Locally Evolving Splicing Systems

Transcription

1 ROMANIAN JOURNAL OF INFORMATION SCIENCE AND TECHNOLOGY Volume 19, Numbe 4, 016, Locally Evolving Splicing Systems Kalpana Mahalingam, Pithwineel Paul Depatment of Mathematics, Indian Institute of Technology, Madas, Chennai-36, India Abstact. An extended H system with locally evolving ules is a model of splicing computation with a special featue that the splicing ules evolve at each step. It was oiginally poved by Paun et al. that all RE languages can be geneated by such systems with a finite set of ules of adius at most 4 and with contexts of length at most 7 fo the insetion-deletion ules by which the splicing ules evolve. We impove this esult by educing the length of the contexts to at most and we also show that such systems with adius at most and contexts of length at most 3 can geneate all RE languages. In both cases, the insetion-deletion ules will inset o delete only one symbol. We also show that if the inseted stings ae of length at most and the deleted stings ae of length 1, then the system can geneate all RE languages with splicing ules of adius 3 and insetion-deletion ules with contexts of length at most. Finally, we show that the length of contexts and of the stings inseted/deleted can be educed to 1 if the ules ae applied in a matix contolled manne. Keywods: Extended H system, splicing, evolving ules, ecusively enumeable. 1 Intoduction The splicing opeation is a fomal model of the ecombinant behaviou of the DNA molecules in the pesence of estiction enzymes. A splicing system, also called H system, consists of an alphabet, a set of splicing ules modeling the ole of enzymes and a set of axioms. Seveal vaiants have been discussed, depending on the natue of the axioms, the set of ules etc. A detailed study can be found in [1, 3, 9]. Of couse, it is desiable to have a finite set of ules and a finite set of axioms. Some impotant vaiants can be found in [4, 5, 7, 10, 1, 13, 15]. Splicing systems with a finite set of axioms and ules can only geneate egula languages [, 11]. Diffeent contolled vesions of finite splicing systems ae able of geneating all ecusively enumeable languages [1, 9]. One such vesion is that of esticted locally evolving splicing systems, intoduced in [8]. Locally evolving splicing systems ae extended H systems whee the splicing ules constantly evolve. The entie set of splicing ules is changed fom one step to anothe by means of insetion/deletion/substitution opeations with espect to given

2 370 Kalpana Mahalingam, Pithwineel Paul contexts. These systems ae shown to geneate all ecusively enumeable languages. In [6], the authos have defined a splicing system that is non-eflexively evolving and poved that such systems ae computationally complete. In this pape, we investigate locally evolving splicing systems as intoduced in [8], whee it was shown that these systems can geneate all RE languages when the adius is 4 and the length of the context of insetion-deletion ules is at most 7. It was conjectued that the adius as well as the length of the context can be educed to. In this pape we povide chaacteizations of RE languages by such systems in two diffeent impoved vesions: the adius at most and the length of the contexts at most 3, espectively when the adius is at most 4 and the length of the context is at most. We also show that instead of using point mutation ules if we use insetion-deletion ules which can inset two symbols and delete one symbol, then the adius can be deceased to 3 and the length of the contexts of insetion-deletion ules can be at most. Finally, we show that, if we apply the ules in a matix contolled way, then the length of the contexts as well as the inseted/deleted stings is 1 while the adius of the splicing ules is 3. This pape is oganized as follows: Section ecalls the necessay definitions. Section 3 illustates the chaacteizations of RE languages with these systems with educed adius and contexts of the ewiting ules. We end the section with a chaacteization of RE languages whee the insetion-deletion ules ae applied in a matix contolled way. Section 4 contains some concluding emaks. Definitions The eade is efeed to [14] fo basic elements of fomal language theoy. We use the convention that V is a finite alphabet, λ denotes the empty sting, V epesents the set of all stings ove V and V + = V \ {λ}. We ecall the following fom [9]. Let #, $ be two symbols not in V. A splicing ule ove V is a sting u 1 #u $u 3 #u 4, whee u i V fo 1 i 4. Fo a splicing ule = u 1 #u $u 3 #u 4 and fo x, y, w, z V we wite, (x, y) (w, z) iff x = x 1 u 1 u x, y = y 1 u 3 u 4 y, w = x 1 u 1 u 4 y, z = y 1 u 3 u x fo x 1, x, y 1, y V. whee An extended H system with locally evolving splicing ules is a constuct 1. V is a finite alphabet,. T V is the teminal alphabet, 3. A 0 is a finite set of stings fom V, γ = (V, T, A 0, A c, E, C 0, P ) 4. A c V is the finite set of cuent axioms, 5. E is an alphabet such that E V, 6. C 0 is an initial sequence of splicing ules, C 0 = ( 1,,, k ), i E #E $ E #E, 1 i k, 7. P is a finite set of ewiting ules of the fom (u, α/β, v) with u, v (E {#, $}), and α, β E {λ}, α β.

