Watson-Crick D0L SystemsValeria MihalacheArto Salomaa

Turku Centre for Computer ScienceTUCS Technical Report No 108April 1997ISBN 951-650-996-7ISSN 1239-1891

AbstractD0L systems constitute the simplest and most widely studied type of Lin-denmayer systems. They have the remarkable property of generating theirlanguage as a (word) sequence and, consequently, are very suitable for model-ing growth properties. In this paper a new type of D0L systems is introduced,where the parallelism presented in L systems is combined with the paradigmof (Watson-Crick) complementarity characteristic for DNA computing. Thenew Watson-Crick D0L systems are still deterministic and generate their lan-guage as a sequence. However, the capability of modeling growth is greatlyenhanced: the new growth functions are not necessarily even ZZ-rational.TUCS Research GroupMathematical Structures of Computer Science

1 IntroductionAdleman demonstrated, [1], how methods of molecular biology can be appliedto solve a computationally di�cult problem. Since then the interest in "DNAcomputing" has been growing rapidly, for instance, see [5], [7] and theirreferences. That DNA computing is relevant also for the theory of formallanguages can be concluded from many chapters of the recent Handbook,[10]. On the other hand, there are still considerable obstructions to creatinga practical molecular computer, and also very pessimistic views have beenexpressed, [4].DNA (deoxyribonucleic acid) is found in all living organisms as the stor-age medium for genetic information. It consists of polymer chains, custom-arily referred to as DNA strands. A chain is composed of nucleotides, alsoreferred to as bases. The chains are also referred to as oligonucleotides, brie yoligos. The four DNA nucleotides or bases are customarily denoted by A(adenine), C (cytosine), G (guanine), and T (thymine). The DNA alphabet�DNA = fA;C;G; Tg will be important in our subsequent considerations.We often use lower-case letters instead of capital ones.Thus, DNA strands may be viewed as words over the DNA alphabet.According to a chemical convention, each strand has a " 5' end" and a " 3'end", for instance,50 ATTAGCAT 30 or 30 TAATCGTA 50;making the words oriented. However, this orientation is irrelevant for ourpurposes and will be ignored in the sequel.The familiar double helix of DNA arises by the bondage of two separatestrands. In the formation of such double strands a phenomenon known asWatson-Crick complementarity comes into the picture. Bonding happens bythe pairwise attraction of bases: A bonds with T , and C bonds with G.This is the reason why the unordered pairs (A;T ) and (C;G) are referredto as complementary pairs of bases. Bonding occurs only if the bases in thecorresponding positions in the two strands are complementary. (Moreover,they have to have opposite orientation but, as already pointed out, we willignore here the orientation.) Thus, the two strands mentioned above willform the double strand ATTAGCATTAATCGTASuch double strands form a data structure of a new type, a data structurecharacteristic for the theory of DNA computing. The very nature of this datastructure is essential for DNA computing. It is the source of the strength of1

