TEST TUBE SYSTEMS WITH CUTTING/RECOMBINATION OPERATIONS Rudolf FREUND Institut fur Computersprachen, Technische Universitat Wien Resselgasse 3, 1040 W

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1 TEST TUBE SYSTEMS WITH CUTTING/RECOMBINATION OPERATIONS Rudlf FREUND Istitut fur Cmputersprache, Techische Uiversitat Wie Resselgasse 3, 1040 Wie, Austria Erzsebet CSUHAJ-VARJ U Cmputer ad Autmati Istitute, Hugaria Academy f Scieces Kede u , 1111 Budapest, Hugary csuhaj@lua.aszi.sztaki.hu Fraz WACHTLER Istitut fur Histlgie ud Embrylgie, Uiversitat Wie Schwarzspaierstr. 17, 1090 Wie, Austria fraz.wachtler@uivie.ac.at We itrduce test tube systems 1;2;3;9 based peratis that are clsely related t the splicig perati 8, i.e. we csider the peratis f cuttig a strig at a specic site it tw pieces with markig them at the cut eds ad f recmbiig tw strigs with specicly marked edigs 7. Whereas i the splicig f tw strigs these strigs are cut at specic sites ad the cut pieces are recmbied immediately i a crsswise way, i CR(cuttig/recmbiati)-schemes cuttig ca happe idepedetly frm recmbiig the cut pieces. Test tube systems based these peratis f cuttig ad recmbiatitur ut t have maximal geerative pwer eve if ly very restricted types f iput lters fr the test tubes are used fr the redistributi f the ctets f the test tubes after a perid f cuttigs ad recmbiatis i the test tubes. 1 Itrducti Test tube systems were itrduced as bilgical cmputer systems based DNA mlecules 1;2;3;9, the practical sluti f NP prblems with such systems was described, ad the theretical features f test tube systems based the splicig perati were ivestigated 4. I this paper we are gig t explre test tube systems based the peratis f cuttig ad recmbiati 7 ad with iput lters which ly allw specic parts f the ctets f the ther test tubes t pass (such lters, fr example, are cceivable fr DNA mlecules by cmbiig the priciples f aity chrmatgraphy ad i-situ-hybridizati). As we shall shw eve very restricted kids f such lters testig fr the existece respectively -existece f specic markigs allw fr reachig the geerative pwer t geerate ay recursively eumerable laguage.

2 The cmputatial uiversality f specic variats f H systems 5;10 ad fr test tube systems based the splicig perati 4 has bee prved recetly. I this paper we shall shw that uiversal test tube systems based cuttig ad recmbiati rules exist fr dieret variats f iput lters. I the secd secti f this paper we dee the tis frm frmal laguage thery eeded i this paper ad itrduce the frmal deitis f cuttig/recmbiati schemes (CR-schemes). I the third secti f this paper we itrduce test tube systems with cuttig/recmbiati rules (CRTTS) ad we prve that CRTTS ca geerate every recursively eumerable laguage. This result als implies the existece f uiversal CRTTS. A shrt summary f the results btaied i this paper ad a verview f future research tpics cclude the paper. 2 Deitis ad Examples I this secti we ly dee sme tis frm frmal laguage thery that we shall eed i this paper. Mrever we recall the deitis fr CR-schemes 7 ad give sme explaatry examples. The free mid geerated by the alphabet V is deted by V, its elemets are called strigs r wrds ver V ; is the empty strig, V + = V fg. The legth f a strig w i V is writte as jwj : A grammar scheme is a triple (V N ;V T ;P); where V N is a (ite) set f symbls, i.e. the alphabet f -termial symbls; V T is a (ite) set f symbls with V N \ V T = ;; i.e. the alphabet f termial symbls; P is a (ite) set f prductis f the frm (; ) ; where 2 (V N [ V T ) + ad 2 (V N [ V T ) : Fr tw wrds x; y 2 (V N [ V T ) ; the derivati relati ` is deed if ad ly if x = uv ad y = uv fr sme prducti (; ) 2 P ad tw strigs u; v 2 (V N [ V T ) ;we the als write x ` y: The reexive ad trasitive clsure f the relati ` is deted by ` : A grammar G is a quadruple (V N ;V T ;P;S); where = (V N ;V T ;P)is a grammar scheme ad S 2 V N ; i a mre geeral way, we ca als take S 2 (V N [ V T ) +. The -free laguage geerated by Gis L(G)= w2v + T js` w : (1) A subset L f V + T is called recursively eumerable if ad ly if there exists a grammar G that geerates L; i.e. L (G) =L: Mrever, L is called recursive if ad ly if bth L ad its cmplemet, V + T L; are recursively eumerable. The family f (-free) recursively eumerable laguages ad the family f (-free) recursive laguages ver the alphabet V T shall be deted by ENUM (V T ) ad REC (V T ) ; respectively.