3 Locally Evolving Splicing Systems 371 The ules in P ae called point mutation ules if the ules ae of the fom α E, β = λ (insetion ules) and α = λ, β E (deletion ules). The splicing ules ae constucted fom the template splicing ules in C 0 using the point mutation ules. The stings inside the system ae modified by the splicing ules constucted as above. The splicing ules, if applicable, must be applied. Initially, stings fom A 0 and A c ae spliced. Late, the stings poduced inside the system ae spliced with the stings in A c using the ules constucted by applying the insetion-deletion ules to the cuent set of splicing ule. The notation = P denotes the usual deivation elation with espect to ules in P. Fo a splicing ule E #E $E #E, we define P () = { = P }. We extend the elation = P to k-tuples of splicing ules by ( 1,,, k ) = P ( 1,,, k ) iff j P ( j), 1 j k. Stating fom C 0, at the time i 1 we can obtain in this fashion a sequence C i = ( i,1,, i,k ) to which the following sequence of splicing ules ae associated. R i = { = i,j fo some 1 j k}. The set R i, contains exactly one descendant of evey ule in C 0 ; out of the possible vaiants which can be obtained due to the possible non-deteminism of using the ules in P, only one is actually chosen. The sets A i fo i 0 ae defined as follows. The initial set A 0 is given. Fo x V and a given set R of splicing ules and fo i 0, define { 1 if B δ i (x, R) = 0 othewise whee B is the condition that thee exists y A i A c and R such that (x, y) (w, z) o (y, x) (w, z), fo some w, z V. Moeove, R i (A i ) = {w V (x, y) (w, z) o (x, y) (z, w) fo R i, x, y A i A c, {x, y} A i }, i 0. Then, A i = {x A i 1 δ i 1 (x, R i 1 ) = 0} R i 1 (A i 1 ), i 1 and the language geneated by γ is defined by L(γ) = ( i 0 A i) T. An extended H system with locally evolving splicing ules γ = (V, T, A 0, A c, E, C 0, P ) is called esticted if cad(a 0 ) = 1, cad(c 0 ) = 1. The system γ is said to be of adius at most m if fo each i 1 if R i = u 1 #u $u 3 #u 4, then u j m, 1 j 4. Note that in a esticted locally evolving system we have exactly one splicing ule at each time. The family of languages geneated by the esticted locally evolving H systems is denoted as EH le (m, c, id), whee m is the maximal adius of the splicing ules, c is the maximal length of the contexts in the insetion/deletion ules, and id is the maximal length of the sting inseted/deleted. In [8], the notation REHE([m]) and in [9] the notation EH (F IN, le([m])) have been used to denote such systems. Fo moe claifications, the eade is efeed to [8, 9]. 3 Results It was shown in [8] that locally evolving splicing systems with adius 4 and the length of the context of insetion-deletion ules is less than 7 can geneate all RE languages. It was conjectued that the adius as well as the length of the context can be educed to. In this section we povide two esults which impove the esult in [8].

4 37 Kalpana Mahalingam, Pithwineel Paul In the next theoem, we pove that the family of languages geneated by esticted locally evolving H systems with splicing ules of adius is equal to the family of ecusively enumeable languages when the insetion-deletion ules used ae point mutation ules with contexts of length 3. Theoem 3.1. EH le (, 3, 1) = RE. Poof: The inclusion EH le (, 3, 1) RE can be poved in a staightfowad but cumbesome way, o we can invoke fo it the Chuch-Tuing thesis. So we only pove the othe inclusion. Let G = (N, T, P 0, S) be a gamma in Kuoda nomal fom. The ules in G ae of the fom A BC, AB CD, A a, A λ whee A, B, C, D N and a T. The ules of P 0 ae labeled in a one-to-one manne. We constuct a esticted extended H system γ = (V, T, A 0, A c, E, C 0, P ) with locally evolving splicing ules such that L(G) = L(γ) as follows: 1. V = N T {X, Y, Z, Y 1, Y, Z, B 0 },. A 0 = {XB 0 SY }, 3. A c = {ZY } {XαZ α N T {B 0 }} {ZY 1, ZCY, Z DY : AB CD} {ZY 1, ZBY, Z CY : A BC} {Z ay : A a} {Z Y : A λ}, 4. E = N T {X, Y, Z, Y 1, Y, Z } {[, 1], [, ], [, 3],..., [, 13] P 0 } {d 1, d, d 3, d 4 } {e 1, e } {f 1, f,..., f 8 } {g 1, g, g 3 } {h 1, h, h 3, h 4 }, 5. C 0 = (#c 1 $Z#Y ). The ule of C 0 is called the template splicing ule and it evolves using the insetion and deletion ules (i.e., point mutation ules) pesent in P. Initially, only the splicing ule #c 1 $Z#Y is pesent in the system. No sting of A 0 and A c of the system γ can be spliced using this ule. Using the point mutation ules in P, the ule of C 0 is modified so that the obtained ules can simulate the ules of P 0. In the constuctions below, the point mutation ules ae placed on the left-hand side and the splicing ules obtained afte applying the point mutation ules to the splicing ule obtained in the pevious step ae placed in the coesponding ight-hand side of the table. The poof is based on the otate-and-simulate technique intoduced in [7]. The splicing ules (A#BY $Z#Y 1 ), (#AY 1 $Z#CY ), (C#Y $Z #DY ) (coesponding to ules SA1 16, SA 18 and SA3 0 espectively) if applied sequentially can simulate the ule : AB CD in the following manne: (XwA BY, Z Y 1 ) (XwAY 1 (Xw AY 1, Z CY ) (XwCY, ZAY 1 (XwC Y, Z DY ) (XwCDY, Z Y, ZBY ); (Rule SA1 16) ); (Rule SA 18) ); (Rule SA3 0). The above mentioned splicing ules ae constucted in SA1, SA, SA3. The ule : AB CD, can be simulated by the application of the splicing ules constucted in (SA1), (SA), (SA3). (SA1) : Stating with the template ule, we constuct the ule A#BY $Z#Y 1.

