Chemical Computing. Peter Dittrich

Transcription

1 in: Unonventional Programming Paradigms (UPP 2004), J.-P. Banatre, J.-L. Giavitto, P. Fradet, O. Mihel, Editors, LNCS 3566, pp , Springer, Berlin, 2005 Chemial Computing Peter Dittrih Bio Systems Analysis Group Jena Centre for Bioinformatis (JCB) and Department of Mathematis and Computer Siene Friedrih-Shiller-University Jena D Jena, Germany Abstrat. All information proessing systems found in living organisms are based on hemial proesses. Harnessing the power of hemistry for omputing might lead to a new unifying paradigm oping with the rapidly inreasing omplexity and autonomy of omputational systems. Chemial omputing refers to omputing with real moleules as well as to programming eletroni devies using priniples taken from hemistry. The paper fouses on the latter, alled artifiial hemial omputing, and disusses several aspets of how the metaphor of hemistry an be employed to build tehnial information proessing systems. In these systems, omputation emerges out of an interplay of many deentralized relatively simple omponents analogized to moleules. Chemial programming enompassed then the definition of moleules, reation rules, and the topology and dynamis of the reation spae. Due to the selforganizing nature of hemial dynamis, new programming methods are required. Potential approahes for hemial programming are disussed and a road map for developing hemial omputing into a unifying and well grounded approah is skethed. 1 Introdution All known life forms proess information on a moleular level. Examples are: signal proessing in bateria (e.g., hemotaxis), gene expression and morphogenesis, defense oordination and adaptation in the immune system, and information broadasting by the endorine system. Chemial proesses play also an important role, when an ant olony seeks a suitable route to a food soure. This kind of hemial information proessing is known to be robust, self-organizing, adaptive, deentralized, asynhronous, fault-tolerant, and evolvable. Computation emerges out of an orhestrated interplay of many deentralized relatively simple omponents (moleules). How an hemistry be employed for omputing? First, it should be noted that hemistry is used for the fabriation of eletroni devies. However, here we are interested in approahes where hemistry stimulates the development of new omputational paradigms. These approahes an be distinguished aording to the following two dimensions: First, real hemial omputing where real

2 moleules and real hemial proesses are employed to ompute. Seond, artifiial hemial omputing where the hemial metaphor is utilized to program or to build omputational systems. The former aims at harnessing new substrates for omputation. The latter takes the hemial metaphor as a design priniple for new software or hardware arhitetures built on onventional silion devies. So, artifiial hemial omputing inludes onstruting hemial-like formal system in order to model and master onurrent proesses, e.g., Gamma [1], CHAM [2]; using the hemial metaphor as a new way to program onventional omputers inluding distributed systems, e.g., smart dust; and taking the hemial metaphor as an inspiration for new arhitetures, e.g., reation-diffusion proessors [3]. 1.1 The Chemial Metaphor Chemistry is a siene of experiment and observation, whih provides a partiular view on our world. Like physis, hemistry deals with matter and energy, but fouses on substanes omposed of moleules and how the omposition of these substanes is hanged by hemial reations. Compared with hemistry, physis is more onerned with energy, fores, and motion, ie. the physial hange of a system. Chemistry looks at the maro and miro level: On the maro level emergent properties and the emergent behavior of substanes are studied, e.g., olor or smell. On the mirosopi level, moleular strutures and reation mehanisms are postulated, whih are taken to explain marosopi observations. Ideally, mirosopi models allow to formally derive marosopi observations. However, this is possible only in limited ases, e.g., no algorithm that omputes the melting temperature of a moleule given its struture is known. In general, hemistry explains a hemial observation using a mixture of mirosopi and marosopi explanations. The diffiulty to predit the marosopi behavior from mirosopi details has its root in the nature of emergene. The time-evolution of a hemial system is a highly parallel self-organization proess where many relatively simple omponents interat in a nonlinear fashion. And it is a entral aim of hemial omputing to harness the power inherent in these self-organization proesses. From a omputer siene perspetive it would be quite appealing to ahieve omputation as an emergent proess, where only mirosopi rules have to be speified and information proessing appears as global behavior. From knowing the biologial arhetype, we an expet a series of interesting properties, suh as, fine grained parallelism without entral ontrol, fault tolerane, and evolvability. There is a wide appliation range, espeially where the harateristis of hemial proesses fit naturally to the desired task, as for example in highly distributed and dynami organi proessor networks or within one omputing node to implement partiular systems like artifiial emotional [4], endorine [5], or immune systems more naturally. It should be mentioned that hemial proesses themselves an be seen as a natural media for information proessing either in vitro or in vivo [6 8]; for a reent disussion of moleular omputing