DNA computing because, in some sense, it makes the powerful twin-shu�elanguage "freely" available. (The universality of the twin-shu�e language isdue to [3], see also [11] and [2]. The interconnection between DNA doublestrands and the twin-shu�e languages was pointed out in [9].) To illustratethe power of complementarity present in double strands, we now brie y recallAdleman's famous experiment, [1], of solving an instance of the Hamiltonianpath problem, HPP.Given a directed graph, we want to solve its HPP. Each vertex is encodedby an oligo of even length. (Thus, in our considerations oligos are simplywords over the DNA alphabet. Adleman used words of length 20.) Eachedge is encoded by an oligo of the same length as was used for the vertices.The encoding of the edges is determined by the encoding of the vertices inthe following fashion. Consider the endomorphism h of ��DNA mapping eachletter into its Watson-Crick complement:h(A) = T; h(T ) = A; h(C) = G; h(G) = C:(We will refer to h as the Watson-Crick morphism.)Let (x; y) be an edge, where x is the oligo encoding the vertex from whichthe edge emanates and, similarly, y encodes the target vertex. Write x inthe form x = x1x2; jx1j = jx2j and, similarly, y = y1y2; jy1j = jy2j: The edge(x; y) is now encoded by the oligo h(x2y1):Adleman's experiment begins by forming a "DNA soup" containing, inlarge quantities oligos of the vertices, as well as of the edges of the graph.The ligation reaction resulting from the Watson-Crick complementarity nowwill link together compatible edges. This domino game continues, identifyinglonger and longer paths. In this way DNA molecules encoding random pathsthrough the graph are formed. By a �ltering procedure not of interest for ushere, one can check whether or not paths satisfying HPP are present.Adleman's experiment shows the computational strength of the Watson-Crick complementarity. It is a consequence of the universality of the twin-shu�e language that any model of DNA computing, where the �ltering pro-cedures available can simulate gsm mappings, is capable of Turing machinecomputations.Although the real practical feasibility of molecular computers remainsstill in doubt, computer scientists and mathematicians are already lookingfor new models of computation along these lines. Such models could be calledWatson-Crick machines. We are not going to specify here explicitly any suchmodel. It seems obvious that in any such model use must be made of thefollowing two advantages stemming from DNA molecules: (i) Watson-Crick2

complementarity which always renders the powerful twin-shu�e languagesavailable, and (ii) the multitude of DNA molecules which brings the massiveparallelism to the computing scene. It seems also that theoretical studies haveso far been focused on the point (i). We will now discuss complementarityas a general language-theoretic principle.2 Complementarity in language theoryThe notion of complementarity based on the idea of Watson-Crick has so farbeen very little observed or investigated in language theory. Yet the notionis very simple and natural. We would like to formulate some of the issues inthe following way.Paradigms of complementarity.(i) A string induces the complementary string, either randomly or guidedby a control device.(ii) The complementarity of two strings leads to some phenomenon suchas bondage. Thus, the occurrence of the phenomenon guarantees thatthe strings involved are complementary.Adleman's experiment makes use of Paradigm (ii). Another type of useof Paradigm (ii), dealing with Prolog, is given in [6]. In the sequel we willbe concerned with Paradigm (i). We have so far considered complementarityonly in connection with the DNA alphabet �DNA. The following generaliza-tion is straightforward.A DNA-like alphabet � is an alphabet with even cardinality 2n; n � 1;where the letters are enumerated as follows:� = fa1; : : : ; an; a1; : : : ; ang:Thus, each of the non-barred letters ai; 1 � i � n; has its barred version ai:We say that ai and ai are complementary. The letter-to-letter endomorphismof �� mapping each letter to the complementary letter is referred to as theWatson-Crick morphism.When the DNA alphabet �DNA is viewed as DNA-like, we consider thepurines A and G as non-barred letters: a1 = A and a2 = G: Hence thepyrimidines T and C are their barred versions: a1 = T and a2 = C:It is both natural and su�cient for our purposes to consider only the"mild" generalization of the alphabet �DNA introduced above. Of course,a more general notion of a DNA-like alphabet would be the pair (�; �);3

where � is a binary relation on �; satisfying some suitably chosen restrictiveconditions.The paradigms of complementarity can now be understood with respectto complementarity in DNA-like alphabets. As already mentioned, we will beconcerned with Paradigm (i). The paradigm can be applied to any language-theoretic device: the transition to the complementary string happens eitherfreely or under some control. However, the paradigm seems very natural inconnection with L systems. According to an L system, the whole string isrewritten. This is also exactly what the Watson-Crick morphism does.We are fully aware that complementarity opens a vast area for language-theoretic research; many kinds of questions can be asked along these lines.Our purpose here is to make only some observations. It seems to us that avery important generalization of the DNA-like alphabet, as well as comple-mentarity based on it, consists of considering a disjoint union of a DNA-likealphabet �1 and another alphabet �2: Only the letters of �1 are a�ectedin the transition to the complementary string, while the letters of �2 re-main unchanged. However, as already pointed out, in this paper the mildgeneralization of the alphabet �DNA, as well as the resulting notion of com-plementarity, will be su�cient for our purposes.We now present an example of a "Watson-Crick 0L system", withoutgiving the full formal de�nition of the notion. Consider �rst an ordinary 0Lsystem, say, the system G with the alphabet � = fa; g; t; cg; the axiom gacand the productionsa! �; a! ca; g! cat; c! ta; t! �:(Observe that we use the DNA alphabet but prefer the lower-case letters.For all unexplained notions about L systems we refer to [8].)The 0L system G is now transformed into a Watson-Crick 0L system Gwby supplementing G with a trigger for complementarity transition. In ourcase the trigger will be the appearance of the subword catcat: The subwordcatcat is not permitted to appear; whenever it would appear in the normalcourse of the 0L derivation, the complementary word must be taken. Thus,gac) catcatais a derivation step according to G but must be replaced by the derivationstep gac)w gtagtataccording to Gw: 4