3 A grammar scheme U with U =(V N ;V T ;P) is called uiversal (fr V T ) if fr every L 2 ENUM (V T ) there exists a wrd A L such that the grammar G L with G L =(V N ;V T ;P;A L ) geerates L: Oe f the imprtat results f frmal laguage thery is that fr every V T such a uiversal grammar U exists. Deiti1. A cuttig/recmbiati scheme (r a CR-scheme) is a quadruple =(V; M; C; R); where V is a ite alphabet; M is a ite set f markigs; V ad M are disjit sets; C is a set f cuttig rules f the frm u l$m v; where u 2 V [ MV ;v 2V [V M; ad m; l 2 M; ad ; $ are special symbls t i V [ M ; R M M is the recmbiati relati represetig the recmbiati rules. Cuttig ad recmbiati rules are applied t bjects frm O (V; M); where we dee O (V; M) =V + [MV [V M [MV M; (2) i.e., as the empty wrd has meaigful represetati i ature, is t csidered t be a bject we have t deal with. Fr x; y; z 2 O (V; M) ad a cuttig rule c = u l$m v we dee x `c (y; z) if ad ly if fr sme 2 V [ MV ad 2 V [ V M we have x=uv ad y = ul; z = mv: Fr x; y; z 2 O (V; M) ad a recmbiati rule r =(l; m) frm R we dee (x; y) `r z if ad ly if fr sme 2 V [MV ad 2 V [ V M we havex=l; y = m; ad z = : Fr a CR-scheme =(V; M; C; R) ad ay laguage L O (V; M) we write (L) = fy j x `c (y; z) r x `c (z; y) fr sme x 2 L; c 2 Cg[ fz j (x; y) `r z fr sme x; y 2 L; r 2 Rg ; (3) ad we dee (L) = S i0 i (L);where 0 (L) =Lad i+1 (L) = i (L)[ i (L) fr all i 0: 2 Thus (L) ctais all bjects btaied by applyig e cuttig r e recmbiati rule t bjects frm L; (L) is the smallest subset f O (V; M) that ctais L ad is clsed uder the cuttig ad recmbiati rules f : There is a clse relatiship betwee CR schemes ad splicig schemes (H schemes): Fr shrt, a splicig rule u 1 v 1 $u 2 v 2 beig applied t tw strigs x 1 u 1 v 1 y 1 ad x 2 u 2 v 2 y 2 yields the tw strigs x 1 u 1 v 2 y 2 ad x 2 u 2 v 1 y 1 which crrespds t cuttig the strigs x 1 u 1 v 1 y 1 ad x 2 u 2 v 2 y 2 it the strigs x 1 u 1 [m] + ; [m] v 1 y 1 ad x 2 u 2 [m] + ; [m] v 2 y 2 by usig the cuttig rules u 1 [m] + $[m] v 1 ad u 2 [m] + $[m] v 2 ad recmbiig them immediately by applyig the recmbiati rule [m] + ; [m] i a crsswise way.