3 see ref. [9]. Here we onentrate on how tehnial eletroni systems an utilize the hemial metaphor. 1.2 The Organization of Chemial Explanations When we study hemistry [10], first we learn how substanes look like. We desribe marosopi properties of the substanes, suh as olor, and how substanes are omposed from elementary objets, the atoms. Seond, we learn how substanes interat, in partiular, we desribe the outome that results from their union. Reative interations among moleules require that these moleules ome into ontat, whih an be the result of a ollision. Third, we learn the detailed dynamial proess of a hemial transformation of substanes. All these steps of desription an be done on a mirosopi and marosopi level. The steps are also not independent: The properties of substanes are often desribed in terms of how a substane reats with other substanes, e.g., when we say fluorine is not healthy in large quantities we desribe the property of fluorine by how it interats with moleules in an organism. In fat, in times when nothing was known about the moleule s struture substanes where lassified aording to their marosopi appearane and reative behavior. Today, lassifiation of substane usually refers to the struture of the moleules, e.g., alohols are haraterized by a funtional OH-group. Sometimes only the omposition of atoms is taken for lassifiation, e.g., hydroarbons. Interestingly and importantly for the suess of the disipline Chemistry is the fat that strutural lassifiation oinide with lassifiations based on behavior and appearane. This phenomenon is not suffiiently explained by the fat that the funtion (ie. physial and reative properties) of a moleule depends on its struture, whih is a form of ausality. Moreover, similarity in struture tends to oinide with similarity in funtion, whih is a form of strong ausality between struture and funtion. Another important observation should be noted: When we ombine some substane in a reation vessel and wait while these substanes reat; as a result only a small subset of moleular speies will appear, whih is usually muh smaller than the set of moleular speies that ould be build from the atoms present in the reation vessel. So there is also a ertain (strong) ausality in the dynamis and a dependeny on initial onditions. Not everything that is possible does appear, though there is also nondeterminism. So, we an say that a hemial system evolves over time in a ontingent way that depends on its history. 1.3 Information Proessing and Computing in Natural System When we intend to take inspiration from hemistry, we have first to investigate where hemial information proessing appears in natural systems. Obviously, living systems are prime andidates, sine information proessing is identified as a fundamental property of life [11].