Formally, the yield relation )w is de�ned from the yield relation ) ofthe 0L system G as follows. Let h be the Watson-Crick morphism. Assumethat � 2 �� is a word not containing catcat as a subword, that is, � 62��catcat��: Then �)w � if �) � and � 62 ��catcat��: Also �)w h(�)if �) �; � 2 ��catcat�� and h(�) 62 ��catcat��:Observe that the relation )w is not de�ned between � and � if either �contains catcat as a subword, or else � ) � and both � and h(�) containcatcat as a subword. It can be veri�ed that in our particular example, startingfrom the axiom gac; such blocking situations do not occur. Indeed, if catcat isa subword of h(�); then gtagta is a subword of �: This is not possible for thesimple reason that g does not appear on the right sides of the productions.The same observation tells us also that, with our productions and triggerfor complementarity transition, blocking can occur only if it occurs in theaxiom, that is, the axiom contains both catcat and gtagta as subwords.Observe further thatgac)w catta)w tacaca)w catacataca;gac)w gtagtat)w catcacat;gac)w gtagtat)w gtagtagt)w gtagtagtgtaare all the derivations according to Gw: Continuing the last one and applying�-rules for t and a between two g's, we obtain derivations with arbitrarilymany complementarity transitions.The complementarity transition dramatically changes the behaviour ofa 0L system, also with respect to fundamental properties. Thus, it is well-known that ifG = (�; h; w) is a 0L system, then for any nonnegative integer nand for any words x1; x2; y1; y2 2 ��, if x1 )n y1 and x2 )n y2, then x1x2 )ny1y2, and conversely (by ")n" we have denoted the relation "derives in nsteps"). This property does not hold anymore for Watson-Crick 0L systems.In the above example, we have ga )w catca and c )w ta. However, dueto the trigger for complementarity transition, gac 6)w catcata, but gac )wgtagtat instead. Conversely, gtagtat cannot be decomposed as to derive fromtwo subwords composing gac.In general, the appearance of a particular subword is one possible way oftriggering the complementarity transition. Natural decision problems in thiscase are: (i) Does a complementarity transition occur in a given Watson-Crick 0L system? (ii) Does a transition occur unboundedly many times inthe same derivation according to a given system?5

