallel biomolecular computation.IJUC, 8(5-6):459–507, 2012.

[76] Csaba Rekeczky, T Serrano-Gotarredona, Tamas Roska,and A Rodriguez-Vazquez. A stored program 2 ndorder/3-layer complex cell cnn-um. In Cellular NeuralNetworks and Their Applications, 2000.(CNNA 2000).Proceedings of the 2000 6th IEEE International Work-shop on, pages 213–217. IEEE, 2000.

[77] Lin Ren, Weibing She, Qingyu Gao, Changwei Pan,Chen Ji, and Irving R Epstein. Retrograde and directwave locomotion in a photosensitive self-oscillating gel.Angewandte Chemie, 2016.

[78] Sukhdev Roy and Mohit Prasad. Novel proposal forall-optical Fredkin logic gate with bacteriorhodopsin-coated microcavity and its applications. Optical En-gineering, 49(6):065201–065201, 2010.

[79] Daniela Rus and Michael T Tolley. Design, fabricationand control of soft robots. Nature, 521(7553):467–475,2015.

[80] Semion K Saikin, Alexander Eisfeld, Stephanie Valleau,and Alan Aspuru-Guzik. Photonics meets excitonics:natural and artificial molecular aggregates. Nanopho-tonics, 2(1):21–38, 2013.

[81] Andrew Schumann. Conventional and unconventionalreversible logic gates on physarum polycephalum. Inter-national Journal of Parallel, Emergent and DistributedSystems, pages 1–14, 2015.

[82] Georg Seelig, David Soloveichik, David Yu Zhang, andErik Winfree. Enzyme-free nucleic acid logic circuits.science, 314(5805):1585–1588, 2006.

[83] Hana Sev?ıkova, Igor Schreiber, and Milos Marek. Dy-namics of oxidation Belousov-Zhabotinsky waves inan electric field. The Journal of Physical Chemistry,100(49):19153–19164, 1996.

[84] Joseph Shamir, H John Caulfield, William Micelli, andRobert J Seymour. Optical computing and the Fredkingates. Applied optics, 25(10):1604–1607, 1986.

[85] Bertram E Shi and Tao Luo. Spatial pattern formationvia reaction-diffusion dynamics in 32× 32× 4 cnn chip.IEEE Transactions on Circuits and Systems I: RegularPapers, 51(5):939–947, 2004.

[86] Jakub Sielewiesiuk and Jerzy Gorecki. Logical functionsof a cross junction of excitable chemical media. TheJournal of Physical Chemistry A, 105(35):8189–8195,2001.

[87] John A Smolin and David P DiVincenzo. Five two-bitquantum gates are sufficient to implement the quantumFredkin gate. Physical Review A, 53(4):2855, 1996.

[88] Oliver Steinbock, Petteri Kettunen, and KennethShowalter. Chemical wave logic gates. The Journal ofPhysical Chemistry, 100(49):18970–18975, 1996.

[89] Oliver Steinbock, Agota Toth, and Kenneth Showal-ter. Navigating complex labyrinths: optimal paths fromchemical waves. Science, pages 868–868, 1995.

[90] William M Stevens, Andrew Adamatzky, Ishrat Jahan,and Ben de Lacy Costello. Time-dependent wave se-lection for information processing in excitable media.Physical Review E, 85(6):066129, 2012.

[91] James Stovold and Simon OKeefe. Simulating neuronsin reaction-diffusion chemistry. In International Con-ference on Information Processing in Cells and Tissues,pages 143–149. Springer, 2012.

[92] James Stovold and Simon OKeefe. Reaction–diffusionchemistry implementation of associative memory neuralnetwork. International Journal of Parallel, Emergentand Distributed Systems, pages 1–21, 2016.

[93] James Stovold and Simon OKeefe. Associative memoryin reaction-diffusion chemistry. In Advances in Uncon-ventional Computing, pages 141–166. Springer, 2017.

[94] Nobuhiko J Suematsu, Yoshihito Mori, TakashiAmemiya, and Satoshi Nakata. Oscillation of speedof a self-propelled belousov–zhabotinsky droplet. TheJournal of Physical Chemistry Letters, 7(17):3424–3428,2016.

[95] Ming-Zhu Sun and Xin Zhao. Multi-bit binary decoderbased on Belousov-Zhabotinsky reaction. The Journalof chemical physics, 138(11):114106, 2013.

[96] Ming-Zhu Sun and Xin Zhao. Crossover structures forlogical computations in excitable chemical medium. In-ternational Journal Unconventional Computing, 2015.

[97] JJ Taboada, AP Munuzuri, V Perez-Munuzuri,M Gomez-Gesteira, and V Perez-Villar. Spiralbreakup induced by an electric current in a belousov–zhabotinsky medium. Chaos: An InterdisciplinaryJournal of Nonlinear Science, 4(3):519–524, 1994.

[98] Motoyoshi Takahashi, Tetsuya Asai, Tetsuya Hirose,and Yoshihito Amemiya. A cmos reaction-diffusiondevice using minority-carrier diffusion in semiconduc-tors. International Journal of Bifurcation and Chaos,17(05):1713–1719, 2007.

[99] Hisako Takigawa-Imamura and Ikuko N Mo-toike. Dendritic gates for signal integration withexcitability-dependent responsiveness. Neural Net-works, 24(10):1143–1152, 2011.

