Applied Mathematics, 13, 4, 6-34 http://dxdoiorg/1436/am1341a15 Published Online October 13 (http://scirporg/journal/am) Equivalence o Subclasses o To-Way Non-Deterministic Watson Crick Automata Kumar Sankar Ray, Kingshuk Chatterjee, Debayan Ganguly Electronics and Communication Science Unit, ISI, Kolkata, India Email: ksray@isicalacin, kingshukchaterjee@gmailcom, debayan3737@gmailcom Received June 5, 13; revised July 5, 13; accepted July 1, 13 Copyright 13 Kumar Sankar Ray et al This is an open access article distributed under the Creative Commons Attribution License, hich permits unrestricted use, distribution, and reproduction in any medium, provided the original ork is properly cited ABSTRACT Watson Crick automata are inite automata orking on double strands Extensive research ork has already been done on non deterministic Watson Crick automata and on deterministic Watson Crick automata Parallel Communicating Watson Crick automata systems have been introduced by E Czeziler et al In this paper e discuss about a variant o Watson Crick automata knon as the to-ay Watson Crick automata hich are more poerul than non-deterministic Watson Crick automata We also establish the equivalence o dierent subclasses o to-ay Watson crick automata We urther sho that recursively enumerable (RE) languages can be realized by an image o generalized sequential machine (gsm) mapping o to-ay Watson-Crick automata Keyords: Non-Deterministic Watson Crick Automata; To-Way Non-Deterministic Watson Crick Automata; RE Languages 1 Introduction The tremendous progress in biotechnology has resulted in decoding o DNA sequences, synthesizing and manipulating DNA, hich lead to its usage in computation by computer scientists As a result sticker systems, splicing systems and carving systems came into existence [1] Many o the NP-complete problems ere solved eiciently using DNA computing The irst, the Adleman experiment as done in 1994 [] As the interest in using DNA in computation increased so did the need or automata hich exploit the properties o DNA The irst such automata hich exploited the DNA property ere the Watson-Crick automata [3] hich are the automata counterpart o the Sticker Systems Essentially Watson-Crick automata are inite automata having to independent heads orking on double strands here the characters on the corresponding positions o the to strands are connected by a complementarity relation similar to the Watson-Crick complementarity relation The movement o the heads although independent o each other is controlled by a single state Details o several variants o non-deterministic Watson-Crick automata have been explored in [4] Deterministic Watson-Crick automata and their variants have been explicitly handled in [5,6] Parallel Communicating Watson-Crick automata ere introduced in [7] and urther investigated in [8] A survey o Watson- Crick automata can be ound in [9] The eect o the complementarity relation on the computing poer o Watson Crick automata is discussed in [1] To-ay inite automaton (FA) is an abstract machine, a generalized version o the inite automaton hich can revisit characters already processed As in FA, in toay FA there are inite number o states ith transitions beteen them based on the current character; but each transition is also labeled ith a value indicating hether the machine ill move its reading head to the let, right, or stay at the same position Equivalently, FAs can be seen as read-only Turing machines ith no ork tape; only a read-only input tape The accepting condition is that hen the reading head alls o the right end o the tape and the state in hich the machine is at that time is inal state then the input ord is accepted A toay Watson Crick automaton (AWK) is similar in concept to a to-ay inite automaton The only dierence beteen them is that in to-ay Watson Crick automata the input tape is double stranded The idea o to-ay Watson Crick automata ere introduced in [4] but no comparison o its poer ith respect to AWK as discussed The importance o AWK is that unlike to-ay FA hich is equal in poer to a FA, AWK are more poerul than AWK hich e establish in this paper
