Logical Gates via Gliders Collisions
LLogical Gates via Gliders Collisions
Genaro J. Mart´ınez , , Andrew Adamatzky Kenichi Morita
14 March 2018 ∗ Escuela Superior de C´omputo, Instituto Polit´ecnico Nacional, M´exico Unconventional Computing Lab, University of the West of England, Bristol,United Kingdom Hiroshima University, Higashi Hiroshima, Japan
Abstract
An elementary cellular automaton with memory is a chain of finite statemachines (cells) updating their state simultaneously and by the same rule.Each cell updates its current state depending on current states of its imme-diate neighbours and a certain number of its own past states. Some cell-state transition rules support gliders, compact patterns of non-quiescentstates translating along the chain. We present designs of logical gates,including reversible Fredkin gate and controlled not gate, implementedvia collisions between gliders.
When designing a universal cellular automata (CA) we aim, similarly to de-signing small universal Turing machines, to minimise the number of cell states,size of neighbourhood and sizes of global configurations involved in a compu-tation [15, 18, 2, 4, 5, 19, 50, 38, 36, 60]. History of 1D universal CA is longyet exciting. In 1971 Smith III proved that a CA for which a number of cell-states multiplied by a neighbourhood size equals 36 simulates a Turing machine[52]. Sixteen years later Albert and Culik II designed a universal 1D CA withjust 14 states and totalistic cell-state transition function [1]. In 1990 Lindgrenand Nordahl reduced the number of cell-states to 7 [21]. These proofs wereobtained using signals interaction in CA. Another 1D universal CA employingsignals was designed by Kenichi Morita in 2007 [43]: the triangular reversibleCA which evolves in partitioned spaces (PCA). ∗ Published in:
Reversibility and Universality: Essays Presented to Kenichi Morita onthe Occasion of his 70th Birthday , A. Adamatzky (ed.), chapter 9, pages 199–220. Springer,2018. A short version is published as: “Conservative Computing in a One-dimensional CellularAutomaton with Memory,”
Journal of Cellular Automata Complex Cellular Automata Repository http://uncomp.uwe.ac.uk/genaro/Complex_CA_repository.html a r X i v : . [ n li n . C G ] M a r n 1998, Cook demonstrated that elementary CA, i.e. with two cell-statesand three-cell neighbourhood, governed by rule 110 is universal [13, 64]. He didthis by simulating a cyclic tag system in a CA. Operations were implementedwith 11 gliders and a glider gun.We will show that an elementary CA with memory, where every cells updatesits state depending not only on its two immediate neighbours but also on its ownpast states, exhibits gliders which collision dynamics allows for implementationof logical gates just with one glider. We demonstrate implementation of not and and gates, delay , nand gate, majority gate, and cnot and Fredkingates. One-dimensional elementary CA (ECA) [63] can be represented as a 4-tuple (cid:104) Σ , ϕ, µ, c (cid:105) , where Σ = { , } is a binary alphabet (cell states), ϕ is a localtransitions function, µ is a cell neighbourhood, c is a start configuration. Thesystem evolves on an array of cells x i , where i ∈ Z (integer set) and eachcell takes a state from the Σ = { , } . Each cell x i has three neighbours in-cluding itself: µ ( x i ) = ( x i − , x i , x i +1 ). The array of cells { x i } represents a global configuration c , such that c ∈ Σ ∗ . The set of finite configurations oflength n is represented as Σ n . Cell states in a configuration c ( t ) are updatedto next configuration c ( t + 1) simultaneously by a the local transition function ϕ : x t +1 i = ϕ ( µ ( x i )). Evolution of ECA is represented by a sequence of finiteconfigurations { c i } given by the global mapping, Φ : Σ n → Σ n . Rule 22 is an ECA with the following local function: ϕ R = (cid:26) , , , , , , . (1)The local function ϕ R has a probability of 37.5% to get states 1 in the nextgeneration and much higher probability to get state 0 in the next generation.Examples of evolution of ECA Rule 22 from a single cell in state ‘1’ and from arandom configurations are shown in Fig. 1. Conventional CA are ahistoric (memoryless): the new state of a cell depends onthe neighbourhood configuration solely at the preceding time step of ϕ . CA with memory (CAM) can be considered as an extension of the standard framework A reproduction of this machine working in rule 110 can be found in http://uncomp.uwe.ac.uk/genaro/rule110/ctsRule110.html a)(b) Figure 1: Typical evolution of ECA rule 22 (a) from a single cell in state ‘1’and (b) from a random initial configuration where half of the cells, chosen atrandom, are assigned state ‘1’. The ECA consists of 598 cells and evolves for352 generations. White colour represents state ‘0’ and dark colour representsstate ‘1’. 