5 Locally Evolving Splicing Systems #c 1 $Z#Y 1. (#c 1, λ/[, 1], $) #c 1 [, 1]$Z#Y. (#, c 1 /λ, [, 1]) #[, 1]$Z#Y 3. (λ, λ/a, #[, 1]) A#[, 1]$Z#Y 4. (A#, λ/b, [, 1]) A#B[, 1]$Z#Y 5. (B[, 1]$, λ/[, ], Z) A#B[, 1]$[, ]Z#Y 6. (B, [, 1]/λ, $[, ]) A#B$[, ]Z#Y 7. (B, λ/y, $[, ]) A#BY $[, ]Z#Y 8. (Y $[, ], Z/λ, #) A#BY $[, ]#Y 9. (Y $[, ], λ/[, 3], #Y ) A#BY $[, ][, 3]#Y 10. ($, [, ]/λ, [, 3]#) A#BY $[, 3]#Y 11. ([, 3]#Y, λ/[, 4], λ) A#BY $[, 3]#Y [, 4] 1. ($, [, 3]/λ, #Y [, 4]) A#BY $#Y [, 4] 13. ($#, Y/λ, [, 4]) A#BY $#[, 4] 14. ($, λ/z, #[, 4]) A#BY $Z#[, 4] 15. ($Z#, λ/y 1, [, 4]) A#BY $Z#Y 1 [, 4] 16. (Z#Y 1, [, 4]/λ, λ) A#BY $Z#Y 1 Only the last splicing ule A#BY $Z#Y 1 can be applied to the stings fom A 0 and A c. Othe ules ae not applicable, since the symbols c 1, [, 1], [, ], [, 3] and [, 4] ae pesent in the splicing ules. Afte the application of the ule A#BY $Z#Y 1, the stings XwAY 1 and ZBY ae poduced. The fist sting again can be spliced with the sting ZY using the splicing ule #AY 1 $Z#Y, but the sting ZBY cannot geneate teminal stings. (SA) : The splicing ule A#BY $Z#Y 1 #AY 1 $Z#Y as follows: is now modified into the splicing ule 0. A#BY $Z#Y 1 1. (λ, λ/[, 5], A#) [, 5]A#BY $Z#Y 1. ([, 5], A/λ, #B) [, 5]#BY $Z#Y 1 3. ([, 5]#, B/λ, Y ) [, 5]#Y $Z#Y 1 4. ([, 5]#, Y/λ, $) [, 5]#$Z#Y 1 5. ([, 5]#, λ/a, $) [, 5]#A$Z#Y 1 6. ([, 5]#A, λ/[, 6], $) [, 5]#A[, 6]$Z#Y 1 7. (λ, [, 5]/λ, #A[, 6]) #A[, 6]$Z#Y 1 8. ([, 6]$, λ/[, 7], Z) #A[, 6]$[, 7]Z#Y 1 9. (A, [, 6]/λ, $[, 7]) #A$[, 7]Z#Y (#A, λ/y 1, $[, 7]) #AY 1 $[, 7]Z#Y (Y 1 $[, 7], Z/λ, #) #AY 1 $[, 7]#Y 1 1. ([, 7]#Y 1, λ/[, 8], λ) #AY 1 $[, 7]#Y 1 [, 8] 13. ($, [, 7]/λ, #Y 1 [, 8]) #AY 1 $#Y 1 [, 8] 14. ($, λ/z, #Y 1 [, 8]) #AY 1 $Z#Y 1 [, 8] 15. (Z#, Y 1 /λ, [, 8]) #AY 1 $Z#[, 8] 16. ($Z#, λ/y, [, 8]) #AY 1 $Z#Y [, 8] 17. (Z#, λ/c, Y [, 8]) #AY 1 $Z#CY [, 8] 18. (#CY, [, 8]/λ, λ) #AY 1 $Z#CY Afte splicing the stings XwAY 1 and ZCY using the ule #AY 1 $Z#CY, the stings XwCY and ZAY 1 ae poduced. (SA3) : Now, #AY 1 $Z#CY is modified into C#Y $Z #DY :

6 374 Kalpana Mahalingam, Pithwineel Paul 0. #AY 1 $Z#CY 1. (λ, λ/[, 9], #A) [, 9]#AY 1 $Z#CY. ([, 9]#, A/λ, Y 1 ) [, 9]#Y 1 $Z#CY 3. ([, 9]#, λ/[, 10], Y 1 ) [, 9]#[, 10]Y 1 $Z#Y 4. ([, 9], λ/c, #[, 10]) [, 9]C#[, 10]Y 1 $Z#CY 5. (λ, [, 9]/λ, C#[, 10]) C#[, 10]Y 1 $Z#CY 6. ([, 10], Y 1 /λ, $) C#[, 10]$Z#CY 7. ([, 10], λ/y, $) C#[, 10]Y $Z#CY 8. ([, 10]Y $, λ/[, 11], Z) C#[, 10]Y $[, 11]Z#CY 9. (#, [, 10]/λ, Y $[, 11]) C#Y $[, 11]Z#CY 10. ([, 11], Z/λ, #C) C#Y $[, 11]#CY 11. ([, 11]#, C/λ, Y ) C#Y $[, 11]#Y 1. ([, 11], λ/z, #Y ) C#Y $[, 11]Z #Y 13. ([, 11]Z #, λ/[, 1], Y ) C#Y $[, 11]Z #[, 1]Y 14. ($, [, 11]/λ, Z #[, 1]) C#Y $Z #[, 1]Y 15. ($Z, λ/[, 13], #[, 1]) C#Y $Z [, 13]#[, 1]Y 16. ([, 13]#, [, 1]/λ, Y ) C#Y $Z [, 13]#Y 17. ([, 13]#, Y /λ, λ) C#Y $Z [, 13]# 18. ([, 13]#, λ/d, λ) C#Y $Z [, 13]#D 19. ([, 13]#D, λ/y, λ) C#Y $Z [, 13]#DY 0. (Z, [, 13]/λ, #DY ) C#Y $Z #DY Now the sting XwCY can be spliced futhe with Z DY using the ule SA3 0 to obtain XwCDY and Z Y. Note that Z Y cannot be used futhe to poduce teminal stings. Since thee exists only one splicing ule at any time in the system, afte completely simulating one ule in P 0, eithe the system simulates anothe ule of P 0 o will teminate the computation. The simulation of ules in P 0 is done at the end of the stings, hence the stings should be ciculaly otated. But, to poceed futhe the system must go back to the template splicing ule. Hence, the splicing ule C#Y $Z #DY is tansfomed into (#c 1 $Z#Y ). (T 1) : In the next step the ule C#Y $Z #DY is tansfomed into the template ule in C C#Y $Z #DY 1. (λ, λ/d 1, C#Y ) d 1 C#Y $Z #DY. (d 1 C#, Y /λ, $) d 1 C#$Z #DY 3. (d 1, C/λ, #$) d 1 #$Z #DY 4. (d 1 #$, Z /λ, #DY ) d 1 #$#DY 5. (d 1 #, λ/c 1, $#) d 1 #c 1 $#DY 6. (d 1 #c 1, λ/d, $#) d 1 #c 1 d $#DY 7. (λ, d 1 /λ, #c 1 d ) #c 1 d $#DY 8. (c 1 d $, λ/d 3, #D) #c 1 d $d 3 #DY 9. (c 1, d /λ, $d 3 ) #c 1 $d 3 #DY 10. (d 3 #, D/λ, Y ) #c 1 $d 3 #Y 11. ($d 3, λ/z, #Y ) #c 1 $d 3 Z#Y 1. ($, d 3 /λ, Z#Y ) #c 1 $Z#Y The constuction of splicing ule to simulate the ule : A BC is simila and hence we omit it. (SB1) : Constuction of splicing ules to simulate the ule : A a:

7 Locally Evolving Splicing Systems #c 1 $Z#Y 1. (#, λ/[, 1], c 1 ) #[, 1]c 1 $Z#Y. (#[, 1], c 1 /λ, $) #[, 1]$Z#Y 3. (#[, 1]$, λ/[, ], Z#) #[, 1]$[, ]Z#Y 4. (#, [, 1]/λ, $[, ]) #$[, ]Z#Y 5. (#, λ/a, $[, ]) #A$[, ]Z#Y 6. (#A, λ/y, $[, ]) #AY $[, ]Z#Y 7. (Y $[, ], Z/λ, #Y ) #AY $[, ]#Y 8. (Y $[, ], λ/[, 3], #Y ) #AY $[, ][, 3]#Y 9. ([, ][, 3]#, λ/a, Y ) #AY $[, ][, 3]#aY 10. ($, [, ]/λ, [, 3]#a) #AY $[, 3]#aY 11. ($, λ/z, [, 3]#a) #AY $Z [, 3]#aY 1. ($Z, [, 3]/λ, #ay ) #AY $Z #ay Similaly, we can constuct the splicing ule #AY $Z #Y which simulates the ule : A λ. (T ) : Now the ule #AY $Z#aY will be tansfomed to the template ule #c 1 $Z #Y. 0. #AY $Z#aY 1. (λ, λ/d 1, #AY ) d 1 #AY $Z#aY. (d 1 #, A/λ, Y $) d 1 #Y $Z#aY 3. (d 1 #, Y/λ, $) d 1 #$Z#aY 4. (d 1 #, λ/d, $Z) d 1 #d $Z#aY 5. (d 1 #, λ/c 1, d $) d 1 #c 1 d $Z#aY 6. (λ, d 1 /λ, #c 1 d ) #c 1 d $Z#aY 7. (c 1 d $, λ/d 3, Z) #c 1 d $d 3 Z#aY 8. (#c 1, d /λ, $d 3 ) #c 1 $d 3 Z#aY 9. (d 3 Z#, a/λ, Y ) #c 1 $d 3 Z#Y 10. ($, d 3 /λ, Z#Y ) #c 1 $Z#Y We now constuct splicing ules necessay fo otating the cuent sting. (ROT 1) : 0. #c 1 $Z#Y 1. (λ, d 4 /λ, #c 1 ) d 4 #c 1 $Z#Y. (d 4 #, c 1 /λ, $Z) d 4 #$Z#Y 3. (d 4 #, λ/α, $Z) d 4 #α$z#y 4. (d 4 #α, λ/y, $Z) d 4 #αy $Z#Y 5. (λ, d 4 /λ, #αy ) #αy $Z#Y In (ROT ) the splicing ule Xα#Z$X# is constucted, which adds the symbol α N T {B 0 } to the ight of the make X.

8 376 Kalpana Mahalingam, Pithwineel Paul (ROT ) : 0. #αy $Z#Y 1. (λ, λ/f 1, #αy ) f 1 #αy $Z#Y. (f 1 #, α/λ, Y ) f 1 #Y $Z#Y 3. (f 1 #, Y/λ, $) f 1 #$Z#Y 4. (f 1, λ/x, #$) f 1 X#$Z#Y 5. (f 1 X#, λ/f, $Z) f 1 X#f $Z#Y 6. (λ, f 1 /λ, X#f ) X#f $Z#Y 7. (X, λ/α, #f ) Xα#f $Z#Y 8. (Xα#, λ/z, f ) Xα#Zf $Z#Y 9. (Zf $, λ/f 3, Z#Y ) Xα#Zf $f 3 Z#Y 10. (Z, f /λ, $f 3 ) Xα#Z$f 3 Z#Y 11. (Z$f 3, Z/λ, #Y ) Xα#Z$f 3 #Y 1. (f 3 #Y, λ/f 4, λ) Xα#Z$f 3 #Y f ($, f 3 /λ, #Y f 4 ) Xα#Z$#Y f ($#, Y/λ, f 4 ) Xα#Z$#f (Z$, λ/x, #f 4 ) Xα#Z$X#f (X#, f 4 /λ, λ) Xα#Z$X# Using ules ROT 1 5 and ROT 16 the symbol α N T {B 0 } adjacent to the make Y in XwαY can be otated to obtain XαwY. (Xw αy, Z Y ) (XwY, ZαY ). The sting XwY can be spliced futhe using ROT 16 but ZαY will neve lead to teminal stings. (Xα Z, X wy ) (XαwY, XZ). Afte otation is complete, the ule Xα#Z$X# is tansfomed into the ule #c 1 $Z#Y. (T 3) : 0. Xα#Z$X# 1. (α#z, λ/f 5, $X) Xα#Zf 5 $X#. (Xα#, Z/λ, f 5 ) Xα#f 5 $X# 3. (X, α/λ, #f 5 ) X#f 5 $X# 4. (λ, λ/f 6, X#f 5 ) f 6 X#f 5 $X# 5. (f 6, X/λ, #f 5 ) f 6 #f 5 $X# 6. (f 6 #, λ/c 1, f 5 ) f 6 #c 1 f 5 $X# 7. (λ, f 6 /λ, #c 1 f 5 ) #c 1 f 5 $X# 8. (c 1 f 5 $, λ/f 7, X#) #c 1 f 5 $f 7 X# 9. (c 1, f 5 /λ, $f 7 ) #c 1 $f 7 X# 10. (f 7 X#, λ/f 8, λ) #c 1 $f 7 X#f (f 7, X/λ, #f 8 ) #c 1 $f 7 #f 8 1. ($, f 7 /λ, #f 8 ) #c 1 $#f ($, λ/z, #f 8 ) #c 1 $Z#f (Z#, λ/y, f 8 ) #c 1 $Z#Y f (Z#Y, f 8 /λ, λ) #c 1 $Z#Y Then eithe the pocess can be epeated by constucting all the splicing ules that simulate ules in P 0 as well as the ules which helps with otation o the following teminating ules can be constucted.