4 Information proessing in living systems an be observed on at least two different levels: the hemial and the neural level. Where the neural level is responsible for ognitive tasks and fast oherent ontrol, suh as vision, planing, and musle ontrol; hemial information proessing is used for regulating and ontrolling fundamental proesses like growth, ontogeny, gene expression, and immune system response. Neurons themselves are based on (eletro-)hemial proesses, and more often than not, hemial proesses are ombined with neuronal proesses resulting in a large-sale omputational result. Real hemial omputing utilizes a series of hemial priniples, whih are also relevant for artifiial hemial omputing, suh as: pattern reognition[12], hange of onformation[13], hemial kinetis [14], formation of (spatial) strutures, energy minimization, and optial ativity [15]. Pattern reognition is a entral mehanism for explaining reations among omplex biomoleules (e.g., transription fators binding to DNA). It is also used in real as well as artifiial hemial omputing approahes, suh as DNA omputing [12] and rewriting systems [1, 16, 17], respetively. 1.4 Appliation of the Chemial Metaphor in Computing There are already a series of approahes in omputer siene that have been inspired by hemistry: An early example are the artifiial moleular mahines suggested by Laing [18]. These mahines onsists of moleules (strings of symbols). Eah moleule an appear in two forms: data or mahine. During a reation, two moleules ome into ontat at a partiular position. One of the moleules is onsidered as the ative mahine, whih is able to manipulate the passive data moleule. The primary motivation for developing these moleular mahines was to onstrut artifiial organisms in order to develop a general theory for living systems (f. [19] for a omparing disussion of more reent approahes in that diretion). A fundamentally different motivation has been the starting point for the development of Gamma by Banâtre and Le Métayer, namely to introdue a new programming formalism that allows to automatize reasoning about programs, suh as automati semanti analysis [20, 1]. Gamma is defined by rewriting operations on multisets, whih mimis hemial reations in a well-stirred reation vessel. Gamma inspired a series of other hemial rewriting systems: Berry and Boudol [2] introdued the hemial abstrat mahine (CHAM) as a tool to model onurrent proesses. Pǎun s P-Systems [16] stress the importane of membranes. Suzuki and Tanaka [21] introdued a rewriting system on multisets in order to study hemial systems, e.g., to investigate the properties of hemial yles [22], and to model hemial-like systems inluding eonomi proesses. Within biologial organisms, the endorine system is a ontrol system, whih transmits information by hemial messengers alled hormones via a broadast strategy. The humanoid robot torso COG [5] is an example where the endorine system has inspired engineering. Artifiial hormones are used to ahieve a oherent behavior among COG s large number of independent proessing elements [5]. In general, hemial-like systems an ontrol the behavior and partiularly

5 emotions in artifiial agents, e.g., the omputer game Creatures [23] and the psyhologial model PSI by Dörner [4]. Further appliation areas of hemial omputing are: the ontrol of morpho-geneti systems, i.e. the ontrol of morphogenesis by artifiial gene expression; in partiular, the ontrol of growth of an artifiial neural networks (f. Astor and Adami [24]); and the ontrol of amorphous omputers [25]. Finally, Husbands et al. [26] introdued diffusing hemial substanes in artifiial neural networks (f. GasNet). 2 Faets of Chemial Computing As exemplified by the previous setion, the world of hemial omputing enjoys already a wide spetrum of approahes. This setion disusses a set of important aspets, whih allow to haraterize hemial omputing in more detail. 2.1 Mirosopi vs. Marosopi Computing Chemial information proessing an be haraterized aording to the level on whih it appears: In approahes like hemial boolean iruits [27], the hemial neuron [14], or the hyperyli memory (Se. 5.2), information is represented by the onentration of substanes and omputation is arried out by an inrease and derease of onentration levels, whih an be regarded as a form of marosopi hemial omputing. Alternatively, in mirosopi hemial omputing, the intermediately stored information and omputational results are represented by single moleules. Examples are DNA omputing [12] and the prime number hemistry (Se. 5.1). The dynamis is usually stohasti, in ontrast to marosopi omputation, whih an be more readily desribed with ordinary differential equations. Nevertheless, mirosopi omputing also an deliver results virtually deterministially, as shown by the prime number hemistry example in Se Deterministi vs. Stohasti Proesses On the moleular level, hemial proesses are stohasti in nature. However, in tehnial appliations deterministi behavior is often required. There are various ways how this an be ahieved: (1) The problem an be stated suh that the order of the sequene of ollisions does not play a role 1. An example is the prime number hemistry where we start with a population that ontains all numbers between 2 and n. The outome will be a reator ontaining all and only prime numbers less or equal n, independently of the sequene of updates. (2) Inreasing the reator size would redue the effet of randomness. If the reator size and together with it the number of moleules of eah moleular type 1 For a theory that onsiders the effet of the order of update see sequential dynamial systems [28, 29]