K S RAY ET AL 7 In this paper, e give a general description o non-deterministic Watson Crick automata and its dierent subclasses in Sections and 3 In Section 4 e describe the tin shule language and state the relation o tin shule language ith RE languages In the olloing section e state the rules governing to-ay non-deterministic Watson Crick automata In Section 6 e give the deinition o the dierent classes (variants) o AWK and investigate the relationship beteen classes o AWK automata We sho that AWK = SWK = 1WK = FWK = FSWK = F1WK similar to the case o Watson Crick automata We urther sho the amily o languages accepted by AWK is context sensitive In Section 7 e sho that to-ay non-deterministic Watson Crick automata are more poerul than non-deterministic Watson Crick automata In Section 9 e urther sho that recursively enumerable (RE) languages can be realized by an image o generalized sequential machine (gsm) mapping o to-ay Watson-Crick automata Basic Terminology or Watson-Crick Automata V is a inite alphabet V denotes the set o all inite ords over V, including the empty ord V V For V, denotes the length o Let uv and vv be to ords and i there is some ord x V, such that v ux, then u is the preix o v, denoted by u v To ords, u and v are preix comparable denoted by u~ p v i u is a preix o v or vice versa Given to alphabets V and U a mapping h: V U, extended to s: V U by h and h x1, x h x1 h x or x1, x V, is called a morphism I ha, or each av, then h is a λ ree morphism A morphism h: V U is called a coding i hau or each av and a eak coding i ha U or each a V I h: V1 V V1 is the morphism deined by ha a or a V1, and ha otherise, then e say that h is a projection (associated ith V1 ) and e denote it by pr V1 For x, y V e deine their shule by xшy x y x y x x x, 1 1 n n 1 n y y y, x, y V 1 i n, n 1 1 n i i, A generalized sequential machine (gsm) is a sequential transducer Such a device is a system g Q, V1, V, q here Q is the set o states, V1, V are the alphabets(input and output alphabets) o the automaton, q Q is the initial state, F Q is the set o inal states and : QV1 P V Q is the transition mapping The deinitions o morphism, gsm and shule are stat- ed in [1] A Watson-Crick automaton is a 6-tuple o the orm M V,, Q, q here V is an alphabet set, Q is a set o states, V V is the complementarity relation similar to Watson Crick complementarity relation and q is the initial state and F Q is the set o inal states The unction contains a inite number o tran- 1 sition rules o the orm q q, hich denotes that the machine in state q parses 1 in upper strand and in loer strand and goes to state q here 1 1 1 1, V is dierent rom is just a pair o strings ritten in that orm instead o 1 1, hereas in the to strands are o same length ie 1 and the corresponding symbols in to strands are complementarity in the sense o relation ρ V a ab, V, a, b V b and WK V V V A transition in a Watson-Crick inite automaton can be deined as ollos: x1u11 V For, such that xu V xu 1 1 1 WK V xu and q, qq, x1 u1 1 x1 u1 1 q q i there is x u xu 1 transition rule q u q in u denotes the transitive and relexive closure o The language accepted by Watson-Crick Automata is 1 L M 1V q q, 1 ith q F, V, WK V For a Watson-Crick Automaton M, ith input 1 WK V here 1 is any string in V and 1 1 q q here q is the initial state and q is
8 K S RAY ET AL a inal state Then q q is a computation 1 1 in M denoted by Another important language associated ith Watson- Crick automaton is deined taking into consideration the transitions and not the language recognised For a Watson-Crick Automaton M V,, Q, q consider a labeling e: Lab, o rules in ith elements in a set Lab For computation : q q, WK V, q F, denoted by e the control ord o, that is the sequence o labels o transition rules used in In this ay the language is obtained L M e : q q, WK V, q F ctr The deinition o L M is stated in [4] ctr 3 Subclasses o Non-Deterministic Watson-Crick Automata (AWK) Depending on the type o states and transition rules