9 Locally Evolving Splicing Systems 377 The teminating ules ae constucted in (T E1) and (T E). The ule XB 0 #$#ZY, constucted fom the template splicing ule in (T E1), emoves XB 0 fom the left hand side of the sting XB 0 wy and the ule constucted in (T E) emoves the make Y. (T E1) : 0. #c 1 $Z#Y 1. (λ, λ/g 1, #c 1 ) g 1 #c 1 $Z#Y. (g 1 #, c 1 /λ, $Z) g 1 #$Z#Y 3. (g 1, λ/x, #$) g 1 X#$Z#Y 4. (g 1 X#, λ/g, $Z#) g 1 X#g $Z#Y 5. (λ, g 1 /λ, X#g ) X#g $Z#Y 6. (X, λ/b 0, #g $) XB 0 #g $Z#Y 7. (g $, λ/g 3, Z#Y ) XB 0 #g $g 3 Z#Y 8. (XB 0 #, g /λ, $g 3 ) XB 0 #$g 3 Z#Y 9. (g 3, Z/λ, #Y ) XB 0 #$g 3 #Y 10. (g 3 #, λ/z, Y ) XB 0 #$g 3 #ZY 11. (#$, g 3 /λ, #ZY ) XB 0 #$#ZY The poduced splicing ule is XB 0 #$#ZY. This ule is applicable to stings XB 0 wy and ZY. (XB 0 wy, ZY ) (XB 0 ZY, wy ) i.e., XB 0 is emoved and wy is spliced futhe. (T E) : The ule #Y $ZY # is constucted fom XB 0 #$#ZY. This ule emoves the make Y. 0. XB 0 #$#ZY 1. (XB 0 #, λ/h 1, $#) XB 0 #h 1 $#ZY. (X, B 0 /λ, #h 1 ) X#h 1 $#ZY 3. (λ, λ/h, X#h 1 ) h X#h 1 $#ZY 4. (h, X/λ, #h 1 ) h #h 1 $#ZY 5. (h #, λ/y, h 1 ) h #Y h 1 $#ZY 6. (λ, h /λ, #Y h 1 ) #Y h 1 $#ZY 7. (Y h 1 $, λ/h 3, #ZY ) #Y h 1 $h 3 #ZY 8. (#Y, h 1 /λ, $h 3 ) #Y $h 3 #ZY 9. (h 3 #, Z/λ, Y ) #Y $h 3 #Y 10. (h 3 #Y, λ/h 4, λ) #Y $h 3 #Y h (h 3 #, Y/λ, h 4 ) #Y $h 3 #h 4 1. ($, h 3 /λ, #h 4 ) #Y $#h ($, λ/z, #h 4 ) #Y $Z#h (Z, λ/y, #h 4 ) #Y $ZY #h (ZY #, h 4 /λ, λ) #Y $ZY # The ule ROT 15 can be applied to stings wy and ZY to obtain w and ZY Y. (w Y, ZY ) (w, ZY Y ). By splicing the sting ZY Y with stings fom A c, teminal stings can neve be eached. If the sting w is in T then w L(G). Since w does not contain any makes, no ule is applicable to w even if it is not a teminal sting. Also the wong application of the insetion-deletion ules cannot geneate splicing ules which lead to teminal stings. The computation stops afte the constuction of ule T E 15 is complete, since the

10 378 Kalpana Mahalingam, Pithwineel Paul ule #Y $ZY # cannot be modified futhe. Thus we can conclude that L(G) = L(γ). This implies RE EH le (, 3, 1). The next esult is a vaiation of Theoem 3.1 in which we give a chaacteization of RE languages when the adius of the system is at most 4 and the length of the context of the insetion-deletion ules is at most with point mutation ules. Thus we have the following. Theoem 3.. RE = EH le (4,, 1). Poof: We only pove the inclusion RE EH le (4,, 1). Let L RE and G = (N, T, P 0, S) be a type-0 gamma in Kuoda nomal fom geneating L. We constuct a esticted extended H system with locally evolving splicing ules γ = (V, T, A 0, A c, E, C 0, P ) whee 1. V = {X, Y, Z, B 0 } N T,. A 0 = {XB 0 SY }, 3. A c = {ZY } {ZvY u v P 0 } {XαZ α N T {B 0 }}, 4. E = N T {X, Y, Z, B 0, c 1, e 1, e,..., e 6, d..., d 6, f 1, f,..., f 5, g 1, g,..., g 6, h 1, h,..., h 6, k 1, k,..., k 7 } {[, 1], [, ], [, 3], [, 4], [, 5], [, 6], [, 7], [, 8] : u v P 0 }, 5. C 0 = (c 1 #Y $Z#). We only constuct the splicing ule which simulates the ule : AB CD and omit the constuctions of ules that simulate A BC, A a and A λ as they ae simila. The ule #ABY $Z#CDY simulates the ule : AB CD P 0. The following insetion-deletion ules with context ae constucted which tansfoms the splicing ule c 1 #Y $Z# into #ABY $Z#CDY. (SA1) : 0. c 1 #Y $Z# 1. (c 1 #, λ/[, 1], Y ) c 1 #[, 1]Y $Z#. (λ, c 1 /λ, #[, 1]) #[, 1]Y $Z# 3. (#, λ/a, [, 1]Y ) #A[, 1]Y $Z# 4. ([, 1]Y, λ/[, ], $Z) #A[, 1]Y [, ]$Z# 5. (#A, [, 1]/λ, Y [, ]) #AY [, ]$Z# 6. (#A, λ/b, Y [, ]) #ABY [, ]$Z# 7. ([, ]$, λ/[, 3], Z#) #ABY [, ]$[, 3]Z# 8. (Y, [, ]/λ, $[, 3]) #ABY $[, 3]Z# 9. ([, 3]Z, λ/[, 4], #) #ABY $[, 3]Z[, 4]# 10. ($, [, 3]/λ, Z[, 4]) #ABY $Z[, 4]# 11. ([, 4]#, λ/c, λ) #ABY $Z[, 4]#C 1. ([, 4]#, λ/[, 5], C) #ABY $Z[, 4]#[, 5]C 13. (Z, [, 4]/λ, #[, 5]) #ABY $Z#[, 5]C 14. ([, 5]C, λ/[, 6], λ) #ABY $Z#[, 5]C[, 6] 15. (Z#, [, 5]/λ, C[, 6]) #ABY $Z#C[, 6] 16. (#C, λ/d, [, 6]) #ABY $Z#CD[, 6] 17. (D, λ/y, [, 6]) #ABY $Z#CDY [, 6] 18. (DY, [, 6]/λ, λ) #ABY $Z#CDY