Clearly, (i) is decidable because the emptiness of the intersection betweena 0L language and a regular language is decidable. Problem (ii) is trickier,and we have no direct answer.3 Watson-Crick D0L systems: general de�ni-tionFrom now on we will focus the attention on D0L systems. We will alsode�ne the notions explicitly. The case of Watson-Crick D0L systems will beparticularly interesting because, similarly as in case of ordinaryD0L systems,the language is generated as a sequence. In what follows, � will be a DNA-likealphabet and h the Watson-Crick morphism.By de�nition, a Watson-Crick D0L system is a pair W = (G;�); whereG = (�; f; w0) is a D0L system (f is the morphism and w0 2 �� is theaxiom) and � : �� ! f0; 1g is a mapping such that �(w0) = �(�) = 0 and,for all words w 2 �+; �(w) = 0 i� �(h(w)) = 1:According to the terminology of the preceding section, the condition�(w) = 1 is the trigger for complementarity transition. If in the course of aderivation a word w is encountered such that �(w) = 1; then w is replacedby h(w): The condition imposed on � guarantees that in such a case neces-sarily �(h(w)) = 0; that is, the blocking situation discussed in the precedingsection never occurs. If f is de�ned in terms of productions, the de�nitionhere is in full accordance with that in the preceding section.Formally, the sequence S(W ) of the system W consists of the wordsw0; w1; w2; : : : ; where for each i � 0;wi+1 = ( f(wi) if �(f(wi)) = 0;h(f(wi)) if �(f(wi)) = 1:The language L(W ) of the system W consists of all words in the sequenceS(W ): Two Watson-Crick D0L systems W1 and W2 are termed sequence(resp. language) equivalent if S(W1) = S(W2) (resp. L(W1) = L(W2)).The de�nition guarantees that, for all i � 0; �(wi) = 0: For instance,consider the DNA alphabet � = fa; g; t; cg and de�ne �(w) = 0 i� w beginswith a purine. In other words, for all w,�(aw) = �(gw) = 0; �(cw) = �(tw) = 1:6

If G = (�; f; w0) is any D0L system such that w0 begins with a or g, then(G;�) is a Watson-Crick D0L system. This de�nition of � is conspicuouslylocal: the value �(w) is determined by the �rst letter of w.The decidability of sequence or language equivalence has been a cele-brated problem for D0L systems. The same problems can be posed forWatson-Crick D0L systems; of course, one has to de�ne � more explicitely.We now give a very natural explicit de�nition.Let � be a DNA-like alphabet. A subset �� of � is termed a selectorif it contains exactly one letter from each pair (a; a); a 2 �: Observe thatthe cardinality of a selector equals the cardinality of � divided by 2: Forinstance, the set of purines fa; gg constitutes a selector for the DNA alphabetfa; g; t; cg:We use the customary notation jwj�� to denote the length of the scatteredsubword of w, consisting of letters of ��: We now de�ne the function � by�(w) = 8><>: 0 if jwj�� � jwj2 ;1 if jwj�� < jwj2 :It is immediately veri�ed that the required condition is satis�ed: �(w) = 0i� �(h(w)) = 1: Thus, each selector can be used to de�ne a Watson-CrickD0L system, provided the axiom is chosen in the proper way. We refer tosuch Watson-Crick D0L systems as selector-based.The condition determining the value of � is global in selector-based sys-tems. Yet the condition is very simple and, thus, the de�nition of the result-ing word sequence is a natural generalization of the ordinary D0L word se-quence. The customary notions and problems dealing with D0L systems arereadily extended to concern (selector-based) Watson-Crick D0L systems. Inparticular, one can consider word length and growth functions. The growthfunctions of a D0L system is de�ned by the condition f(i) = jwij; wherew0; w1; w2; : : : is the sequence of the system. In case of D0L systems thisde�nition leads to a simple matrix representation of f . Consequently, be-cause the classical Perron-Frobenius theory is applicable, the theory of D0Lgrowth functions is quite well understood, [8], [11]. The same does not holdfor Watson-Crick D0L systems. Even in the simple case of selector-basedsystems the complementarity transition invalidates the matrix representa-tion. The following examples are intended to illustrate this phenomenon.They show also that (selector-based) Watson-Crick D0L systems constitutea really substantial generalization of D0L systems. That they constitute ageneralization is obvious: ordinary D0L systems are obtained as a specialcase by using only non-barred letters, both in the system and in the selector.7