there are our types or subclasses o Watson-Crick Automata Watson-Crick Automaton M V,,Q, q is 1) stateless (NWK): I it has only one state, ie Q F q ; ) all-inal (FWK): I all the states are inal, ie Q F; 3) simple (SWK): I at each step the automaton reads either rom the upper strand or rom the loer strand, ie or any transition rule 1 q q, either 1 or ; 4) 1-limlited (1WK): I or any transition rule q 1 q q, e have 1 1 Theorem 1: Simple and 1 limited Watson-Crick automata accept the same amily o languages as the amily o languages accepted by Watson-Crick automata ith arbitrary transition rules The proo o Theorem 1 is in [4] Theorem : Non-deterministic Watson-Crick automata are equivalent ith non-deterministic simple Watson-Crick automata Corollary 1: Non-deterministic Watson-Crick automata are equivalent ith non-deterministic 1-limited Watson-Crick automata The proo o Theorem and Corollary 1 are given in [4] 4 Tin-Shule Language Consider an alphabet V and its barred variant, V a a V The language TSV xv xшx is called the tin-shule language over V (For a string x V, x denotes the string obtained by replacing each symbol in x ith its barred variant) For the morphism h: V V V deined by ha, or a V and h a, or a V Clearly the equality is TSV EQh, prv Theorem 3: Each recursively enumerable language L T can be ritten in the orm L pr T TSV R, here V is an alphabet and R is a regular language In this representation, the language TS V depends on the language L This can be avoided in the olloing ay: Let a coding be : V,1, or instance, 1 i th ai, here ai is the i symbol o V in a speciied ordering The language R is regular A generalized sequential machine gsm can simulate the intersection ith a regular language, the projection prt as ell as the decoding o elements in TS V Thus e obtain: Corollary : For each recursively enumerable language L there is a gsm g L such that L g TS L, 1 Thereore, by using a sequential transducer hich can be a deterministic one, e can obtain all recursively enumerable language, starting rom the unique language TS Proos o Theorem 3 and Corollary are in [1],1 5 To-Way Non-Deterministic Watson Crick Automata (AWK) To-ay non-deterministic Watson Crick automata system is a 6 tuple, M V #,$,, Q, q here V is a set o alphabet, #,$ V are the beginning and the end marker respectively; that is the ord to be recognized is provided as an input to the automaton in the orm # $ Q is a set o states, 1 V V and #,#, $,$ and 1 is the complementarity relation and q is the initial state and F Q is the set o inal states is the inite number o transition rules; 1, 1) either o the orm q q, hich denotes, dir that the machine in state q parses 1 in upper strand in dir 1 direction and in loer strand in dir direction and goes to state q here 1, V,, d ir L, R, here L signiies that the head is reading the ord in the let direction signiies that the head is reading the ord in right direction and i a head reads the empty ord it remains in its
K S RAY ET AL 9 current position denoted by x1, ) or o the orms q q, here x, dir x1, x #, and, dir L, R, ith restrictions that hen x1 or x the corresponding or dir L and hen x 1 or x the corresponding dir or dir 1 Moreover, there cannot be transi- 1, tion rules having the orm q q here, dir 1 1 # here 1 V and L or # here V and dir L or both These rules ensure that the reading heads do not go past the input ord on the let side or the heads do not move hen it reads empty ord Moreover once a head goes past the right end o the tape it cannot comeback Accepting conditions W 1 is accepted by M i, starting in state q (initial # 1 $ 1 state) ith and on the dou- # $ WK V ble stranded input tape and the to heads at the let end o # 1 $ and # $ respectively, M eventually enters a inal state at the same time both the heads all o the right hand side o the double stranded input tape The ord 1 is rejected i one o the olloing 3 conditions occurs: 1) The to-ay WK automaton goes into a loop hich is identiied in a similar ay as loops in to-ay FAs are identiied ) When both the heads all o the right