3f CA where every cell x i is allowed to remember some period of its previousevolution. CAM was introduced by Sanz, see overview in [51]. To implement amemory function we need to specify the kind of memory φ , as follows: φ ( x t − τi , . . . , x t − i , x ti ) → s i . (2)The parameter τ < t determines the depth, or a degree, of the memory andeach cell s i ∈ Σ being a state function of the series of states of the cell x i withmemory up to time-step. To execute the evolution we apply the original ruleas: ϕ ( . . . , s ti − , s ti , s ti +1 , . . . ) → x t +1 i . (3)The main feature in CAM is that the mapping ϕ remains unaltered, whilehistoric memory of all past iterations is retained by featuring each cell in thecontext of history of its past states in φ . This way, cells canalise memory to themap ϕ . For example, we can consider memory function φ as a majority memory φ maj → s i , where in case of a tie given by Σ = Σ in φ we will take the lastvalue x i . In this case, function φ maj represents the classic majority function forthree values [37] as follows:( a ∧ b ) ∨ ( b ∧ c ) ∨ ( c ∧ a ) (4)that represents the cells ( x t − τi , . . . , x t − i , x ti ) and define a temporal ring of s cells,before reaching the next global configuration c . φ R maj :4ECA with memory (ECAM) rule φ R maj :4 employ the majority memory ( maj )and degree of memory τ = 4.Figure 2 shows a typical evolution of ECAM rule φ R maj :4 from a randominitial condition. There we observe emergence of gliders travelling and collidingwith each other. localisation period shift velocity mass g L − − /
11 38 g R
11 2 2 /
11 38Table 1: Properties of gliders in ECAM rule φ R maj :4 .The set of gliders G φ R maj :4 = { g L , g R } are defined in a square of 11 × φ R maj :4 from a random initial configurationwith the ratio of 37% on a ring of 968 cells for 1,274 generations. Filter and aselection of colors are used to improve the view of particles.5 R g L Figure 3: g L and g R gliders in ECAM rule φ R maj :4 . We assume presence of a glider at a given site of space and time is a logical ‘1’(
True ) and absence is ‘0’ (
False ). Logic gates f ( a, b ) → ¬ a | ¬ b | a ∧ b arerealised via binary collisions of gliders in rule φ R maj :4 . Figure 4bc shows how and-not gate is realised. Fig. 4d demonstrates implementations of and gate.Also a delay operator can be implemented by delaying glider a and conservingmomentum of glider b as shown in Fig. 4eA nand gate can be cascaded from majority and not gates. To design nand gate, we fix third value in majority gate to produce and gate as illus-trated in Fig 5b. Later a not gate is cascaded to get a nand gate (Fig. 5a).Figure 6a represents a value ‘0’ as one glider and a value ‘1’ as a coupleof gliders. Figure 6b presents a scheme of nand gate in φ R maj :4 . We havethree-input values/gliders that will be evaluated by one control glider travellingperpendicularly to input gliders. The control glider transforms f ( a, b, c ) into amajority function. Further, other glider acts as an active signal in not gate.6 b ¬ b ¬ aa b (a) a ⇥ ¬ b (b) ¬ a ⇥ b (c) a b (d) delay a (e) Figure 4: Examples of logical gates implemented via glider collisions. Presenceof a glider is logical
Truth ‘1’, absence logical