In the following example we consider the DNA-like alphabet � = fa1; a2;a3; a1; a2; a3g and the selector set �� = fa1; a2; a3g: Our selector-basedWatson-Crick D0L system has the axiom a1a2a3 and its morphism is de�nedby the rulesa1 ! a1; a2 ! a2; a3 ! a3; a1 ! a1 a2; a2 ! a2; a3 ! a33:The beginning of the sequence S(W ) looks as follows, where bold charactersindicate that complementarity transition has taken place:a1a2a3; a1 a2a33; a1 a22a33; a1a32a33; a1 a23a93; a1 a24a93; a1 a25a93;a1 a26a93; a1 a27a93; a1 a28a93; a1a92a39; a1 a29a273 ; a1 a210a273 ;after which there are 16 words before the next complementarity transition.Observe that only the barred letters induce growth. The length sequenceis strictly growing:3;5; 6;7;13; 14; 15; 16; 17; 18;19;37;38; : : :The growth uctuates between linear growth (from n to n+1) and exponen-tial jumps (from 1 + 2 � 3n to 1 + 3n + 3n+1). Indeed, when f(i) is of theform 1+2�3n ; n � 0; then f(i+1) = 1+3n+3n+1: When f(i) (> 3) is notof the form 1 + 2� 3n, then f(i+1) = f(i) + 1: Thus, the strictly increasinggrowth function f assumes all values � 3, except the values strictly betweenthe numbers 1 + 2 � 3n and 1 + 3n + 3n+1; for all n � 0. Consequently, thegrowth function f(i) is linearly bounded, indeed f(i) < 4(i+ 1) for all i � 0, and(ii) the di�erences f(i+1)�f(i) have no �nite upper bound, that is, thereis no integer k such that f(i+ 1)� f(i) � k holds for all k.No function satisfying (i) and (ii) can be a D0L growth function. Indeed,if a linearly bounded function is a D0L growth function, then it can bedecomposed (see [8] or [10]) into linear functions of the form ai+ b: At leastone component of the decomposition would have to satisfy (ii), which is notpossible.In fact, even a stronger result concerning the growth function f of our ex-ample system can be obtained: f is not a ZZ-rational function. (D0L growthfunctions are a special case of IN-rational functions. IN-rational functionscoincide with growth functions of HD0L systems, that is, morphic imagesof D0L systems. ZZ-rational functions coincide with di�erences of two IN-rational functions. See [8] or [10] for further details.) This stronger result8

can be established by the following simple argument. If f were ZZ-rational,then also the function f1(i) = f(i+1)� f(i) would be ZZ-rational. However,f1 assumes the same value (= 1) in arbitrarily long intervals without beingultimately constant. By the "Lemma of Long Constant Intervals", f1 cannotbe ZZ-rational. We want to present these results as a formal theorem.Theorem. The class of growth functions of selector-based Watson-CrickD0L systems includes strictly the class of D0L growth functions. More-over, the former class contains functions that are not ZZ-rational.Our intention in the sequel is to give an explicit formula for the abovegrowth function. To this aim, denote by in the (n+1)-st "exponential jump-ing" position that occurs in the values of f; for arbitrary n � 0. Moreprecisely, in is such that(1) f(in � 1) = 1 + 2 � 3n,(2) f(in) = 1 + 3n + 3n+1:Particular values are i0 = 1; i1 = 4; i2 = 11:The value of the growth function f with respect to the argument in isf(in + k) = 1 + 3n + 3n+1 + k; for 0 � k � 2 � 3n � 1f(in + 2� 3n) = 1 + 2 � 3n+1(the upper limit for k was obtained as (1+ 2� 3n+1 � 1)� (1 + 3n +3n+1) =2� 3n � 1:)From here we deduce the recurrent relation for the sequence of numbersin; n � 0; i0 = 1;in+1 = in + 2� 3n + 1; for any n � 0:The solution of this recurrence is in = i0 + 2 n�1Xk=0 3k + n; i.e. in = 3n + n forany n � 0:The growth function f is therefore given byf(i) = 8>>>>>><>>>>>>: 3; if i = 0;1 + 3n + 3n+1 + k; if i = 3n + n+ k;for any n � 0; 0 � k � 2 � 3n � 1,1 + 2� 3n+1; if i = 3n + n+ 2� 3n,for any n � 0:9