[100] Tommaso Toffoli. Reversible computing. In Interna-tional Colloquium on Automata, Languages, and Pro-gramming, pages 632–644. Springer, 1980.

[101] Kazuhisa Tomita and Ichiro Tsuda. Chaos inthe Belousov-Zhabotinsky reaction in a flow system.Physics Letters A, 71(5):489–492, 1979.

[102] Rita Toth, Christopher Stone, Andrew Adamatzky, Bende Lacy Costello, and Larry Bull. Experimental valida-tion of binary collisions between wave fragments in thephotosensitive Belousov–Zhabotinsky reaction. Chaos,Solitons & Fractals, 41(4):1605–1615, 2009.

[103] Rita Toth, Christopher Stone, Ben de Lacy Costello,Andrew Adamatzky, and Larry Bull. Simple collision-based chemical logic gates with adaptive computing.Theoretical and Technological Advancements in Nan-otechnology and Molecular Computation: Interdisci-plinary Gains: Interdisciplinary Gains, page 162, 2010.

13

[104] Stephanie Valleau, Semion K Saikin, Man-Hong Yung,and Alan Aspuru Guzik. Exciton transport in thin-film cyanine dye j-aggregates. The journal of chemicalphysics, 137(3):034109, 2012.

[105] Vladimir K Vanag, Anatol M Zhabotinsky, and Irv-ing R Epstein. Pattern formation in the Belousov-Zhabotinsky reaction with photochemical global feed-back. The Journal of Physical Chemistry A,104(49):11566–11577, 2000.

[106] V. A. Vavilin, A. M. Zhabotinsky, and AN Zaikin. Effectof ultraviolet radiation on oscillating oxidation reactionof malonic acid derivatives, 1968.

[107] Alejandro Vazquez-Otero, Jan Faigl, Natividad Duro,and Raquel Dormido. Reaction-diffusion based compu-tational model for autonomous mobile robot explorationof unknown environments. IJUC, 10(4):295–316, 2014.

[108] Michael Wehner, Ryan L Truby, Daniel J Fitzgerald,Bobak Mosadegh, George M Whitesides, Jennifer ALewis, and Robert J Wood. An integrated designand fabrication strategy for entirely soft, autonomousrobots. Nature, 536(7617):451–455, 2016.

[109] Hai-Rui Wei and Fu-Guo Deng. Compact quan-tum gates on electron-spin qubits assisted by diamondnitrogen-vacancy centers inside cavities. Physical Re-view A, 88(4):042323, 2013.

[110] Josef-Stefan Wenzler, Tyler Dunn, Tommaso Toffoli,and Pritiraj Mohanty. A nanomechanical Fredkin gate.Nano letters, 14(1):89–93, 2013.

[111] Arthur T Winfree and Wolfgang Jahnke. Three-dimensional scroll ring dynamics in the Belousov-Zhabotinskii reagent and in the two-variable oregonatormodel. The Journal of Physical Chemistry, 93(7):2823–2832, 1989.

[112] David Harlan Wood and Chen Junghuei. Fredkin gatecircuits via recombination enzymes. In Congress on evo-

lutionary computation, 2004.[113] Victor V Yashin and Anna C Balazs. Modeling poly-

mer gels exhibiting self-oscillations due to the Belousov-Zhabotinsky reaction. Macromolecules, 39(6):2024–2026, 2006.

[114] Hiroshi Yokoi, Andy Adamatzky, Ben de Lacy Costello,and Chris Melhuish. Excitable chemical mediumcontroller for a robotic hand: Closed-loop experi-ments. International Journal of Bifurcation and Chaos,14(09):3347–3354, 2004.

[115] Ryo Yoshida. Self-oscillating gels driven by theBelousov–Zhabotinsky reaction as novel smart materi-als. Advanced Materials, 22(31):3463–3483, 2010.

[116] Ryo Yoshida, Satoko Onodera, Tomohiko Yamaguchi,and Etsuo Kokufuta. Aspects of the Belousov-Zhabotinsky reaction in polymer gels. The Journal ofPhysical Chemistry A, 103(43):8573–8578, 1999.

[117] Kenichi Yoshikawa, Ikuko Motoike, T. Ichino, T. Ya-maguchi, Yasuhiro Igarashi, Jerzy Gorecki, andJoanna Natalia Gorecka. Basic information process-ing operations with pulses of excitation in a reaction-diffusion system. IJUC, 5(1):3–37, 2009.

[118] George T Zardalidis and Ioannis Karafyllidis. Designand simulation of a nanoelectronic single-electron uni-versal Fredkin gate. IEEE Transactions on Circuits andSystems I: Regular Papers, 51(12):2395–2403, 2004.

[119] AM Zhabotinsky. Periodic processes of malonic acidoxidation in a liquid phase. Biofizika, 9(306-311):11,1964.

[120] Guo-Mao Zhang, Ieong Wong, Meng-Ta Chou, and XinZhao. Towards constructing multi-bit binary adderbased on Belousov-Zhabotinsky reaction. The Journalof chemical physics, 136(16):164108, 2012.

[121] Shi-Biao Zheng. Implementation of Toffoli gates with asingle asymmetric Heisenberg x y interaction. PhysicalReview A, 87(4):042318, 2013.