Formal

Languages

Theory of

CodesCombinatorics

on words

Molecular

Computing

Formal

Languages

Molecular

Computing

Theory of

Codes

Combinatorics

on words

THESIS

On the power of classes of

splicing systems

Dottoranda: Rosalba Zizza (XIII ciclo)Supervisori: Prof. Giancarlo Mauri

Prof.ssa Clelia De Felice (Univ. di Salerno)

“ Formal Language Theory and DNA:an analysis of the generative capacity

of specific recombinant behaviors”

SPLICING

Modelli non convenzionali di

calcolo

Tom Head 1987 (Bull. of Math. Biology)

LINEARELINEARE

CIRCOLARECIRCOLARE

Una motivazione generale per lo splicing

Gearchia di Chomsky Splicing systems

RE mT

CS LBA

CF PDA

REG DFAF1 , F2 {FIN , RE, CS, CF,REG}

H(F1 , F2) ; C(F1 , F2 )

1) Processo generativo del linguaggio

2) Prove di consistenza del sistema splicing

LINEAR SPLICING

restriction

enzyme 1

ligase enzyme

restriction

enzyme 2

Paun’s definition

Linear splicing systems Linear splicing systems (A= finite alphabet, I A* initial language)

SPA = (A, I, R) R A* | A* $ A* | A* rules

x u1u2 y, wu3u4 z A*

r = u1 | u2 $ u3 | u4 R

x u1 , u2 y wu3 , u4 z x u1 u4 z , wu3 u2 y

DefinitioDefinitionsns

Splicing language L(SPA) , H(F1, F2)

Some known resultsSome known results [Head; Paun; Pixton; 1996-]

• Fin H(Fin, Fin) Reg

• Fin H(Fin, Reg) Re

• H(Reg, Fin) = Reg

Problem (HEAD)

Can we decide whether a regular language

is generated by a finite splicing system?

[P. Bonizzoni, R.Z., Tech Rep. 254-00 DSI, submitted]

L(SH ) L(SPA ) L(SPI )[P. Bonizzoni, C. Ferretti, G. Mauri, R.Z., Grammar Systems 2000, IPL ‘01]

Comparing the three definitions of (finite) splicing

Theorem

L regular language 0-generated

L generated by finite splicing

Monoide sintattico:

Rappresentazione di L attraverso classi di congruenza

Proprieta’ delle classi di congruenza...

regole splicing

CIRCULAR SPLICING

restriction

enzyme 1

restriction

enzyme 2

ligase enzyme

Circular languages: Circular languages: definitions and definitions and examplesexamples

• Conjugacy relation on A* w, w A*, w ~ w w=xy, w = yx

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* ~

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

CL

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

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

Il nostro “approccio”...

Linguaggi Circolari

Linguaggi Formali chiusi sotto coniugazione

Regolari

Regolari

Paun’s definition

Circular splicing systems Circular splicing systems (A= finite alphabet, I A~ initial language)

SCPA = (A, I, R) R A* | A* $ A* | A* rules

~hu1u2 ,~ku3u4 A~

r = u1 | u2 $ u3 | u4 R

u2 hu1 u4ku3 ~ u2 hu1 u4ku3

Definition

In the literature... In the literature...

Other models, additional hypothesis (on R)

Other definitions of circular splicing

(Head, Pixton)

Splicing language

C(SCPA)

Problem 1

Problem 2

Characterize circular regular languages generated by finite circular

splicing

Structure of circular regular languages (regular languages

closed under conjugacy relation)

Circular finite splicing languages Circular finite splicing languages and Chomsky hierarchyand Chomsky hierarchy

CS~

CF~

Reg~

~((aa)*b)

~(aa)*~(an bn)

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

ContributionsContributions

Reg~

Fingerprint closedstar languages

X*, X regulargroup code

Cir (X*)X finite

cyclic languages

weak cyclic,altri esempi ~ (a*ba*)*

[P. Bonizzoni, C. De Felice, G. Mauri, R.Z., Words99, DNA6 (2000), submitted]-Reg~ C(Fin, Fin)

-Comparing the three def. of circular splicing systems C(SCH ) C(SCPA ) C(SCPI )

“Consistence” easily follows!!!

then the circular language generated by SCPA is ~ X*

The unique problem is the generation

of all words of the language

L A* star language = L regular, closed under

conjugacy relation, L=X*, with X regular

Proposition

Why studying star languages?Why studying star languages?Given SCPA=(A,I,R), if I ~ X*, ~ X* star language

Proposition

Theorem

X* star language AND fingerprint closed

~X* generated (by splicing)

X regular group code. For any automaton A and for any cycle c in A, c X*.

(X* is fingerprint closed)

X* star language, X finite set

~ X* generated (by splicing)

The case of one-letter The case of one-letter alphabetalphabet

Each language on a* is closed under

conjugacy relation

TheoremL a* is CPA generated L = L 1 (aG ) +

• L 1 is a finite set

• n : G is a set of representatives of the elements in a subgroup G’ of Zn

• max{ m | am L 1 } < n = min{ ag | ag G } = min aG

Uniform languages characterization

J N, L = AJ = j J Aj = {w A * | |w|=j}

Complexity description / minimal splicing systemComplexity description / minimal splicing system

TheoremL a* generated by a finite circular (Paun) system, then L is generated by ({a}, I, R) with

I = L1 aG R= { an | 1 $ 1 | an }

Examples• L = { a 3 , a 4 } { a 6 , a 14, a

16 }+

I={I={a 3 , a 4 , a 6 , a 14, a 16 } R={} R={a6 | 1 $ 1 | a6 } }

• L = { a 3 , a 4 , a5 , a7 } {a8 , a9 , a10 , a12 , a13 , a14 , a15 }+

I={I={a3 , a4, a 5, a7, a8, a9, a10, a12 , a13 , a 14, a15 } R={} R={a8 | 1 $ 1 | a8 } }

Given L a* , we CAN NOT DECIDE whether

L is generated by a circular (Paun) splicing system

(Rice’s theorem)

Problem:Problem: Given L a* , regular ,

can we decide whether L is generated by a circular (Paun) splicing system?

Probably YES !!!

L = { a 3 , a 4 } { a 6 , a 14, a 16 }+

Sketch:

G’ = {0, 2, 4} subgroup of Z6

• |Fl |=1 , Fl ={qn }

• p | n :

np

{ a 3 , a 4 , a 6 }

a 11

a 12

G ={6, 14, 16 }

Computational power of Pixton’s systemsComputational power of Pixton’s systems

SCPI = (A, I, R)

A~

(, ; ), (, ; ) R

~ h h

~h ,~ h

h

Pixton’s definition R A* A* A* rules

h

C(SCH ) C(SCPA ) C(SCPI ) ~Reg Remind

Pixtonrecombinant

process

~ ((A2)* (A3)*) ~Reg \ C(SCPI )

F Class of circular regular languages generated by Pixton

• X* generated by regular group codes F• All known examples of regular splicing languages F

• ~ A* \ a+ = ~(a*ba*)* (star free language)

• ~(aa)*b

• ~{w A* | h,k N : |w|a =2h+1, |w|b =2k+1}

• ~(aa)*a, ~(ab)*a ~(ab)*b

(Linear splicing) Inclusion results

(Circular splicing)

Characterization of regular (finite) splicing languages

Fingerprint closed star languages, cyclic languages,

weak cyclic languages, unary languages (CODES)

Pixton systems (subclasses or regular languages)

(Linear splicing) Problems on descriptional complexity

(Formal languages) Characterization of circular regular languages

Unary Languages:

linear splicing vs. circular splicing

(Circular splicing) Pixton systems