6 tends to infinity, the moleules onentrations tend to a deterministi dynamis. In this ase, the dynamis of the onentrations an be represented by a differential equation and simulated by numerial integration of this equation. (3) A well-defined deterministi update sheme an be used. For example we an hek one reation rule after another in a fixed predefined sequene, e.g., early ARMS [21] and MGS [17] 2 Doing this, we gain determinism and might gain effiieny, but we loose aspets of the hemial metaphor and may introdue artifats by the update sheme, e.g., when the rule order plays a signifiant role. This might be reasonable from a omputing point of view, but is unnatural from a hemial point of view. 2.3 Closed vs. Open Systems In thermodynamis, a system that an exhange mass and energy with its environment is alled open.when mass is not exhanged the system is alled losed If the system annot exhange anything, it is alled isolated. In hemial omputing we also enounter losed and open systems, whose harateristis are quite different. In a losed system, moleules do not leave the reation vessel. There is no dilution flow. Reation rules must be balaned, whih means that the mass on the left hand side must be equal to the mass on the right hand side. So, a moleule an only disappear by transforming it via a reation into other moleules. In an isolated system, stable dissipative strutures an not appear; they an only appear as transient phenomena loally. The prime number hemistry is an example for a losed and isolated system. There is no dilution flow and moleules are transformed by the mass-onserving rule: a + b a + b/a for b being a multiple of a. The hyperyli memory is an example for an open system. Moleules onstantly vanish and are regenerated from an impliitly assumed substrate, whih is available at a onstant onentration from the environment. Before the query, the system is in a quasi-stationary state, whih is a dissipative struture that requires a onstant regeneration of all of its omponents. The hyperyli memory is also an example where there is a so alled nonseletive dilution flow, where the rate of deay is proportional to the onentration of a moleule, or more preisely, the onentration of moleules in the dilution flow is the same as in the reation vessel. Systems with seletive dilution flows are not disussed here, but it should be noted that by introduing a seletive dilution flow, we an move gradually from an open to a losed system and an apture aspets from both. Does it make sense to onsider open systems with a non-seletive dilution flow, where we have to regenerate onstantly moleules we wish to have in the reator? From a formal point of view, both might be equivalent: In an open system, a stable solution is a self-regenerating set of moleules; while in a losed system, a stable solution is just a set of moleules, whih do not reat further 2 Note that both mentioned systems (ARMS, MGS) allow also a randomized natural update sheme.

7 to form other moleules (nevertheless there might be a reversible dynamis). So from this point of view, taking a losed systems approah appears more reasonable, beause the solution is more stable. We do not have to fear that information gets lost by the dilution flow and we do not have to are for regenerating moleules. However, when using an open system approah we arrive at more robust and flexible organi systems. Open reation systems are espeially suitable, where the substrate is unreliable and highly dynami. Consider for example a omputational substrate that is under onstant hange, where nodes are added and removed at a relatively high rate, e.g., the network of ativated ellular phones. In suh a system, there is no plae that exists for long. When a ellular phone is swithed off, the moleules residing in that plaes vanish, too, whih auses from a hemial point of view a general, non-seletive dilution flow. Thus stable strutures must onsists of moleules that onstantly reprodue themselves as a whole; aording to the theory of hemial organization [30], they must enompass a self-maintaining set of moleules. 2.4 Computing with Invisible Networks What is the differene between hemial omputing and an artifiial neural network (ANN)? In both approahes, a network is speified by a set of omponents (moleules/neurons), a set of interations (reations/onnetions), and a desription of the behavior (dynamis/firing rule). In ontrast to hemial omputing, an ANN is usually aompanied by a learning proedure. However, learning an be added to hemial omputing by means of evolutionary omputation [31, 32] or by transferring learning tehniques from omputational intelligene, e.g., Hebbian learning. In partiular, this should be straight forward for expliitly defined hemial systems operating marosopially, whih are quite similar to dynamial neural networks (see Se. 5.2 or ref. [14]). But there are some remarkable differenes: When we onsider a reation system with impliitly defined moleules and rules like the prime number hemistry in Se. 5.1, we an easily obtain giant networks that are invisible. When we look inside a reation vessel, no omponent that represents a onnetion an be seen. Even the nodes of the hemial network annot be easily identified beause they are not spatially differentiated from eah other, sine a hemial node may be represented by a olletion of moleules that are instanes of one moleular speies. The prime number hemistry is an example where a ouple of simple rules imply a giant network, muh larger than a human brain, e.g., for n = Another important differene to ANNs should be mentioned: When exeuting a hemial omputation, only a subnetwork is ative at a ertain point in time, whih is illustrated by Fig. 1. Sine the size of a reation vessel is limited, it an only ontain a fration of moleules from the set of all possible moleules. These present moleules together with all reations that an our among them an be regarded as the ative reation network. Due to internal or external dynamis, the set of moleular speies in the reation vessel an hange, and thus the