hand side o the input tape and the machine is in a non inal state 3) I the machine comes to a halt (ie there are no transition rules that can be applied or that particular state in hich the machine is) beore the heads all o the right hand side o the input tape ie mathematically # 1 $ # $ LM 1 V q q, ith q F, V, 1 WK V 6 Subclasses o To-Way Non-Deterministic Watson-Crick Automata (AWK) Depending on the type o states and transition rules there are our types or subclasses o to-ay Watson-Crick Automata similar to Watson Crick automata -ay Watson-Crick Automaton M V,, Q, q is 1) stateless (NWK): I it has only one state, ie Q F q ; ) all-inal (FWK): I all the states are inal, ie Q F; 3) simple (SWK): I at each step the automaton reads either rom the upper strand or rom the loer strand, ie 1, or any transition rule q q either, dir 1 or ; 4) 1-limlited (1WK): I or any transition rule 1, q q, e have 1 1, dir Many combinations o these classes can also be obtained such as all-inal simple to-ay WK automata (FSWK), all inal 1 limited to-ay WK automata (1FWK), stateless 1 limited to-ay WK automata (1NWK) etc Theorem 4: Simple and 1 limited to-ay Watson Crick automata accept the same amily o languages as the amily o languages accepted by to-ay Watson Crick automata ith arbitrary transition rules The proo o theorem is similar to the proo done in [4] or Theorem 1 Let M V,, Q, q be a non-deterministic to-ay Watson Crick automaton We introduce a 1 limited to-ay Watson Crick automaton M ' V,, Q, q, F, For each transition rule t o the orm 1, q q in here 1 aa 1 a n, dir here 1 n and b 1 b b m here m and n m, the condition nm is imposed because rules ith m n is already in the 1-limited orm and no urther modiication is required or them We introduce ne rules in o the orm a1, q qt,1, ai 1, qti, qt,i 1, 1 i n1,,, qt,n q t,1 b1, dir, q, j t, j 1, 1 t q bj 1, dir j m,, q t, m1 q bm, dir All the ne states are introduced in Q along ith states in Q From the construction o M hich is obtained rom M it is obvious that both M and M recognize the same language So AWK are subset o 1WK and rom the deinition o 1WK and AWK e kno
3 K S RAY ET AL that 1WK are subset o AWK So AWK and 1WK are equivalent ie they accept the same amily o languages A similar proo can also be established or SWK Thereore e can say, AWK = SWK = 1WK Theorem 5: All inal to-ay Watson Crick automata accept the same amily o languages as the amily o languages accepted by to-ay Watson Crick automata ith arbitrary transition rules Let M V,, Q, q be a to-ay non-deterministic Watson Crick automaton We introduce an all inal to-ay Watson Crick automaton M V,, Q, q, Each transition rule t o the orm 1, q q, dir in δ here aa a 1 1 n here 1 n and bb 1 bm here m alls under one o the ive classes The classes are deined as ollos: Class 1: Transition rules o the orm 1, q q in here 1 aa 1 an here, dir 1 n and bb 1 b m here m and a and b $, ie 1 and do not have $ at their ends n n 1 Class : Transition rules o the orm q q in here 1 aa 1 an here 1 n and bb 1 b m here m, and an, bn $, ie 1 and both have $ at their ends Class 3: Transition rules o the orm 1 q q in here 1 aa 1 an here, dir 1 n and bb 1 bm here m, a n $ and bn $ ie 1 has $ at its end and does not have $ at its end 1, Class 4: Transition rules o the orm q q in here 1 aa 1 an here 1 n and bb 1 bm here m, a n $ and bn $ ie 1 does not have $ at its end and has $ at its end Class 5: Either transition rules o the orm $, L q q in or transition rules o the orm, $, L q q in or transition rules o the orm $, L, q q in $, L The transition rules o M are modiied as ollos to orm the transition rules o M Transition rules o M hich all in class 1 and class 5 are kept same in M For transition ru les o M hich belong to class to instances can occur; 1 case 1: For transition q q ', here q is a inal state In this case the transition rules are kept same in M 1 case : For transition q q, here q is a non inal state In this case the transition rules o M are modiied as ollos or M 1 For each transition ru le q q in M be- longing to class here q is