Transcript Nessun titolo diapositiva - Dipartimento Di Informatica

“New results on finite H-systems”
Budapest, 29/30 November 2002
Rosalba Zizza
Jointly with
Paola Bonizzoni, Clelia De Felice, Giancarlo Mauri
Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. of Milano - Bicocca, ITALY
Dipartimento di Informatica e Applicazioni, Univ. of Salerno, ITALY
Partially supported by:
- MIUR Project “Formal Languages and Automata: Theory and Applications”
- 60% Project “Linguaggi Formali e Modelli di Calcolo”
- the contribution of EU Commission under The 5th Framework Programme, Project MolCoNet (IST-2001-32008)
LINEAR SPLICING
DNA Strand 1
restriction
enzyme
ligase enzyme
DNA Strand 2
restriction
enzyme
ligase enzyme
Paun’s linear splicing operation (1996)
: (x u1u2 y, wu3u4 z)
r = u1 | u2 $ u3 | u4 rule
(x u1 u4 z , wu3 u2 y)
sites
x
u1
x
u1
u2
x
w
u3
u4
z
u4
u2
u1
y
u4
y
z
w
u3
w
u3
u2
z
Pattern
recognition
cut
paste
z
Paun’s linear splicing system (1996)
SPA = (A, I, R)
A=finite alphabet; I A* initial language; RA*|A*$A*|A* set of rules;
L(SPA) = I  (I)  2(I)  ... = n0 n(I) splicing language
Example
(aa)*b =L(SPA) , I={b, aab} , R={1| b$ 1| aab}
H(F1, F2) = {L=L(SPA) | SPA = (A,I,R), IF1, R  F2,
H(F1, F2)
I\R
FIN
REG
LIN
CF
CS
RE
FIN FIN,REG
FIN,RE
FIN,RE
FIN,RE
FIN,RE
FIN,RE
REG
REG
REG,RE
REG,RE
REG,RE
REG,RE
REG,RE
LIN
LIN,CF
LIN,RE
LIN,RE
LIN,RE
LIN,RE
LIN,RE
CF
CF
CF,RE
CF,RE
CF,RE
CF,RE
CF,RE
CS
CS,RE
CS,RE
CS,RE
CS,RE
CS,RE
CS,RE
RE
RE
RE
RE
RE
RE
RE
(aab , aab) = (aaaab, b)
F1, F2 families in the Chomsky hierarchy}
[Head, Paun, Pixton,
Handbook of Formal Languages, 1996]
{ L | L=L(SPA), I regular, R finite } = Regular
{ L | L=L(SPA), I, R finite sets }  Regular
(aa)*  L(SPA)
(proper subclass)
In the following…
Finite linear splicing system: SPA = ( A, I, R) with A, I, R finite sets
Problem 1
Problem 2
Characterize regular languages
generated by finite linear Paun splicing systems
Given L regular,
can we decide whether L  H(FIN,FIN) ?
Computational power of splicing languages and
regular languages: a short survey…
 Head 1987 (Bull. Math. Biol.): SLT=languages generated by Null Context splicing systems (triples (1,x,1))
 Gatterdam 1992 (SIAM J. of Comp.): specific finite Head’s splicing systems
 Culik, Harju 1992 (Discr. App. Math.): (Head’s) splicing and domino languages
 Kim 1997 (SIAM J. of Comp.): from the finite state automaton recognizing I to the f.s.a. recognizing L(SH)
 Kim 1997 (Cocoon97): given LREG, a finite set of triples X, we can decide whether  IL s.t. L= L(SH)
Pixton 1996 (Theor. Comp. Sci.): if F is a full AFL, then H(FA,FIN)  FA
Mateescu, Paun, Rozenberg, Salomaa 1998 (Discr. Appl. Math.): simple splicing systems
(all rules a|1 $ a|1, aA); we can decide whether LREG, L= L(SPA ), SPA simple splicing system.
Head 1998 (Computing with Bio-Molecules): given LREG, we can decide whether L= L(SPA ) with
“special” one sided-contexts rR: r=u|1 $ v|1 (resp. r=1|u $ 1|v), u|1 $ u|1R (resp. 1|u $ 1|uR)
Head 1998 (Discr. Appl. Math.): SLT=hierarchy of simple splicing systems
Bonizzoni, Ferretti, Mauri, Zizza 2001 (IPL): Strict inclusion among finite splicing systems
Head 2002 Splicing systems: regular languages and below (DNA8)
Main Difficulty
Rules for generating...
c
z
u
c
v’
v
z
u
u
u’
c
v
v’
v
TOOLS: Automata Theory
Syntactic Congruence (w.r.t. L) [x]
x L x’ 
[ w,z A* wxz  L  wx’z L]
syntactic monoid M(L)= A*/ L

C(x,L) = C(x’,L)
Context of x and x’
L regular  M (L) finite
Minimal Automaton
Constant [Schützenberger, 1975]
w  A* is a CONSTANT for a language L if C(w,L)=Cl (w,L)  Cr (w,L)
Left context
Right context
Partial results
[Bonizzoni, De Felice, Mauri, Zizza (2002)]
L=L(A ) , A = (A, Q,, q0 ,F) minimal
Marker w[x]
deterministic
[x]
>
q0
w
>
>
>
>
qF
only here
>
L(w[x])={y’1wx’ y’2 L|(q0 ,y’1 w x’ y’2)=qF, x’  [x]} finite splicing language
Marker Language
Note that we can
-ERASE Locally reversible Hypotheses,
-- qF  F
Reflexive splicing system
SPA = (A, I, R)
u1 | u2 $ u3 | u4 R
Remark
[Handbook 1996]
finite + (reflexive hypothesis on R)

u1 | u2 $ u1 | u2 , u3 | u4 $ u3 | u4 R
[Handbook 1996]
Finite Head splicing system
Finite Paun splicing system,
reflexive and symmetric
Reflexive splicing system
L is a reflexive splicing language
Theorem

[Handbook 1996]
L=L(SPA), SPA reflexive splicing system
[Head, Splicing languages generated by one-sided context (1998)]
L is a regular language generated by a reflexive SPA=(A, I, R) , where
rR: r=u|1 $ v|1 (resp. r=1|u $ 1|v)

 finite set of constants F for L s.t. the set L\ {A*cA* : c  F} is finite
We can decide the above property,
but only when ALL rules are either r=u|1 $ v|1 or
r=1|u $ 1|v
Our result
[Bonizzoni, De Felice, Mauri, Zizza, submitted (2002)]
Lemma L is a regular reflexive splicing language  finite splicing system
SPA=(A, I, R) s.t. L=L(SPA) and
each site is a constant for L
Theorem L is a regular reflexive splicing language 
L is a split-language.
Not all one-sided contexts
Extend Head’s result
Reflexive splicing
languages
Alternative, constructive, effective
proof for constant languages
Decidability property
Contain some constant languages, but
also reflexive splicing languages
Marker languages
Split-languages
T finite subset of N,
Constant language
{mt | mt is a constant for a regular language L, t  T}
L(mt) = {x mt y L| x,yA*}
L is a split language

L = X

Finite set, s.t. no word in X
has mt as a factor
t  T L(mt)

(j,j’)L(j,j’)
Union of
constant languages
mt
m(j,1)
m(j,2)
L1m t L2 = L1 m(j,1) m(j,2) L2
L1 m(j,1) m(j’,2) L’2 
L’1m(j’,1) m(j,2) L2
m(j’,1)
m(j’,2)
mt’
L’1 m t’ L’2 = L’1m(j’,1) m(j’,2) L’2