8 a a a b b f f d f b b a f a f e d g e d g time Fig. 1. Illustration of the invisible network, whose ative part hanges over time. A bold harater denotes a moleular speies that is present in the reator. These speies imply the urrently ative network highlighted by solid arrows. Note that a harater in the reation network denotes a moleular speies, whereas the same harater in the skethed reation vessel denotes a onrete moleule (or instane) of that speies. ative network evolves over time, too (Fig. 1). This phenomenon is aptured theoretially by a movement through the set of hemial organizations [33, 30]. 3 Chemial Programming Programming a hemial omputer means to define a hemial system, whih is often also referred to as a reation system or an artifiial hemistry [34]. There are two fundamentally different approahes to hemial programming: (1) automati programming by optimization, e.g. by means of evolutionary omputation [31, 32], and (2) engineering by a human programmer, e.g. [1, 25]. Both approahes require speifying the following three aspets of the target hemistry: (1) Moleules: In the first step, we have to speify how the moleules should look like. Should they be symbols or should they posses a struture, e.g., a sequene of haraters [35], a hierarhial expression [36], or a graph like struture [37]. If moleules posses a struture, the definition of the reation rules and the dynamis an refer to this struture, whih allows to define large (even infinite) reation systems, as exemplified by the prime number hemistry in Se If the moleules are symbols, we have to speify the set of possible moleules expliitly by enumeration of all possible moleules, e.g., M = {a,b,,d}. If moleules posses a struture, we an define the set of all possible moleules impliitly, e.g., M = {1,2,...,10000}.

9 (2) Reations: In the next step, we have to speify what happens when moleules ollide. Real moleules an ollide elastially or they an ollide ausing a reation, whih transforms the moleules. In a simple abstration, a reation rule is just a pair of two multisets of moleules, whih speifies what kind of moleules an be transformed and replaed by what kind of moleules, e.g., a well known reation rules is ({H 2,H 2,O 2 }, {H 2 O,H 2 O}), whih is written in hemial notation equivalently as 2H 2 + O 2 2H 2 O. In general, reation rules an beome more ompliated and an arry further information, suh as parameters speifying kineti onstants or environmental onditions under whih this reation an our. Analogously to moleules, reation rules an be speified expliitly like 2H 2 +O 2 2H 2 O, or impliitly, as in the prime number hemistry. (3) Dynamis: Finally, we have to speify the dynamis, whih inludes the geometry of reation vessel. Do we assume a well-stirred vessel or vessel with a spatial struture? How do moleules ollide? How are the reation rules applied, e.g., deterministially or stohastially? Well-stirred deterministi reation systems are usually simulated by integrating ordinary differential equations. Stohasti systems an be simulated by expliit stohasti ollisions of individual moleules or by more advaned disrete event based methods like the Gillespie algorithm [38]. Spatial strutures are usually introdued by some sort of ellular automata (e.g., lattie moleular automata [39]) or by ompartments like in amorphous omputing [25], membranes in P-systems [16] or mobile proess aluli [40], or topology in MGS [17]. 3.1 Strategies for Chemial Programming We distinguish different strategies of hemial programming aording to the omponents a programmer an manipulate: (1) Define moleules and reations expliitly: The programmer has to speify the set of moleules as a set of symbols and the reation rules as a set of expliit transformation rules. An example for this approah is the hyperyli memory (Se. 5.2), the metaboli robot [41], and evolved hemial systems like those reported by Ziegler and Banzhaf [31]. (2) Define moleules and reations impliitly: The programmer speifies moleules and reation rules impliitly like demonstrated by the prime number hemistry in Se For defining reation rules impliitly, the moleules an not be just a list of symbols, rather they must posses a struture to whih the definition of the reation rules an refer to (see Se. 3(1)). This approah is quite general, but beause of its generality additional priniple for guiding the programmer are required. (3) Change-moleules-only priniple: Again, the reation rules are impliitly defined but fixed (predefined) and annot be hanged by the programmer. The programmer or an evolutionary proess has just to selet the right moleules and the dynamis, inluding the topology of the reation spae. This resembles the way real hemial omputers are programmed, e.g., seleting appropriate DNA strands for solving a Hamiltonian path problem as in the famous example by Adleman [12]. Although the programmer has limited hoies (ompared to