a non inal state, 1 q q here and $ $ are 1 1 introduced in M and there is no transition rom q in M These ne rules in M ensure that i the heads go o the right end o the tape i n M hen M is in a non inal state then M ould go to state q and ould not accept the string as there is no transition rom q ie the above stated rules ensure the heads do not all o the right end o the tape or M hen M does not accept the ord As M is all inal i the heads go o the right end o the tape it ill accept the given string For transition rules o M hich belong to class 3 the olloing modiications are needed Class 3 also has to instances similar to class 1 case 1: For transition q q, here q is, dir a inal state In this case the transition rules are kept same in M 1 c ase : For transition q q, here q is, dir a non inal state In this case the transition rules o M are modiied as ollos or M For each transition rule 1 q q in M be-, dir longing to class 3 here q is a non inal state, 1 q q here 1 1 $ is introduced in, dir M here q denotes that the head on the upper strand has gone past the right end marker $ in the original machine M on application o the above transition rule Only rules having λ on the upper strand are applied to q because in the actual machine M i the above rules o class 3 are applied then the upper head ould have gone past the right end o the tape So only rules
K S RAY ET AL 31 having λ on the upper head can be applied to the machine M As M replicates M similar thing is done in M too Thus, all the transition rules that can be applied to q in M ith on the upper strand and aa 1 a n and an $ in the loer strand can also be applied to q in M Rules having on the upper strand and a1a a n and an $ the loer strand here e tran in th sition goes to a inal state are applied to q u $ Finally or rules ith on the upper strand and aa 1 an and an $ in the loer strand here the transition goes to a non inal state, the rules o the, orm q l here qu $ are intro-, R $ duced in M and th ere are no transition rules rom q ul$ These rules ensure that hen M reaches the end o the string on a non inal state then M goes to q ul$ and M does not accept the string as there is no tra nsirom q ul$ ie the above stated rules ensure the tion heads do not all o the right end o the tape or M hen heads o M all o the right end and the state to hich M goes is non inal Class 4 rules are handled in a similar ay to class 3 rules It is obvious rom the transition rules introduced in M that M accepts the same amily o languages as M Thus, FWK = AWK Theorem 6: All inal 1 limited to-ay Watson Crick automata accept the same amily o languages as the amily o languages accepted by 1 limited toay Watson Crick automata ith arbitrary transition rules Let M V,, Q, q, F, be a to-ay 1 limited non-deterministic Watson Crick automaton We introduce an all inal 1 limited to-ay Watson Crick automaton M V,, Q, q, Each transition rule t o 1, the orm q q in here 1 1, dir alls under one o the our classes The classes are deined as ollos: Class 1: Transition rules o the orm 1, q q in here 1 $ and $, dir $, R Class : Transition rules o the orm q q in,, Class 3: Transition rules o the orm q q in $, R $, R Class 4: Transition rules o the orm q q in,, or transition rules o the orm q q in $, R The transition rules o M are modiied as ollos to orm the transition rules o M Transition rules o M hich all in class 1 and class 4 are kept same in M For transition ru les o M hich belong to class have to instances $, R case 1: For transition q q, here q is a, inal state In this case the transition rules are kept same in M $, R case : For transition q q, here q is a, non inal state In this case the transition rules o M are modiied as ollos or M $, R For each transition rule q q in M be-, longing to class here q is a non inal state, q $, L xr, q b and q bul$ q here x V,, are introduced in M q denotes that the head on the upper strand has gone past the right end marker $ in the original machine M Only rules having on the upper strand can be applied to q (or reasons similar to reasons stated in proo