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(circular)

circular

Paola Bonizzoni, Clelia De Felice, Giancarlo Mauri, Rosalba Zizza

Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. di Milano - Bicocca ITALY

Dipartimento di Informatica e Applicazioni, Univ. di Salerno, ITALY

Circular splicing, definitions

State of the art

Our contributions

Works in progress

We apologize...

<<An important aspect of this year’s meeting can be summed

up us: SHOW ME THE EXPERIMENTAL RESULT! >>

(T. Amenyo, Informal Report on 3rd Annual

DIMACS Workshop on DNA Computing, 1997)

theoretical results

Before Adleman experiment (1994)...

Tom Head 1987 (Bull. of Math. Biology)

“ Formal Language Theory and DNA:

an analysis of the generative capacity of

specific recombinant behaviors”

Unconventional

models of computation

SPLICING

Circular languages: definitions and examples

w, w A*, w~ w w=xy, w = yx

- Conjugacy relation on A*

abaa

Example

abaa, baaa, aaab,aaba are conjugate

- A~ = A* ~ =set of all circular words

~w = [w]~ , w A*

- Circular language

C A ~ set of equivalence classes

A*

A* ~

Cir(L) = {~w | w L} (circularization of L)

L

L

C

(A linearization of C, i.e. Cir(L)=C )

{w A*| ~w C}=Lin(C)

C

(Full linearization of C)

FA~ ={ C A~| L A*, Cir(L) = C, L FA, FA Chomsky hierarchy}

Theorem [Head, Paun, Pixton]

C Reg ~ Lin (C) Reg

(A= finite alphabet, I A~ initial language)

Paun’s definition

SCPA = (A, I, R)

R A* | A* $ A* | A* rules

r = u1| u2 $ u3 | u4 R

~hu1u2,

~ku3u4

A~

u2 hu1

u4ku3

~ u2 hu1 u4ku3

Definition

A circular splicing language C(SCPA) (i.e. a circular language generated

by a splicing system SCPA ) is the smallest circular language containing

I and closed under the application of the rules in R

Other definitions of splicing systems

(A= finite alphabet, I A~ initial language)

Head’s definition

SCH = (A, I, T)

T A* A* A* triples

A~

~hpxq,

~kuxv

(p, x, q ), (u,x,v) T

vkux

~hpx vkux q

qhpx

Pixton’s definition

SCPI = (A, I, R)

R A* A* A* rules

A~

~h,

~h

(, ;), (, ; ) R

~ h h

h

h

Characterize

C(Reg, Fin)

FA~ C(Fin, Fin)

class of circular languages C= C(SCPA) generated by SCPA with I and R both finite sets.

Theorem [ Paun96]

F{Reg~, CF~, RE~}

R +add. hyp. (symmetry, reflexivity, self-splicing)

C(F, Fin) F

Theorem [Pixton95-96]

F{Reg~, CF~, RE~}

R Fin+add. hyp. (symmetry, reflexivity)

C(Reg~, Fin)Reg~,

C(F, Reg) F

Circular finite splicing languages and Chomsky hierarchy

CS~

CF~

C(Fin, Fin)

Reg~

~((aa)*b)

~(an bn)

~(aa)*

I= ~ab ~1, R={a | b $ b | a}

I= ~aa ~1, R={aa | 1 $ 1 | aa}

Our contributions

Reg~ C(Fin, Fin)

C(Fin, Fin)

Fingerprint closed

star languages

Reg~

X*, X regular

group code

cyclic

languages

Cir (X*)

X finite

weak cyclic,

other examples

~ (a*ba*)*

Comparing the three definitions of splicing systems

C(SCH ) C(SCPA ) C(SCPI )

~ (a*ba*)*, ~ ((aa)*b)

= ... ?

Star languages

L A* is star language if L is regular, closed under

conjugacy relation and L=X*, with X regular

Proposition:

SCPA=(A,I,R), I Cir(X*) C(SCPA) Cir (X*)

“Consistence” easily follows!!!

Examples

- (b*(ab*a)*)* = X*

X=b ab*a

X= a*ba*

- (a*ba*)* = X*

q0

q0

y

x

z

x’

y’

z’

Fingerprint closed languages

Definition

For any cycle c, L contains the Fingerprints of c

Fingerprint of a cycle

cnc L

power of the cycle, where the internal cycles are crossed a finite number of times

i n y , j n x

c=(x(y(zz’)jy’)i x’)nc

Fingerprint closed star languages C(Fin,Fin)

Sketch

Take SCPA = (A, I, R) with

I=Cir({successful path containing fingerprint of cycles})

R={1 | 1 $ 1 | ƒ | ƒ fingerprint of cycle c, for any cycle c}

Star languages fingerprint closed

(for example X=b ab*a)

- X*, X regular group code

(for example X=Ad )

- X finite, Cir(X*)

Star languages not fingerprint closed

(a*ba*)*but not generated!!!

Not Star Languages in C(Fin, Fin)

new!

Cyclic Languages

Definition

Cyclic(z) ={(~(z* p)) | p Pref (Lin( ~z))}

Example

z = abc A*

Lin ( ~z) =Lin (~ abc) ={abc, bca,cab}

Pref(Lin ( ~z)) =Pref(Lin (~ abc)) =Pref({abc, bca,cba}) = {a, ab, b, bc, c, ca}

Cyclic(abc)= ~(abc)*a~(abc)*ab

~(abc)*b ~(abc)*bc

~(abc)*c ~(abc)*ca

For any z, |z|>2, z unbordered word, then

Cyclic(z) C(Fin,Fin)

i.e. z uA* A*u

The proof is quite technical ...

Example (continued)

Cyclic (abc) is generated by SCPA = (A,I,R) where I,R are defined as follows

I={~ ((abc)i p | 0 i 3, p Pref(Lin(~ (abc))) }

R={z ab | z $ z | ca z, z ab | z $ z b | c z, z ca | z $ z $ bc z,

z a | z $ z | b z, z b | z $ z $ c z , z c | z $ z | a z }

Other circular regular splicing languages

- ~(abc)*a~(abc)*ab ~(abc)*b ~(abc)*bc ~(abc)*c ~(abc)*ca

~(abc)*ac

Cyclic(abc)

weak cyclic languages

- Cyclic (abca) .... bordered word...

Works in progress

- Characterize Reg~ C(Fin, Fin)

- Characterize FA~ C(Fin, Fin)

- C(SCPI) = Star languages

- Additional hypothesis

- r= u1| u2 $ u3 | u4 in R
- Reflexive: r’ = u1| u2 $ u1| u2
- Symmetric: r” = u3 | u4 $ u1| u2
- Self-splicing: From ~ xu1u2yu3u4 ,
- with r,r” as above, generates ~u4 xu1 , ~u2yu3 .

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