Formal Languages Theory of Codes Combinatorics on words Molecular Computing
Mar 27, 2015
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