4 I the fllwig we shall restrict urselves t cuttig rules f the frm u [m] + $[m] v; i.e. the markigs geerated by the cuttig rule are the psitive [m] + ad the egative [m] versi f [m] : I this case the derivati f the tw parts xu [m] + ; [m] vy frm the bject xuvy by the cuttig rule u [m] + $[m] v; i.e. the derivati xuvy `u [m] + $[m] v xu [m] + ; [m] vy i a mre depictive way ca be expressed by xu j [m] vy ` xu [m] + ; [m] vy : The fllwig example shws the chemical backgrud f the tatis itrduced abve, i.e. the markigs [m] + ad [m], respectively, crrespd t the psitive ad egative charges f is: Example 1. Csider the salt mlecule NaCl; which i water dissipates t the is Na+ ad Cl. This reacti crrespds t applyig the frmal cuttig rule Na [m] + $[m] Cl t the mlecule strig NaCl; which yields the frmal derivati step Na j [m] Cl ` Na[m] + ;[m] Cl : Obviusly the tw parts Na[m] + ;[m] Cl ca be recmbied t NaCl by the recmbiati rule [m] + ; [m] ; i.e. Na[m] + ;[m] Cl ` NaCl: 2 3 CR Test Tube Systems I this secti we itrduce test tube systems 1;2;3;9 that are based the frmal peratis f cuttig ad recmbiati rules as itrduced i the previus secti. The idea f test tube systems is t describe cmputatial devices where the cmputatis i each test tube are based specic peratis ad ay cmputati is de i a distributed way. As a cmmuicati step the resultig ctets f the test tubes the is redistributed accrdig t specic cstraits, i.e. the ctets f each test tube is distributed t all test tubes accrdig t specic iput lters agai, whereas the rest remais i the test tube. These ideas have already bee frmalized fr the splicig perati 4 ;i the fllwig we dee test tube systems where the peratis that ca take place i e test tube are cuttigs ad recmbiatis ad ivestigate the geerative pwer f these systems with the peratis f cuttigs ad recmbiatis. We shall shw that every recursively eumerable laguage ca be geerated by such a test tube system which ly eeds a very special restricted kid f iput lters. Mrever, this result als guaratees the existece f a uiversal test tube system with cuttigs ad recmbiatis. Deiti2. A test tube system with cuttigs ad recmbiatis (a CRTTS

5 fr shrt) is a quituple (V; M; A; ; ; I) ; where 1. V is a (ite) set f symbls; 2. M is a (ite) set f markigs; M ad V are disjit sets; 3. A is a (ite) set f axims, which are elemets frm O (V; M); 4. ; 1; is the umber f test tubes; 5. is a (ite) sequece ( 1 ; :::; ) ; where i =(C i ;R i ) is a ite set f test tube peratis f cuttigs ad recmbiatis, respectively, i the test tube T i ; i.e. C i is a (ite) set f cuttig rules ver (V; M) ad R i is a (ite) set f recmbiati rules ver (V; M); i =(V; M; C i ;R i )is the crrespdig CR-scheme; 6. I =(I 1 ; :::; I ) ; where I i O (V; M) is the iput lter fr the test tube T i ; 1 i : The cmputatis i the system ru as fllws: At the begiig f the cmputati the axims are distributed ver the test tubes T i accrdig t the crrespdig iput lters I i, i.e. T i starts with A \ I i : Nw let L i be the ctets f T i at the begiig f a derivati step. The i each test tube the CR-scheme i perates L i ; i.e. we btai i (L i) : The ext substep is the redistributi f i (L i)ver all test tubes accrdig t the crrespdig iput lters. Frm i (L i) ly the part i (L i) \ I j that passes the iput lter I j is distributed t the test tubes T j ; 1 j ; whereas the S rest i (L i) 1j ( i (L i) \ I j ) remais i T i : The al result f the cmputatis i csists f all strigs frm V + that ca be extracted frm the al test tube T 1 : Mre frmally, aistataeus descripti (ID fr shrt) f the system is a -tuple (L 1 ; :::; L ) with L i O (V; M); 1 i ; where L i describes the ctets f test tube T i at the begiig f a derivati step. The iitial ID is (A \ I 1 ; :::; A \ I ) ; i.e. at time t = 0 the test tubes T i ctai the axims A \ I i : Let (L 1 (t) ; :::; L (t)) dete the ID at time t; the e derivati step with the system yields the ID (L 1 (t +1); :::; L (t + 1)) ; where L i (t +1) = = S 1j j (L j(t)) \ I i [ S i (L i (t)) 1j ( i (L i (t)) \ I j ) S 1j j (L j (t)) \ I i [ i (L i (t)) i (L S i (t)) \ 1j I j : (4)

6 We als write (L 1 (t) ; :::; L (t)) ` (L 1 (t +1); :::; L (t + 1)) : The laguage geerated by ; L();is deed by L () = S 1 t=0 (L 1 (t) \ V + ) : 2 A miimal requiremet the feasability f the iput lters I i is their recursiveess, i.e. we demad that it is decidable whether a strig ca pass the lter r t. Yet i rder t btai mre iterestig results we havetput restrictis the iput lters: Deiti3. A subset f O (V; M) is called a simple (V; M) 2 -lter if it equals 1. V + r 2. fmg V fr sme m 2 M r 3. V fmg fr sme m 2 M r 4. fmg V fg fr sme m; 2 M: A simple (V; M) 2 -lter is called a simple (V; M) 1 -lter, if it is t f the frm fmg V fg : Ay ite ui f simple (V; M) i -lters, i 2f1;2g;is called a(v; M) i -lter. 2 I the fllwig example we shw hw the laguage a 2 j 1 ca be geerated by acrtts with (V; M) 1 -lters: Example 2. Let =(V; M; A; 8;;I) be the CRTTS with V = fa; B; F; X; Y g ;M= [x] + ;[x] jx2fb; c; d; e; f; l; ; r;s;tg ; A = fxaaby; XaaY; XBY; Fg ; =( 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 );I=(I 1 ;I 2 ;I 3 ;I 4 ;I 5 ;I 6 ;I 7 ;I 8 ); 1 = ;; [s] + ;[e] ; 2 = X [e] + $[e] aa; [s] + $[s] F ; [f] + ;[t] ; 3 = aa [f] + $[f] BY; F [t] + $[t] ;; ; 4 = aa [b] + $[b] BY; XB [c] + $[c] Y ;; ; 0 1 X [d] + $[d] aaa; X [d] + $[d] aay ; 5 A [b] + ; [c] ; [c] + ; ; [d] 0 1 w [r] + $[r] ay j w 2faa; ab; Bag [ 6 A; Xaa [] + $[] Y ;; 7 = X [l] + $[l] v jv 2faaa; aab; BaY g ; [r] + ; [] ;

7 8 = ;; [] + ; [l] ; I 1 = V [s] + [ [e] V ;I 2 = V [f] + [ [t] V [V [t] + ; I 3 = I 4 =I 6 = V + ;I 5 =V [b] + [V [c] + [ [c] V ; I 7 =V [r] + [ [] V ;I 8 =V [] + [ [l] V : The geerati f the wrds a 2 i this CR test tube system briey ca be described i the fllwig way: I geeral, let us assume we have already btaied the wrd Xa 2 BY fr sme 1 i test tube T 4 (rigially we start with the axim XaaBY ), where the fllwig cuttigs ca take place: Xa 2 j BY ` Xa 2 [b] + ;[b] Xa 2 [b] + ;[c] BY ad XB j Y ` [c] XB [c] + ;[c] Y ` Xa 2 Y by the recmbiati rule [b] Y : I T 5 we the btai [b] + ; [c] ; ad by cuttig with e f the rules X [d] + $[d] aaa; X [d] + $[d] aay we get X j [d] a 2 Y ` X [d] + ; [d] a 2 Y : The recmbiati rule [c] + ; [d] the yields XB [c] + ;[d] a 2 Y ` XBa 2 Y: I sum, we have rtated the symbl B frm the ed t the begiig f the blck f symbls a: The rules i the test tubes T 6 ; T 7 ad T 8 have the eect that a symbl a at the ed f the wrd t the left f the symbl Y is elimiated, yet istead tw symbls a are added at the begiig, i.e. frm Xa k Ba m Y we btai Xa k+2 Ba m 1 Y: A full cycle f rtatis therefre dubles the umber f symbls a; i.e. frm XBa 2 Y we btai Xa 2+1 BY: The test tubes T 3 ;T 2 ; ad T 1 