11 Locally Evolving Splicing Systems 379 Thus the obtained splicing ules ae of the fom #uy $Z#vY which simulates the ules : u v P 0. Theefoe the stings XwuY and ZvY can be spliced to obtain XwvY and ZuY. (Xw uy, Z vy ) (XwvY, ZuY ). The fist sting XwvY can be spliced futhe but ZuY will neve lead to a teminal sting. (T 1) : We now poceed futhe to constuct the template splicing ule c 1 #Y $Z#Y fom the ule SA #ABY $Z#CDY 1. (Z#, λ/e 1, CD) #ABY $Z#e 1 CDY. (#e 1, C/λ, DY ) #ABY $Z#e 1 DY 3. (#e 1, D/λ, Y ) #ABY $Z#e 1 Y 4. (Z#, λ/e, e 1 Y ) #ABY $Z#e e 1 Y 5. (#e, e 1 /λ, Y ) #ABY $Z#e Y 6. ($Z, λ/e 3, #e ) #ABY $Ze 3 #e Y 7. (e 3 #, e /λ, Y ) #ABY $Ze 3 #Y 8. (Y $, λ/e 4, Ze 3 ) #ABY $e 4 Ze 3 #Y 9. (e 4 Z, e 3 /λ, #Y ) #ABY $e 4 Z#Y 10. (Y, λ/e 5, $e 4 ) #ABY e 5 $e 4 Z#Y 11. (e 5 $, e 4 /λ, Z#) #ABY e 5 $Z#Y 1. (#A, B/λ, Y e 5 ) #AY e 5 $Z#Y 13. (#, A/λ, Y e 5 ) #Y e 5 $Z#Y 14. (#, λ/e 6, Y e 5 ) #e 6 Y e 5 $Z#Y 15. (e 6 Y, e 5 /λ, $) #e 6 Y $Z#Y 16. (λ, λ/c 1, #e 6 ) c 1 #e 6 Y $Z#Y 17. (c 1 #, e 6 /λ, Y ) c 1 #Y $Z#Y We now constuct splicing ules which otate the symbols between the makes X and Y of the sting XwY whee w (N T {B 0 }). At fist, α N T {B 0 } is cut fom the ight hand side of the sting XwαY. This opeation can be done by application of the splicing ule #αy $Z#Y in ROT ROT 1 : 0. c 1 #Y $Z# 1. (λ, λ/d 1, c 1 #) d 1 c 1 #Y $Z#. (d 1, c 1 /λ, #Y ) d 1 #Y $Z# 3. (d 1 #, λ/α, Y ) d 1 #αy $Z# 4. (d 1 #, λ/d, αy ) d 1 #d αy $Z# 5. (λ, d 1 /λ, #d ) #d αy $Z# 6. (d α, λ/d 3, Y ) #d αd 3 Y $Z# 7. (#, d /λ, αd 3 ) #αd 3 Y $Z# 8. (d 3 Y, λ/d 4, $) #αd 3 Y d 4 $Z# 9. (#α, d 3 /λ, Y d 4 ) #αy d 4 $Z# 10. (d 4 $, λ/d 5, Z) #αy d 4 $d 5 Z# 11. (αy, d 4 /λ, $d 5 ) #αy $d 5 Z# 1. (d 5 Z, λ/d 6, #) #αy $d 5 Zd 6 # 13. (Y $, d 5 /λ, Zd 6 ) #αy $Zd 6 # 14. (d 6 #, λ/y, λ) #αy $Zd 6 #Y 15. ($Z, d 6 /λ, #Y ) #αy $Z#Y

12 380 Kalpana Mahalingam, Pithwineel Paul Afte application of the ule #αy $Z#Y to stings XwαY and ZY, the stings XwY and ZαY ae poduced. Using the ule ROT the sting XwY is spliced with XαZ to obtain XαwY and XZ. (ROT ) : 0. #αy $Z#Y 1. (λ, λ/k 1, #α) k 1 #αy $Z#Y. (k 1 #, α/λ, Y ) k 1 #Y $Z#Y 3. (k 1 #, Y/λ, $) k 1 #$Z#Y 4. (k 1, λ/x, #$) k 1 X#$Z#Y 5. (k 1 X, λ/α, #$) k 1 Xα#$Z#Y 6. (k 1 X, λ/k, α#) k 1 Xk α#$z#y 7. (λ, k 1 /λ, Xk ) Xk α#$z#y 8. (k α, λ/k 3, #$) Xk αk 3 #$Z#Y 9. (X, k /λ, αk 3 ) Xαk 3 #$Z#Y 10. (k 3 #, λ/k 4, $) Xαk 3 #k 4 $Z#Y 11. (Xα, k 3 /λ, #k 4 ) Xα#k 4 $Z#Y 1. (α#, λ/z, k 4 ) Xα#Zk 4 $Z#Y 13. (k 4 $, λ/k 5, Z) Xα#Zk 4 $k 5 Z#Y 14. (#Z, k 4 /λ, $k 5 ) Xα#Z$k 5 Z#Y 15. (k 5, Z/λ, #Y ) Xα#Z$k 5 #Y 16. (k 5 #, λ/k 6, Y ) Xα#Z$k 5 #k 6 Y 17. (k 6 Y, λ/k 7, λ) Xα#Z$k 5 X#k 6 Y k (k 6, Y/λ, k 7 ) Xα#Z$k 5 #k 6 k (k 5 #, k 6 /λ, k 7 ) Xα#Z$k 5 #k 7 0. ($, k 5 /λ, #k 7 ) Xα#Z$#k 7 1. ($, λ/x, #k 7 ) Xα#Z$X#k 7. (X#, k 7 /λ, λ) Xα#Z$X# The splicing ules constucted at the end of (ROT 1) and (ROT ) otate one symbol inside the makes X and Y. Afte completely otating one symbol, the ule Xα#Z$X# will again be tansfomed into the ule c 1 #Y $Z#. The constuctions of the template ule fom the otation ule and that of the teminating ule fom the template ule ae simila to that of the constuction in Theoem 3.1 and hence we omit it. If the sting geneated afte application of the teminating ule is ove teminals, then w L(G). The splicing ules constucted fom the template contains at least one of the makes X and/o Y. Hence, even if w is not a teminal sting, it cannot be spliced futhe, since it does not contain any makes. Since, the teminating ule cannot be modified futhe and the computation stops afte application of this ule, L(γ) = L(G). The splicing ules constucted have adius 4, so we can conclude that RE EH le (4, 3, 1). In the next esult, instead of using point mutation ules, we use insetion ules which can inset at most two symbols, but the deletion ules emove only one symbol. In Theoem 3. the adius of the splicing ules is 4, but the insetion-deletion ules ae point mutation ules, having contexts of length. If insetion ules ae allowed to inset at most two symbols, then the adius of the splicing ules can be educed to 3. Since the constuction in the poof is simila to that of the above esults, we omit it. Theoem 3.3. RE = EH le (3,, ). In the next esult, we investigate the computational powe of esticted locally evolving splicing systems, whee the insetion-deletion ules ae applied in a matix