10 the previous setting), the expressive power is the same, if a universal hemistry is used. A universal hemistry is defined as a hemistry that inludes every possible (let s say, finite) reation network topology. Suh hemistry an, for example, easily be defined based on lambda-alulus [36] or ombinators [42]. However, these abstrat formalisms stemming from theoretial omputer siene an not be easily and intuitively handled by programmers, other approahes are more feasible for pratial appliation (see Banâtre, Fradet, and Radena in this volume). (4) Multi-level hemial programming: As before, the programmer selets appropriate moleules and dynamis. At the same time, the physis an be manipulated, too, but at a slower rate. For example, the alulus speifying impliitly the set of possible moleules and the set of possible reations an be altered and extended. By this, the funtion (meaning) of moleules an beome more transparent (syntati sugar). On a higher level of abstration, moleules may be assembled to higher lusters resembling maro moleules or modules, whih an again serve as building bloks for impliit definitions of other moleules, reation rules, and dynamis. Therefore, a programmer operates on different levels, suh as: Level 0: manipulation of the physis, e.g., ombinator rules. Level 1: seleting (defining) the right moleules and reation rules, and dynamis in the ontext of the hosen physis, e.g., seleting appropriate ombinators. Level 2: Speifying higher level lusters, modules, maro-moleular omplexes, e.g., based on membrane omputing onepts. 4 Conlusion and Challenges This essay disussed several aspets of artifiial hemial omputing. It has been shown that hemial-like systems possess a number of interesting properties, whih appear espeially feasible in domains like distributed omputing, ambient omputing, or organi omputing. Furthermore, a ouple of appliation senarios have been desribed, inluding a first suessful ommerial appliation [23]. Taking heed of these fats, the hemial metaphor appears as a paradigm, whih will qualitatively enrih our repertoire of programming tehniques. The road map of hemial omputing inludes a series of hallenges: (1) Effiieny: How to obtain runtime and memory effiient hemial programs and their exeution on eletroni hardware? (2) Salability: How do hemial omputing paradigms sale up? (3) Programmability: How to program a hemial omputer? (4) Adaptability and robustness: How to ahieve self-adapting, learning, and reliable hemial omputing systems? (5) Theory: How to desribe hemial omputing proesses theoretially? Here, hemial organization theory [30] appears as a promising approah, espeially when dealing with onstrutive hemial systems. Other soures for a theoretial base are lassial dynamial systems theory and approahes from omputer siene like rewriting aluli [1, 2, 16, 40] and temporal logi. Furthermore we may investigate fundamental question onerning the power and limits of hemial omputing by questions like: Can the hemial metaphor lead to new omputational systems with abilities