False ‘0’. (a) Scheme of thegate. (bc) and-not gate, (d) and , (e) delay . Figures 7 and 8 show the implementation of nand gate in φ R maj :4 , en-coded in its initial condition set of gliders in six positions. As was shown inthe scheme (Fig. 6c), gliders coming from the left side represent binary valuesand a glider coming from the right acts as a control glider for the values of the majority gate. The control glider continues its travel undisturbed, after inter-action with input gliders, and therefore it can be recycled in further majority gate. A concept of ballistic logical gates represent Boolean values by balls which pre-serve their identity during collision but change their velocity vectors; the ballsare routed in the computing space using mirrors [58]. The billiard ball model was advanced by Margolus in his designs of partitioned CA to demonstrate alogical universality of a billiard-ball model and to implement Fredkin gate withsoft spheres in billiard-ball model [28, 29].Ballistic collisions of gliders, or particles, permit to represent functions oftwo arguments (general signal-interaction scheme is conceptualized in Fig. 9a),as follows [58]: f ( u, v ) is a product of one collision (Fig. 9b);2. f i ( u, v ) (cid:55)→ ( u, v ) identity (Fig. 9c);3. f r ( u, v ) (cid:55)→ ( v, u ) reflection (Fig. 9d); where Fig. 9a represents a collision not preserving identity of gliders u and v ;Fig. 9b shows a collision where identities of input gliders are preserved.Below we show that ECAM rule φ R maj :4 reproduces ballistic collisions.Figure 10 demonstrates identity collisions f i ( u, v ) (cid:55)→ ( u, v ). The collisions be-tween gliders are illustrated in (Fig. 10). The collisions are of soliton nature [20,7 AJ cbaba NOT a AND (a)
MAJ ba NOT
NAND gate a b c maj nand (b)
Figure 5: Gate nand made of majority and not gates. (a) Schematics ofgates. (b) Design of nand gate. value 1value 0 (a)
NAND gate M A J abc operator NOT (b) Figure 6: Gate majority . (a) Binary values represented by gliders, (b) schemeof an nand gate using majority and not gates.8 (a)
000 1 1 (b)
Figure 7: Gate nand implemented with majority and not gates in ECAM rule φ R maj :4 following the scheme proposed in Fig. 6b. (a) Input values f (0 , , f (0 , , (a)
00 0 11 (b)
Figure 8: Gate nand implemented with majority and not gates in ECAM rule φ R maj :4 following the scheme proposed in Fig. 6b. (a) Input values f (1 , , f (1 , , vf (a) u v v u f (b) u vuv (c) u vu v (d) Figure 9: Schematics of ballistic collisions implemented with gliders.56]. A carry-ripple adder [55] can be implemented in ECAM Rule φ R maj :4 bysolitonic reactions with pairs of gliders and identity function (Fig. 10a).Figure 11 illustrates elastic collisions f r ( u, v ) (cid:55)→ ( v, u ) with a pair of gliders.The pairs of gliders reflect in the same manner in all collisions (Fig. 11a). Theseelastic collisions are robust to phase changes and initial positions of gliders(Fig. 11b).The ballistic interaction gate (Fig. 12a) is invertible (Fig. 12b) [10]. There-fore inverse and elastic collisions in φ R maj :4 may model the interaction gate:gliders g R and g L are equivalent to balls. To allow gliders to return to the orig-inal input locations we may constrain them with boundary conditions (Figs. 10and 11) or route them with mirrors. A conservative logic gate is a Boolean function that is invertible and preservesignals [17]. Fredkin gate is a classical conservative logic gate. The gate realisesthe transformation ( c, p, q ) (cid:55)→ ( c, cp + ¯ cq, cq + ¯ cp ), where ( c, p, q ) ∈ { , } .Schematic functioning of Fredkin gate is shown in Fig. 13 and the truth tableis in Table 2. c p q x y z → a) (b) Figure 10: Implementation of ballistic collisions in ECAM rule φ R maj :4 . (a)20 gliders are synchronised to simulate the function identity f i ( u, v ) (cid:55)→ ( u, v ),(b) 18 gliders preserve their identity during collisions. ECAM is a ring of 418cells. It evolved in 1697 time steps. 12 a) (b) Figure 11: Implementation of ballistic collisions in ECAM rule φ R maj :4 . (a)Pairs of six gliders are synchronised to simulate the reflections or elastic collisions f r ( u, v ) (cid:55)→ ( v, u ), (b) 16 pairs of gliders preserve their identity during collisions.ECAM is a ring of 418 cells. It evolved in 1697 time steps.13 y xy ¯ xyx ¯ yxy (a) xyxy ¯ xyx ¯ yxy (b) xy xy ¯ xyx ¯ yxy (c) Figure 12: In ballistic collisions we can implement an (a) interaction gate, (b)its inverse, and (c) its billiard ball model realisation. cpq xyz (a) ab ab (b) ab ab (c) Figure 13: Fredkin gate. (a) Scheme of the gate, (b) operation for control value1, (c) operation for control value 0.Fredkin gate is universal because one can implement a functionally completeset of logical functions with this gate (Fig. 14). Other gates implemented withFredkin gate are shown below: • c = u , p = v , q = 1 (or c = u , p = 1, q = v ) yields the implies gate y = u → v ( z = u → v ). • c = u , p = 0, q = v (or c = u , p = v , q = 0) yields the not implies gate y = ( v → u ) ( z = ( v → u )) [17]. To simulate a Fredkin gate in ECAM rule φ R maj :4 , we utilise a set of collisionsto preserve the reactions and the persistence of data, these basic collisions havea specific task and actions which are specified as follows. • Mirror reflects a glider, the mirror should be deleted, not to become anobstacle for other signals. • Doubler splits a signal into two signals. • Soliton crosses two gliders preserving its identity but might change itsphase. • Splitter separates two gliders into gliders travelling in opposite directions.14 x = cpq y = cp + ¯ cqz = ¯ cp + cq (a) u u ¯ uc = u, p = 1 , q = 0 u (b) u uu ¯ uc = u, p = 1 , q = 0 (c) uv ¯ u + v u + vu c = u, p = v, q = 1 (d) uv u + v ¯ u + v u c = u, p = 1 , q = v (e) uv uvu ¯ uv c = u, p = 0 , q = v (f) uv uvu ¯ uv c = u, p = v, q = 0 (g) Figure 14: Realisation of Boolean functions using Fredkin gate. (a) Fredkingate, (b) not gate, (c) fanout gate, (d-e) or gate, and (f-g) and gate [17].15 Flag is a glider that is generated depending of an input value given. • Displacer moves a glider forward.Fredkin gates implemented in non-invertible systems were proposed and sim-ulated by Adamatzky in a non-linear medium — Oregonator model of Belousov-Zhabotinsky [7]. p cq mirrormirror c c ~cq c ~ cpcq cpc z =~ cp + cqp cq z yc = x cp ~ cpy =~ cq + cp L f R f Figure 15: A scheme of Fredkin gate implementation in ECAM rule φ R maj :4 via glider collisions. Figure 15 displays the schematic diagram proposed to simulate Fredkin gates inECAM rule φ R maj :4 . There are three inputs ( c, p, q ) and three outputs ( x, y, z ).During the computation auxiliary gliders are generated. They are deleted before16eaching outputs. Gliders travel in two directions in a 1D chain of cells, theycan be reused only when crossing periodic boundaries or via combined collisions.We use two gliders as flags, travelling from the left ‘ L f ’ and from the right ‘ R f ’.Flags are activated depending on initial values for c or q as followsIf c = 0 then R f = 1,If q = 0 then L f = 1,In any other case L and R = 0.Mirrors M are defined as follows:If c = p = q = 1 then M = 2,If c = p = 1 then M = 2,If c = q = 1 then M = 1,If p = 1 then M = 1,In any other case M = 0.Distances between gliders are fixed as positive integers determined for anumber of cells in the state ‘0’, as n (cid:95) ∀ n ≥
0. During the computation wesplit gliders when two gliders travel together, the split gliders travel in oppositedirections. We use two gliders as mirrors to change the direction of an argumentmovement. We use displacer to move an output glider and adjust its distancewith respect to other. cpq xyz L f R f M111 111 0 0 2110 110 1 0 2101 101 0 0 1100 100 1 0 0011 011 0 1 0010 001 1 1 1001 010 0 1 0Table 3: Following the scheme in Fig. 15 we specify a sequence of collisions thatare controlled with flag gliders ( L f , R f ) and mirrors (M) glider.Table 3 shows values of inputs, outputs, flags, and mirrors. First columnrepresents inputs , the second column outputs , third column are values of aflag activated in the left side, fourth column are values of the flag activated inthe right side, and the last column shows the number of necessary mirrors.Space-time configurations of ECAM rule φ R maj :4 implementing Fredkingate for all non-zero combinations of inputs are shown in Figs. 16–19.17 =1 p =1 c =1 mirror mirror (a) q =1 p =0 c =1 mirror (b) Figure 16: Fredkin gate in ECAM rule φ R maj :4 . (a) INPUT c = 1, p = 1, q = 1, OUTPUT x = 1, y = 1, z = 1, (b) INPUT c = 1, p = 0, q = 1, OUTPUT x = 1, y = 0, z = 1. 18 =0 p =1 c =1 mirror mirrorL f (a) q =0 p =0 c =1 L f (b) Figure 17: Fredkin gate in ECAM rule φ R maj :4 . (a) INPUT c = 1, p = 1, q = 0, OUTPUT x = 1, y = 1, z = 0, (b) INPUT c = 1, p = 0, q = 0, OUTPUT x = 1, y = 0, z = 0. 19 =1 p =1 c =0 R f (a) q =1 p =0 c =0 R f (b) Figure 18: Fredkin gate in ECAM rule φ R maj :4 . (a) INPUT c = 0, p = 1, q = 1, OUTPUT x = 0, y = 1, z = 1, (b) INPUT c = 0, p = 0, q = 1, OUTPUT x = 0, y = 1, z = 0. 20 =0 p =1 c =0 mirror R f L f Figure 19: Fredkin gate in ECAM rule φ R maj :4 . INPUT c = 0, p = 1, q = 0, OUTPUT x = 0, y = 0, z = 1. 21 F F F F FF F F F FF F F F F FF F F F FF F F F F F
Figure 20: Cascaded Fredkin gates (It is a modification of Fredkin array pro-posed [30]).