o Theorem 5) Thus, all the transition rules that can be applied to q ith on the upper strand and $ in the loer strand ar e applied to q For rules having on the upper strand and $ in the loer strand here the transition goes to a inal state are applied to q u $ Finally or rules ith on the upper strand and $ in the loer strand here the transition goes to a non inal state in M, the rules o the orm,, q qb ul$ and q bul$ qul$, here $, L xr, x V are introduced in M These rules ensure that hen M reaches the end o the string on a non inal state, M does not accept the string as there are no transit ions rom state q ul $ Class 3 is handled in a similar ay as class It is obvious rom the transition rules introduced in M that M accepts the same amily o languages as M Thus, 1FWK = 1WK Corollary 3: All inal 1 limited to-ay Watson Crick automata accept the same amily o languages as the amily o languages accepted by arbitrary to-
3 K S RAY ET AL ay Watson Crick automata ith arbitrary transition rules Proo: From Theorem 4 e kno AWK = 1WK and rom Theorem 6 e obtain 1FWK = 1WK Thus combining both the results e get 1FWK = AWK Thus rom the above Theorems e can state that AWK = 1FWK = 1WK = SWK = FSWK = FWK 7 Poer o To-Way Non-Deterministic WK Automata In this section e irst sho that AWK are subset o AWK Then e urther sho that this subset relation is proper ie AWK are more poerul than AWK Theorem 7: AWK AWK The theorem says that non-deterministic Watson Crick automata are subset o to-ay non-deterministic Watson Crick automata Proo: Let M V,, Q, q be a non-deterministic Watson Crick automaton here V is a set o alphabet, Q is a set o states, V V is the complementarity relation and q is the initial state and F Q is the set o inal states is the inite number o transition rules 1 o the orm q q, here 1, V We introduce a to-ay non-deterministic Watson Crick automaton M V #,$,, Q, q, q, here V is a set o alphabet, #,$ V are the beginning and t he end marker respectively, that is, the ord to be recognized is provided as an input to the automaton in the orm # $ QQq, q, #,#, $,$ is the co mplementarity relation and q is the initial state and q is a inal state is the inite number o transition rules o the orm 1 1) For each rule q q in introduce 1 q q in # ) q q # 3) For each state x F in M introduce $ x q in $, R From the construction o M it is evident that all that ill be accepted by M ill be accepted by M Theorem 8: One-Way To headed inite automata are equivalent to AWK An inormal proo o this theorem is in [4] Example 1 Let #,$,, Q, q be a AWK M V here V a, b,#,$ V are the beginning and the end marker respectively, that is, the ord to be recognized is provided as an input to the automaton in the orm # $ Q is a set o states, Q q, q1, q, q3, q4, is the identity complementarity relatio n and q is the initial state and F q 4 is the set o inal states is the inite number o transition rules In this example the mirror language L { a, b and R L a, b and here R denotes the reverse o is accepted using to- ay Watson Crick automaton The transition rules o M are as ollo #, R q q1, # xr, q1 q1, xv,, $, L q1 q,, xl, q, q xv, xr, #, R q q3, $, L xr, q3 q3, xv,, $, R q3 q4 $, R Theorem 9: One-ay inite automata ith heads cannot accept the mirror language The above theorem is stated in [11] Theorem 1: AWK are more poerul than AWK ie AWK AWK Proo Fr om Theorem 8 e kno that AWK is equivalent to 1-ay to headed inite automata and rom Theorem 9 e kno that 1-ay to headed inite automata cannot recognize the mirror language Thus AWK cannot recognize the mirror language But in Example 1 e have shon that to-ay AWK can accept the mirror language and in theorem 7 e have shon that AWK AWK ie AWK accepts all the amily o languag es hich are accepted by AWK Moreover it also accepts the mirror language hich AWK cannot accept Thus AWK accepts at least one language more than AWK Hence e conclude that the accepting poer o to-ay AWK is more than AWK Mathematically AWK AWK, ie the subset relation is proper Theorem 11: Family o languages accepted by WK automata is context sensitive A linear bounded Turing machine (LBA) can simulate the actions o to-ay Watson Crick automaton As the language accepted by LBA is context sensitive so the