ally allw us t btai the termial strigs a 2 : I T 3 we get Xa 2 j [f] BY ` Xa 2 [f] + ;[f] BY ad F j F [t] + ; [t] : The bjects Xa 2 [f] + ;F[t] + ;ad [t] are passed t T 2 ; [t] where we have Xa 2 [f] + ;[t] ` Xa 2 ;X j a X[e] 2` + ;[e] a 2 ;as [e] well as j F [t] + ` [s] + ; [s] F [t] + : Passig the bjects [e] a 2 ad [s] + t [s] T 1 ; we ally btai the termial wrd a 2 by [s] + ; [e] a 2 ` a 2 : 2 Remark 1. I geeral, the frmal deitis allw a iite umber f bjects t be geerated i e derivati step, which is a uatural situati fr practical implemetatis. A mre practical assumpti wuld be that

8 istead f i (L i)ay arbitrary (ite) subset f i (L i) ca evlve i the test tube T i durig a cmputati perid. The ly this subset is distributed t all test tubes accrdig t the iput lters. I fact, fr all examples ad all cstructis i the prfs f this paper such aiterpretati f the cmputatis i the test tubes still wuld allw us t geerate all desired bjects, althugh it wuld ever be clear, whe these bjects wuld evlve. I a practical evirmet the umber ad the size f bjects that ca be geerated als depeds the amut f rigial material f axims we take at the begiig. Mrever, if parts f (the subset f) i (L i) are t be redistributed ver dieret test tubes it is ly ecessary t assume that ay allwed distributi f the whle material will pssibly happe; i practical implemetatis f test tube systems a itermediate amplicati 2;9 f the material may already guaratee that eugh material is distributed t all the pssible test tubes. Similar ideas as fr the cstructi f the CRTTS i the precedig example ca be used fr prvig the geeral result established i the fllwig therem: Therem 1. Fr every recursively eumerable laguage L; L V + T cstruct a CRTTS L with (V; M) 1 -lters which geerates L: ; we ca Prf. Let L be give by a grammar G 0 L =(V0 N ;V T;P 0 ;S 0 ); i.e. L (G 0 L )=L: Withut lss f geerality we ca csider the laguage L fhg istead f L; where h=2(v 0 N [V T) is a ew symbl; let G L =(V N ;V T ;P;S) be a grammar such that L (G L )=Lfhgad mrever fr each derivati f ay wrd wh 2 L fhg the symbl h is geerated i the last step f this derivati ad des t ccur i ather setetial frm f this derivati. The eective cstructi f G L frm G 0 L is bvius by usig cmm prf techiques frm the thery f frmal laguages ad therefre mitted. Mrever, fr each (; ) 2 P withut lss f geerality we ca assume 1 jj2 ad 0 jj2:nw let P 0 = P [f(u; U ) j U 2 (V N [ V T )g = f( i ; i ) j1 i mg ad L be the CRTTS L =(V; M; A; 3m +5;;I) with V = V N [ V T [fhg[fb; F; X; Y g ; M = [x] + ; [x] j x 2fb; c; d; e; f; h; s; tg[fi l ;i ;i r j1 i mg ; A =fxsby; F; XBY g[fx i Y j 1 i mg; =( 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; :::; 3m+4 ; 3m+5 ) ; 1 = ;; [s] + ; [e] ; 0 X [e] + $[e] a ja 2V 2 [f] + ;[t] 3 = a [f] + $[f] [ [s] + $[s] hby j a 2 V [ F [t] + $[t] F ; 1 A ; ;; ;

9 0 1 w [ b] + $[ b] BY j w 2 V 2 [fxgv [ 4 A; XB [c] + $[c] Y ;; 0 1 X [d] + $[d] v jv 2V 3 [V 2 fyg[v fyg ; 5 A [b] + ;[c] ; [c] + ; ;[d] I 1 = V [s] + [ [e] V ;I 2 =V [f] + [ [t] V [V [t] + ; I 3 =I 4 =V + ;I 5 =V [b] + [V [c] + [ [c] V ; ad fr all i with 1 i m; 0 1 w [i r ] + $[i r ] i Y jw 2fXBg[V 2 [ 3i+3 A fub; Bu j u 2 V gg [ X i [i ] + $[i ] Y ;;; 0 X [il ] + 1 $[i l ] v jv 2V 3 [V 2 fbg[v fbgv[ B 3i+4 fbgv 2 ; [V fby g[fbgv fyg[fby g [i r ] + ; [i ] 3i+5 = ;; [i ] + ; [i l ] ; I 3i+3 = V + ;I 3i+4 = V [i r ] + [ I 3i+5 = V [i ] + [ [i l ] V : [i ] V ; C A ; Ay setetial frm w ccurig i a derivati i G L is represeted by a rtated versi f the frm XvBuY; where w = uv; i L ; the symbl B markig the begiig f the wrd w i its rtated versi i XvBuY ca be rtated i the test tubes T 4 ad T 5 as it was already explaied i the previus example. The al extracti f the termial wrds i Lh is de i the test tubes T 3 ;T 2 ; ad T 1 i a similar way as i the previus example. I the test tubes T 3i+3 ;T 3i+4 ; ad T 3i+5 the applicati f a prducti ( i ; i )issimulated i that way that i is elimiated at the right ed ad i is iserted at the left ed, i.e. frm XvBu i Y we btai X i vbuy :By the cuttig rules i T 3i+3 we btai Xz j [i r] i Y ` Xz [i r ] + ;[i r ] i Y ad X i j [i ] Y rules yield ` T 3i+5 we btai X i [i ] + ;[i ] Xz [i r ] + ;[i ] X i [i ] + ;[i l ] Y ; i T 3i+4 the cuttig ad recmbiati Y ` XzY; X j zy ` X [i l ] + ; [i l ] [i l] zy ; ad i zy ` X i zy: By simulatig the additial uit prductis (U; U ) ; every symbl U 2 (V N [ V T ) ca be rtated, i.e. frm

10 XvBuUY we btai XU vbuy: I this way we ca rtate the setetial frm uv represeted by a strig f the frm as Xv 2 Buv 1 Y; v = v 1 v 2 ; util the psiti where we wat t apply a prducti (; ) is just at the left f the symbl Y; i.e. util we have btaied XvBuY: 2 Whe usig (V; M) 2 -lters, all the test tubes T 3i+3 ;T 3i+4 ad T 3i+5 ; 1 i m; as well as T 4 ad T 5 cstructed i the prf f Therem 1 ca be merged it ly tw test tubes T 0 2 ad T 0 3 ; respectively: Therem 2. Fr every recursively eumerable laguage L; L V + T ; we ca cstruct a CRTTS 0 L with (V; M) 2-lters ad ly three test tubes which geerates L: Prf. Let L be give by the grammar G L ad let P 0 be deed as i the prf f Therem 1. We w csider the CRTTS 0 L = (V; M; A; 3; (0 1 ;0 2 ;0 3 );(I0 1 ;I0 2 ;I0 3 )) ; where M; A are deed as i the prf f Therem 1 ad 0 1 ;; = [s] + ; [e] ; [f] + ; [t] ; 0 2 X [i = l ] + $[i l ] v jv 2V 3 [V 2 fbg[v fbgv [fbgv 2 [ V fby g[fbgv fyg;1img[ w [i r ] + $[i r ] i Y j w 2fBg[fBu j u 2 V g[v 2 ;1im [ X i [i ] + $[i ] Y j1 i m [ [s] + $[s] F [ X [d] + $[d] v jv 2 V 3 [V 2 fbg[v fby g [ w [b] + $[b] BY j w 2 V [ XB [c] + $[c] Y [ X [e] + $[e] w jw 2V 3 [V 2 fhg[v fhbg [ a [f] + $[f] hby j a 2 V [ F [t] + $[t] ;; ; 0 3 ;; = [i ] + ;[i l ] ; [i r ] + ;[i ] ; [c] + ;[d] ; [b] + ;[c] ; I 0 1 [e] = V [f] + [ V [s] + [ [t] V ;I 0 2 =V+ ; I 0 3 [d] = V [b] + [V [c] + [ [c] V [ S 1im [i l ] V [i r ] + [ V [i ] + [ [i ] V : By the cuttig rules i T 0 2 we btai X j ubv i Y ` X [i l ] + ; [i l ] ubv i Y [i l] ad [i l ] ubv j [i r] i Y ` [i l ] ubv [i r ] + ; [i r ] i Y ; as well as i additi

11 X i j [i ] Y ` X i [i ] + ;[i ] X i [i ] + ;[i l ] ubv [i r ] + ` X i ubv [i r ] + ad Y ;by the recmbiati rules i T 0 3 X i ubv [i r ] + ; [i ] we get X i ubvy; i.e. i sum frm XuBv i Y we derive X i ubvy thus simulatig the applicati f the prducti ( i ; i ): Rtatig the symbl B wrks i a similar way, thus yieldig XBwY frm XwBY; ad termial strigs w 2 V + T are btaied i T 0 1 by passig the bjects [e] w [f]+ as well as [s] + ad [t] frm T 0 2 t T 0 1 : The use f the (V; M) 2 [e] -lters as f the (V; M) 2 -lters [d] V [b] + ad [i l ] Y ` V [f] + i I 0 1 as well V [i r ] + ; 1 i m; respectively, which are t (V; M) 1 -lters, guaratees that ly bjects with crrespdig markigs the left-had side ad the right-had