We demonstrated how to implement a functionally complete set of Boolean func-tions in a one-dimensional cellular automaton with binary cell states, three-cellneighbourhood and memory depth four. We also shown that Fredkin gate canbe realised the this automaton. Let us compare complexity of our designs withpreviously published models of universal one-dimensional cellular automata.CA simulating Turing machine, proposed by Smith III in 1971 [52], satisfiedthe condition: a number of cell-states multiplied by a neighbourhood size equals η = 36. A universal 1D CA designed by Albert and Culik II [1] has 14 statesand totalistic cell-state transition rule, that is for their automaton η = 42. TheLindgren-Nordahl CA [21] has 7 states, assuming three-cell neighbourhood wehave η = 21. ECAM implementation of a Fredkin gate, proposed in presentpaper, has two cell states, three cell neighbourhood, and memory depth 4, thisimplies η = 24. Thus, in terms of a neighbourhood size and a number of statesour automaton is less efficient than the CA proposed by Lindgren-Nordahl [21],however we use gliders as signals. Gliders are discrete analogs of solitons, there-fore our design, in principle, can be considered as a blueprint for physical im-plementation, as we will discuss below. Cook’s design [13, 64] of a universal CAvia cyclic tag system is the most optimal in terms of neighbourhood size andnumber of states, η = 6, however the implementation involves 11 gliders and aglider gun. Our design of Fredkin gate utilises just two types of gliders.Polymer chains that support propagation of travelling localisations (solitons,kinks, defects) are potential substrates for implementation of glider-based Fred-kin gate. There are many potential candidates, here we discuss actin filaments.Actin is a protein presented in all eukaryotic cells in forms of globular actin (G-22ctin) and filamentous actin (F-actin), see history overview in [53]. Filamentousactin is a double helix of F-actin unit chains. In [6] we proposed a model of actinpolymer filaments as two chains of one-dimensional binary-state semi-totalisticautomaton arrays and uncovered rules supporting gliders, discrete analogs ofionic waves propagating in actin filaments [59].Let us evaluate a physical space-time requirement for implementation ofFredkin gate on actin filaments. Assume a unit of F-actin corresponds to a cellof 1D CA. Maximum diameter of an actin filament is 8 nm [40, 54]. An actinfilament is composed of overlapping units of F-actin. Thus, diameter of a singleunit is c. 4 nm. A glider in our ECAM model occupies 10 cells, that makes asize of a signal in actin filaments 40 nm. Maximum distance between inputs inECAM Fredkin gate is 200 cells. This makes gate size 880 nm.With regards to speed of Fredkin gate realisation, being unaware of exactmechanisms of travelling localisations on actin filaments we can propose specula-tive estimates. Assume the underpinning mechanism of generating a localisationis an excitation of F-actin molecule (single unit of actin polymer). An excitationin a molecule takes place when an electron in a ground state absorbs a photonand moves up to a high yet unstable energy level. Later the electron returnsto its ground state. When returning to the ground state the electron releasesphoton which travels with speed × ˚A per second. F-actin molecule (oneunit of actin filament) is a polymer chain of at most 3K nodes. Thus the F-actinunit can be spanned by an excitation in at most 10 − sec. That can be adoptedas a physical time step equivalent to one step of ECAM evolution. The ECAMFredkin gate completes its operation on actin filament in 600 time steps, thatis at most 10 − sec, i.e. 0.1 picosecond of real time.Experimental implementation of the Fredkin gate, including cascading of thegates, on actin filaments, or any other soliton-supporting polymer chain makesa very challenging topic of future studies. There methodological approaches tocontrol and monitor attosecond scale molecular dynamics [14, 8, 48, 11] howeverit is not clear if they can be applied to actin polymers. The problems to overcomeinclude input of data in acting filament with a single F-actin unit precision,keeping the polymer chain stable and insulated from thermal noise, readingoutputs from the polymer chain. However exact experimental implementationremains uncertain.But, although this can be confined in a physical computing device severallimitations should be improved: avoid the use of mirrors and the use of flags. Anoption is that they could be manipulated into in a ring (or virtual CA collider[32, 27]). References [1] Albert, J. & Culik II, K. (1987) A simple universal cellular automaton andits one-way and totalistic versions,
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Appendix – Binary collisions in φ R majmaj