Just as in case of D0L systems, rewriting being deterministic gives riseto certain periodicities in the Watson-Crick D0L sequence.Suppose that a word occurs twice in the sequence of a Watson-CrickD0L system, that is, wi = wi+j for some i; j � 0; j 6= 0: Let w0 = f(wi): Dueto the determinism, then also f(wi+j) = w0: Therefore, if �(w0) = 0; thenwi+1 = w0; wi+j+1 = w0; whereas if �(w0) = 1; then wi+1 = w0; wi+j+1 = w0:Thus wi = wi+j implies wi+1 = wi+j+1; whence the words start repeatingperiodically in the sequence. It then follows that the language generated bysuch a Watson-Crick D0L system is �nite. Moreover, the converse of thisproperty is true as well, i.e., the language generated by a Watson-Crick D0Lsystem is �nite if and only if a repetition occurs in the sequence of the system.However, due to the replacing of a string by its complementary in thegenerating process, the periodicity features here are not as strong as for usualdeterministic Lindenmayer systems, where we have that the partial alphabets�i = alph(wi) (the smallest set of letters � such that wi 2 ��), i � 0; forman ultimately periodic sequence, and also pre�xes and su�xes of any chosenlength form an ultimately periodic sequence. As far as Watson-Crick D0Lsystems are concerned, this is not the case anymore. As a proof observethat in the above example, we have that the alphabets �in�1 = fa1; a2; a3g,whereas for any positive integer k 6= in � 1; �k = fa1; a2; a3g (the notationin is with the same meaning as above, i.e., in = 3n+n, for any n � 0.) Moreprecisely, �i = �j for any i; j of the form i = 3n+n�1; j = 3m+m�1;wheren;m � 0 arbitrary, and �i = �j for any i; j > 0 such that i 6= 3n + n � 1;j 6= 3m+m�1; no matter which are n;m � 0. Then clearly �i; i � 0; do notform an ultimately periodic sequence. Since either �i = �j or �i \ �j = ;;for any i; j � 0; the non-periodicity of the alphabets �i implies the non-periodicity of pre�xes and su�xes of any chosen length.4 Plain Watson-Crick D0L SystemsWe already came to the conclusion that selector-based systems constitute avery natural special case of Watson-Crick D0L systems. Finally, we considera really basic variant of selector-based systems: the alphabet is the DNAalphabet itself.A selector-based Watson-Crick D0L system is referred to as plain if itsalphabet is the DNA alphabet fa; g; t; cg; and the selector is the set of purinesfa; gg:That we use purines as selectors is of course no loss of generality with re-spect to any important formal properties. On the other hand, plain Watson-Crick D0L systems can no longer simulate arbitrary D0L systems, for in-10

stance, with respect to growth properties. Yet they can be surprisingly com-plex, as seen in the following example.Consider the plain Watson-Crick D0L systemW with the axiom atg andproductions a! at; t! t; g ! g; c! c10:According to our choice of the selector set (purines), the complementaritytransition has to take place whenever t and c would have the majority ina word. If we were dealing with an ordinary D0L system, the generatedsequence would be atig; i � 1; and the letter c would not be reachable.However, for our Watson-Crick D0L system W , the sequence S(W ) is muchmore complicated. The �rst few words in the sequence are listed below. Asbefore, bold characters indicate that complementarity transition has takenplace. Recall that (a; t) and (c; g) are the complementary pairs:atg; at2g; ta3c; a(ta)3g10; at(tat)3g10; at2(tat2)3g10;ta3(ata3)3c10; a(ta)3(ta2(ta)3)3g100:In the example in the preceding section, where the alphabet was bigger,we could a�ect the alternation between linear and exponential growth ina more conspicuous fashion. Here the construction is more complicated.Whenever a "bad" letter starts to be overwhelming, the complementaritytransition changes it to a "good" letter. This may cause a loss in the othercomplementary pair which, however, is smaller than the gain.We are basically interested only in the growth function of the system Wand, hence, the order of letters in a word is immaterial. We now continuethe sequence S(W ), indicating only the number of occurrences of each letter.According to this convention, the last word obtained above is written in theform a19t15g100; and the sequence continues:a19t15g100; a19t34g100; a19t53g100; a19t72g100;a19t91g100; a19t110g100; a129t19c100; a148t129g1000; : : : ;after which the growth is again linear before the next complementarity tran-sition.This example is di�erent from the one in the preceding section. Here alsothe linear growth becomes faster, along with the exponential growth in c.Although the word sequence is very complicated from the point of view of Lsystems, the growth function of W is still probably a D0L growth function.However, we do not enter here any discussion about the growth functionsand other properties of plain Watson-Crick D0L systems.11