11 superior to onventional approahes, or even to systems that an not be realized by onventional approahes? It is evident that the future will witness further integration of onepts from the natural sienes and omputer siene, whih will redue the differenes between the living and the tehnologial world. Like living systems, omputing systems in the future will onsist of deentralized and highly distributed omponents that interat with inreasing autonomy and flexibility. For harnessing their potential it will be ruial to obtain new, organi methods for their onstrution and ontrol. Chemial omputing, whih has been employed by nature with great suess, offers a promising paradigm. 5 Appendix: Examples 5.1 Prime Number Chemistry Banâtre and Le Metayer [1] suggested the numerial devision operator as an impliit reation mehanism, whih results in a prime number generating hemistry defined as follows (see ref. [41, 43] for details): the set of all possible moleules are all integers greater one and smaller n+1: M = {2,3,4,...,n}. The reation rules are defined by a devision operation: R = {a + b a + a,b, M, = a/b, a mod b = 0} = { , , ,...}. So, two moleules a and b an reat, if a is a multiple of b. For the dynamis, we assume a well-stirred reation vessel. The state of the reation vessel of size M is represented by a vetor (or equivalently by a multi-set) P = (p 1,p 2,...,p M ) where p i M. The dynamis is simulated by the following stohasti algorithm: (1) hose two integers i,j {1,...,M},i j randomly. (2) if there is a rule in R where p i + p j mathes the left hand side, replae p i and p j by the right hand side. (3) goto 1. Assume that we initialize the reation vessel P suh that every moleule from M is ontained in P, then we will surely reah a stationary state where all moleules from P are prime numbers and every prime number greater one and less or equal n is ontained in P. The outome (prime numbers present in P) is deterministi and in partiular independent from the sequene of reations, where the atual onentration of eah prime number an vary and depends on the sequene of reations. Now assume that P is smaller than M, e.g., M = 100 and n = The outome (moleular speies present in P) is not deterministi. It depends on the sequene of updates, e.g., P = (20,24,600) an result in the stable solutions P = (20,24,30) or P = (20,24,25). Note that the behavior (ability to produe prime numbers) depends ritially on the reator size M [41, 43]. 5.2 Hyperyli Assoiative Memory Assume that we have an unreliable media, where all moleules deay sooner or later. In order to store data over a longer period, moleules have to be reprodued. Simple self-repliating moleules are not suffiient, sine, as disussed by

12 Fig. 2. Hyperyli assoiative memory. Left: Illustration of the reation network. An arrow represents a atalyti interation where both reatants at as atalysts and are not used up. Only the ative network is shown. Right: Example of a stohasti simulation of a query. 400 moleules of type q 1 are inserted. Reator size M = Eigen and Shuster [44], in a limited volume, self-repliating moleules ompete for resoures and an not oexist stably (exponential growth and no interation assumed). In the following example [41], three units of data {d 1,d 2,d 3 } are stored in three different moleules {w 1,w 2,w 3 }. In order to query the memory, there are three input moleules {q 1,q 2,q 3 }. Our demanded speifiations are: the hemial system should store the data for a long period of time, under onstant dilution of the moleules. The system should produe d i provided q i as input. We assume the following reations: R = {w 1 + w 2 w 1 + w 2 + w 2,w 2 + w 3 w 2 + w 3 + w 3,w 3 + w 1 w 1 + w 2 + w 1,w 1 + q 1 w 1 + q 1 + d 1,w 2 + q 2 w 1 + q 2 + d 2,w 3 +q 3 w 1 +q 3 +d 3 }. For the dynamis, we assume a well-stirred reation vessel that ontains a onstant number of M moleules. The state of the vessel is represented by a vetor (or equivalently by a multi-set) P = (p 1,p 2,...,p M ) where p i M. The dynamis is simulated by the following stohasti algorithm: (1) hose three integers i,j,k {1,...,M},i j randomly. (2) if there is a rule p i + p j p i + p j + x in R, replae moleule p k by x. (3) goto 1. This kind of stohasti algorithm is equivalent to the deterministi repliator equation and atalyti network equation. It is also used in several other works [45, 35,46]. Figure 2 shows an example of a simulation where 400 moleules of type q 1 are inserted into a reator that ontains approximately the same amount of eah information moleule {w 1,w 2,w 3 }. Interation of q 1 with w 1 results in the prodution of d 1. Sine all moleules are subjet to a dilution flow and q 1 is not produed, q 1 and d 1 are washed out while the onentrations of {w 1,w 2,w 3 } stabilize again. Aknowledgment: I am grateful to F. Centler, N. Matsumaru, and K.-P. Zauner for helpful omments. This work was supported by the Federal Ministry of Eduation and Researh (BMBF) Grant A to Friedrih Shiller University Jena.

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