13 Locally Evolving Splicing Systems 381 contolled manne. Matix insetion-deletion systems ae investigated in [10]. In the following esult, we show that, if matix contolled insetion-deletion ules ae used, then locally evolving splicing systems can geneate all RE languages, whee the splicing ules ae of adius 3, and insetion-deletion ules ae point mutation ules with contexts of length 1. In matix insetion-deletion system, a set of ules ae applied in a sequential manne, i.e., if the ules of the matix R n = [ n1, n,..., nm ], n, m N is applied, it means that the ules n1, n,..., nm ae applied in sequence. We denote by Mat k EH le (m, c, id) the family of languages geneated by matix esticted locally evolving splicing systems, whee k epesents the maximum numbe of insetion-deletion ules equied to tansfom one splicing ule to anothe, white the othe paametes ae as above. Theoem 3.4. RE = Mat 10 EH le (3, 1, 1). Poof: We only pove the inclusion RE Mat 10 EH le (3, 1, 1). Let G = (N, T, P 0, S) be a gamma in Kuoda nomal fom with the ules in P 0 labeled in one-to-one manne. We constuct a esticted extended H system γ = (V, T, A 0, A c, E, C 0, P ) with locally evolving splicing ules such that L(G) = L(γ). V = N T {X, Y, Z, B 0 }, A 0 = {XB 0 SY }, A c = {ZY } {XαZ α N T {B 0 }} {ZCDY i : AB CD} {ZBCY i : A BC} {ZaY i : A a}, E = N T {X, Y, Z} {c 1, c } {d 1, d } {e 1, k 1 } {f 1, f, f 3 } {g 1, g } {h 1 }, C 0 = (c 1 #$Z#Y ). In this poof, the insetion-deletion ules wok in the matix contolled manne. Thus, P = {R i, R i i P } {R α, R α α V } {R B0, R B 0 } is the set containing the matices. We discuss the constuction of #ABY $Z#CDY fom c 1 #Y $Z#. Note that the ules of P 0 ae labeled with i and fo each ule thee exist matices R i and R i. The sequential application of the ules of R i constucts the splicing ule, which simulates the ule i P 0 and the sequential application of ules fom R i econstucts the template splicing ule fom the splicing ule constucted fo simulating the i labeled ules. (R1) : The i : AB CD ule is simulated by the splicing ule #ABY $Z#CDY. Afte applying the ules in R i = [ i1, i,..., i6 ], the template splicing ule foms the splicing ule #ABY $Z#CDY. 0. c 1 #$Z#Y 1. i1 : (#, λ/c, Y ) c 1 #$Z#CY. i : (C, λ/d, Y ) c 1 #$Z#CDY 3. i3 : (#, λ/a, $) c 1 #A$Z#CDY 4. i4 : (A, λ/b, $) c 1 #AB$Z#CDY 5. i5 : (B, λ/y, $) c 1 #ABY $Z#CDY 6. i6 : (λ, c 1 /λ, #) #ABY $Z#CDY (T 1) : Afte the complete simulation of the ule i : AB CD, the template splicing ule is constucted. If the ules of R i = [ i1, i,..., i8 ] ae applied, then the template splicing ules ae poduced fom #ABY $Z#CDY.

14 38 Kalpana Mahalingam, Pithwineel Paul 0. #ABY $Z#CDY 1. i1 : (Z, λ/d 1, #) #ABY $Zd 1 #CDY. i : (B, Y/λ, $) #AB$Zd 1#CDY 3. i3 : (A, B/λ, $) #A$Zd 1#CDY 4. i4 : (#, A/λ, $) #$Zd 1#CDY 5. i5 : (#, C/λ, D) #$Zd 1#DY 6. i6 : (#, D/λ, Y ) #$Zd 1#Y 7. i7 : (λ, λ/c 1, #) c 1 #$Zd 1 #Y 8. i8 : (Z, d 1/λ, #) c 1 #$Z#Y We skip the constuction fo the ules A BC, A a and A λ as they ae simila to those consideed in pevious poofs. We now constuct the necessay otation ules. (ROT 1) : Fo each α N T {B 0 }, we constuct splicing ule #αy $Z#Y fom the template splicing ule c 1 #Y $Z# by application of the ules in R α = [ α1, α,..., α5 ]. 0. c 1 #$Z#Y 1. α1 : (#, λ/e 1, $#) c 1 #e 1 $Z#Y. α : (e 1, λ/α, $) c 1 #e 1 α$z#y 3. α3 : (α, λ/y, $) c 1 #e 1 αy $Z#Y 4. α4 : (#, e 1 /λ, α) c 1 #αy $Z#Y 5. α5 : (λ, c 1 /λ, #) #αy $Z#Y (ROT ) : Now the ule Xα#Z$X# is constucted fom #αy $Z#Y to complete the otation of the symbol α N T {B 0 }. This is possible only afte the application of ules in R α = [ α1, α,..., α10]. 0. #αy $Z#Y 1. α1 : (λ, λ/f 1, #) f 1 #αy $Z#Y. α : (#, α/λ, Y ) f 1 #Y $Z#Y 3. α3 : (#, Y/λ, $) f 1 #$Z#Y 4. α4 : ($, Z/λ, #$) f 1 #$#Y 5. α5 : (#, Y/λ, λ) f 1 #$# 6. α6 : ($, λ/x, #) f 1 #$X# 7. α7 : (#, λ/z, $) f 1 #Z$X# 8. α8 : (f 1, λ/α, Z) f 1 α#z$x# 9. α9 : (f 1, λ/x, α) f 1 Xα#Z$X# 10. α10 : (λ, f 1 /λ, X) Xα#Z$X# (T ) : The ule c 1 #Y $Z# is constucted fom Xα#Z$X# by applying R α = [ α1, α,..., α9].