Let us have a look at one more example. Let us consider the plain Watson-Crick D0L system W with the axiom ac and productionsa! at; g ! g; t! t; c! c3:Again, we use purines (a and g) as selectors. The word sequence of W is (westick with the convention of using bold characters whenever a complemen-tarity transition occurs in the generated sequence):ac; tcg3; ag3c3; tc4g9; ag12c9; tc13g27; : : :Observe that at each rewriting step a complementarity transition takes placein the fragment of W 's sequence pointed out above! We can actually provethat this happens for any rewriting step, in the (in�nite) word sequence ofW .Let w = tcigj be an arbitrary word satisfying i+1 < j � 3i: Then f(w) =tc3igj ; and since jf(w)jfa;gg = j < 3i + 1 = jf(w)jfc;tg; a complementaritytransition has to occur when rewriting w according to the productions of W .That is, if w is a word in the sequence of W , then the word following w willbe w0 = ag3icj: Now f(w0) = ag3i+1c3j, with jf(w0)jfa;gg = 3i + 2 < 3j =jf(w0)jfc;tg, whence the word following w0 in a word sequence of W has to bethe complementary of f(w0), i.e. w00 = tc3i+1g3j: Denoting k = 3i+1; l = 3j;then w00 = tckgl, and, moreover, k and l satisfy the same conditions as iand j satisfy in w; that is, k + 1 < l � 3k: This fact, together with theobservation that the second word generated in the sequence of W is of theform tcigj, for i = 1; j = 3; and with i+ 1 < j � 3i; proves our claim thata complementarity transition takes place at any rewriting step in generatingthe word sequence of W .References[1] L. M. Adleman, Molecular computation of solutions to combinatorialproblems. Science, 266 (1994), 1021-1024[2] J. Engelfriet, Reverse twin shu�es. Bulletin of EATCS 60 (1996), 144[3] J. Engelfriet, G. Rozenberg, Fixed point languages, equality languagesand representation of recursively enumerable languages. J. Assoc. Com-put. Mach. 27 (1980), 499-518[4] J. Hartmanis, On Weight of Computations. EATCS Bulletin 55 (1995),136-138 12

[5] L. Kari, DNA computing: the arrival of biological mathematics. Math-ematical Intelligencer, to appear[6] V. Mihalache, Prolog Approach to DNA Computing. Proceedings IEEEICEC'97, Indianapolis, 1997, 249-254[7] G. P�aun, A. Salomaa, DNA computing based on the splicing operation.Mathematica Japonica 43 (1996), 607-632[8] G. Rozenberg, A. Salomaa, The Mathematical Theory of L Systems,Academic Press, New York, London, 1980[9] G. Rozenberg, A. Salomaa, Watson-Crick complementarity, universalcomputations and genetic engineering. Leiden University, Computer Sci-ence Technical Report 96-28 (1996)[10] G. Rozenberg, A. Salomaa (ed.), Handbook of Formal Languages, Vol.1-3. Springer-Verlag, Berlin, Heidelberg, New York (1997)[11] A. Salomaa, Jewels of Formal Language Theory. Computer SciencePress, Rockville, Md. (1981)

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Turku Centre for Computer ScienceLemmink�aisenkatu 14FIN-20520 TurkuFinlandhttp://www.tucs.abo.�University of Turku� Department of Mathematical Sciences�Abo Akademi University� Department of Computer Science� Institute for Advanced Management Systems ResearchTurku School of Economics and Business Administration� Institute of Information Systems Science