K S RAY ET AL 33 amily o languages accepted by to-ay Watson Crick automaton is also context sensitive 8 Characterization o Recursively Enumerable (RE) Languages in Terms o AWK Automata In this section e discuss AWK in th e light o the RE languages We sho each language in the amily o RE is the image o a gsm mapping o a language in AWK Theorem 1: TS V AWK(ctrl) The proo o this theorem is in [4] Theorem 13: For each recursively enumerable lan- guage L there is a gsm g L such that L = g L (AWK (ctrl)) Proo: We have already shon in Theorem 7 AWK AWK and it is stated in [4] that TS V AWK(ctrl) an d rom corollary e kno that each language in the amily o RE is the image o a gsm mapping o a language in TS {,1} As TS V AWK(ctrl) and AWK AWK, so e can state, each language in the amily o RE is the image o a gsm mapping o a language in AWK(ctrl) 9 Conclusion In this paper, e discuss about the poer o a variant o non-deterministic Watson Crick automata knon as AWK We describe their structure and accepting conditions We introduce dierent subclasses o AWK similar to AWK and sho the equivalence o some o those subclasses We urther establish the act that AWK are more poerul than AWK Based on the relation beteen AWK and AWK e sho that a gsm mapping o AWK results in the generation o each language in the amily o the recursively enumerable languages REFERENCES [1] L C S Calude and G Paun, Compu ting ith Cells and Atoms: An Introduction to Quantum, DNA and Membrane Computing, Taylor & Francis Publishers, London, 1 [] L M Adleman, Molecular Computation o Solutions to Combinatorial Problems, Science, Ne Series, Vol 6, No 5187, 1994, pp 11-14 http://dxdoiorg/1116/science7973651 [3] R Freund, G Paun, G Rozenberg and A Saloma, A, Watson-Crick Finite Automata, Proceedings o the 3rd DIMACS Workshop on DNA Based Computers, Philadelphia, 1997, pp 97-38 [4] G Paun, G Rozenberg and A Salomaa, DNA Computing: Ne Computing Paradigms, Springer-Verlag, Berlin, 1998 [5] E Czeizler, E Czeizler, L Kari and K Salomaa, Watson- Crick Automata: Determinism and State Complexity, Proceeding o: 1th International Workshop on Descriptional Complexity o Formal Systems, DCFS, 16-18 July 8, pp 11-133 [6] E Czeizler, E Czeizler, L Kari and K Salomaa, On the Descriptional Complexity o Watson-Crick Automata, Theoretical Computer Science, Vol 41, No 35, 9, pp 35-36 http://dxdoiorg/1116/jtcs951 [7] E Czeizler, E Czeizler, Parallel Communicating Watthe Poer o Parallel Communicating son-crick Automata Systems, In: Z Fulop Z Esik (Ed), Proceedings o 11th International Conerence, AFL 5, 5, pp 83-96 [8] E Czeizler, On Watson-Crick Automata Systems, Theoretical Computer Science, Vol 358, No 1, 6, pp 14-147 [9] E Czeizler, A Short Survey on Watson-Crick Automata, Bulletin o the EATCS, Vol 88, 6, pp 14-119 [1] D Kuske and P Weigel, The Role o the Complementa- rity Relation in Watson-Crick Automata and Sticker Systems, Developments in Language Theory, Vol 334, Lecture Notes in Computer Science, Springer, Berlin, 4, pp 7-83 [11] M Holzer, M Kutrib and A Malcher, Multi-Head Finite Automata: Characterizations, Concepts and Open Problems, Proceedings International Workshop on the Complexity o Simple Programs, Cork, 6-7 December 8, pp 93-17
34 K S RAY ET AL Abbreviations AWK: non-deterministic Watson-Crick automata NWK: stateless non-deterministic Watson-Crick auto- mata FWK: all inal non-deterministic Watson-Crick automata SWK: simple non-deterministic Watson-Crick automata 1WK: 1-limited non-deterministic Watson-Crick automata AWK: to ay non-deterministic Watson-Crick automata NWK: to ay stateless non-deterministic Watson- Crick automata FWK: to ay all inal non-deterministic Watson- Crick automata SWK: to ay simple non-deterministic Watson- Crick automata 1WK: to ay 1-limited non-deterministic Watson- Crick automata TS V : tin-shule language RE: recursive enumerable gsm: generalized sequential machine