side ca pass frm T 0 2 t T 0 1 ad T 0 3 ; whereas bjects with markigs that d t t tgether remai i T 0 2 : 2 The existece f uiversal grammar schemes w implies the existece f uiversal CRTTS with (V; M) i -lters, i 2f1;2g: Crllary 1. Fr every alphabet V T ad every uiversal grammar scheme U fr V T ; U =(V U ;V T ;P U ); we ca eectively cstruct a uiversal CRTTS U;i ; U;i = (V i ;M i ;A i ; i ; i ;I i ), with (V; M) i -lters, i 2 f1;2g; such that ay L 2 ENUM (V T ) is geerated by the CRTTS L;i with (V; M) i -lters with L;i = (V i ;M i ;A i [fxa L BY g ; i ; i ;I i ); where A L is the axim fr which the grammar G L = (V N ;V T ;P;A L ) geerates L; i the case i = 2; 2 =3;i.e. 2 csists f ly three cmpets. Prf. We just have t apply the cstructis i the prfs f Therem 1 ad Therem 2, respectively, t the uiversal grammar scheme U fr V T with the ly excepti that we d t take XSBY as a axim i A: Istead f this, i every special CRTTS L;i with (V; M) i -lters the startig axim A L is take. 2 4 Summary ad Future Research The cstructi f mlecular cmputers based test tubes has bee csidered by usig dieret peratis the test tubes 1;2;3;9. Test tube systems based the splicig perati were shw t allw the cstructi f uiversal mechaisms 4. I the precedig secti we have shw the (theretical) pssibility hw t btai uiversal bilgical mlecular cmputers based test tube systems with cuttig ad recmbiati rules. Our deitis ca easily be exteded i rder t cver a large variety fsuch test tube sytems 6 ; hece ur results shuld als be true fr varius pssible practical implemetatis f

12 such systems. O the ther had, we ly prvide a kid f prgrammig laguage fr mlecular cmputers; feasible slutis fr specic prblems shuld take advatage f the specic features f CRTTS withut ly relyig the geeral methds used i the prfs f the results give i this paper. Ackwledgemets The rst tw authrs wat t gratefully thak Gherghe Pau fr days (ad ights) f itesive discussis the tpics related with splicig ad test tube systems ad they als appreciate fruitful discussis with Lila Kari sme f the tpics csidered i this paper. The wrk f the secd authr was supprted by Hugaria Scietic Research Fud OTKA N. T Refereces 1. L. M. Adlema, Mlecular cmputati f slutis t cmbiatrial prblems, Sciece 226, 1021 { 1024 (1994). 2. L. M. Adlema, O cstructig a mlecular cmputer, mauscript (1995). 3. D. Beh, C. Duwrth, R.J. Lipt, ad J. Sgall, O the cmputatial pwer f DNA, t appear (1996). 4. E. Csuhaj-Varju, L. Kari, ad Gh. Pau, Test tube distributed systems based splicig, Cmputers ad Articial Itelligece, Vl. 15 (2), (1996). 5. R. Freud, L. Kari, ad Gh. Pau, DNA cmputig based splicig: The existece f uiversal cmputers, Tech. Reprt 185-2/FR-2/95, TU Wie, Istitute fr Cmputer Laguages (1995). 6. R. Freud ad F. Freud, Test tube systems r Hw t bake a DNA cake, Prc. Wrkshp Grammar Systems: Recet Results ad Perspectives, t appear (1996). 7. R. Freud ad F. Wachtler, Uiversal systems with peratis related t splicig, Cmputers ad Articial Itelligece, Vl. 15 (4), (1996). 8. T. Head, Frmal laguage thery ad DNA: A aalysis f the geerative capacity f specic recmbiat behavirs, Bull. Math. Bilgy 49, 737 { 759 (1987). 9. R. J. Lipt, Speedig up cmputatis via mlecular bilgy, mauscript (1994). 10. Gh. Pau, Regular exteded H systems are cmputatially uiversal, J. Autmata, Laguages ad Cmbiatrics, Vl. 1 (1), (1996).