15 Locally Evolving Splicing Systems Xα#Z$X# 1. α1 : (#, λ/k 1, Z) Xα#k 1 Z$X#. α : ($, λ/x, #) Xα#k 1 Z$# 3. α3 : ($, λ/z, #) Xα#k 1 Z$Z# 4. α4 : (#, λ/y, λ) Xα#k 1 Z$Z#Y 5. α5 : (X, α/λ, #) X#k 1 Z$Z#Y 6. α6 : (X, λ/c 1, #) Xc 1 #k 1 Z$Z#Y 7. α7 : (λ, X/λ, c 1 ) c 1 #k 1 Z$Z#Y 8. α8 : (#, k 1 /λ, Z) c 1 #Z$Z#Y 9. α9 : (#, Z/λ, $) c 1 #$Z#Y Now we constuct splicing ules that emove the makes fom the given sting and teminates the computation pocess. (T E1) : The make X is emoved afte application of the splicing ule XB 0 #$#ZY which is constucted afte application of R B0 = [ B01,..., B07] fom c 1 #Y $Z#. 0. c 1 #$Z#Y 1. B01 : (#, λ/g 1, $) c 1 #g 1 $Z#Y. B0 : ($, Z/λ, #) c 1 #g 1 $#Y 3. B03 : (#, λ/z, Y ) c 1 #g 1 $#ZY 4. B04 : (c 1, λ/b 0, #) c 1 B 0 #g 1 $#ZY 5. B05 : (c 1, λ/x, B 0 ) c 1 XB 0 #g 1 $#ZY 6. B06 : (λ, c 1 /λ, X) XB 0 #g 1 $#ZY 7. B07 : (#, g 1 /λ, $) XB 0 #$#ZY (T E) : Similaly, #Y $ZY # is constucted to emove the make Y. The ules in R B 0 = [ B 01,..., B 09 ] tansfoms XB 0#$#ZY into #Y $ZY #. 0. XB 0 #$#ZY 1. B : (Y, λ/h 01 1, λ) XB 0 #$#ZY h 1. B : (Z, Y/λ, h 0 1) XB 0 #$#Zh 1 3. B : (#, Z/λ, h 03 1) XB 0 #$#h 1 4. B : (#, λ/y, $) 04 XB 0#Y $#h 1 5. B : ($, λ/z, #) 05 XB 0#Y $Z#h 1 6. B : (Z, λ/y, #) 06 XB 0#Y $ZY #h 1 7. B : (X, B 07 0/λ, #) X#Y $ZY #h 1 8. B : (λ, X/λ, #) 08 #Y $ZY #h 1 9. B : (#, h 09 1/λ, λ) #Y $ZY # Since the ules constucted at the end of each section ae same as in the poof of Theoem 3. and A 0 contains the sting XB 0 SY, it follows fom Theoem 3. that L(γ) = L(G). 4 Conclusions In this pape, we have shown that ecusively enumeable languages can be geneated by esticted locally evolving splicing systems with educed contexts of insetiondeletion ules and adius. It was conjectued [9] that the adius as well as the length of the context can be educed to. Seveal esults suppoting this conjectue wee given,

16 384 Kalpana Mahalingam, Pithwineel Paul without completely solving the conjectue, as the esults mentioned in this pape may not be optimal. Refeences [1] CALUDE C. and PAUN G.: Computing with cells and atoms: An intoduction to quantum, DNA and membane computing, CRC Pess, (000). [] CULIK K. II and HARJU T.: Splicing semigoups of dominoes and DNA, Discete Applied Mathematics, 31 (1991), [3] FREUND R., KARI L. and PAUN G.: DNA computing based on splicing: the existence of univesal computes, Theoy of Computing Systems, 3 (1999), [4] JEGANATHAN L. and RAMA R.: Matix splicing systems, Intenational Jounal of Compute Mathematics, 87 (010), [5] KRITHIVASAN K., CHAKARAVARTHY V.T. and RAMA R.: Aay splicing systems, LNCS, 118 (1997), [6] LOOS R.: An altenative definition of splicing, Theoetical Compute Science, 358 (006), [7] PAUN G.: Regula extended H systems ae computationally univesal, Jounal of Automata, Languages and Combinatoics, 1(1996), [8] PAUN G., ROZENBERG G. and SALOMAA A.: Computing by splicing: Pogammed and evolving splicing systems, IEEE Intenational Confeence on Evolutionay Computing, Indianapolis, (1997), [9] PAUN G., ROZENBERG G. and SALOMAA A.: DNA computing. New computing paadigms, Spinge, (1998). [10] PETRE I. and VERLAN S.: Matix insetion-deletion systems, Theoetical Compute Science, 456 (01), [11] PIXTON D. : Regulaity of splicing languages, Discete Applied Mathematics, 69 (1996), [1] RAMA R. and KRISHNA S.N.: Contextual aay splicing systems, SPIRE/CRIWG, (1999), [13] RAMA R. and RAGHAVAN U.: Splicing aay systems, Intenational Jounal of Compute Mathematics, 73 (1999), [14] SALOMAA A.: Fomal languages, Academic Pess, New Yok, (1973). [15] SANTHANAM R. and KRITHIVASAN K.: Gaph splicing systems, Discete Applied Mathematics, 154 (006),