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“Towards a characterization of regular languages generated by finite splicing
systems: where are we?”
Ravello, 19-21 Settembre 2003
Paola Bonizzoni, Giancarlo Mauri
Dipartimento di Informatica Sistemistica e Comunicazioni,
Univ. of Milano - Bicocca, ITALY
Clelia De Felice, Rosalba Zizza
Dipartimento di Informatica e Applicazioni,
Univ. of Salerno, ITALY
COFINauditorium
COFINauditorium working on
splicing themes
after this talk
: (x u1u2 y, wu3u4 z)r = u1 | u2 $ u3 | u4 rule
(x u1 u4 z , wu3 u2 y)
Paun’s linear splicing operation (1996)
cut
paste
Pattern recognition
x y w z
sites
u1 u2 u3 u4
xw
zy
u1
u2 u3
u4
x u1 zu4 w u3 u2 y
Example
mesto, passo s| s $ s | t
me s s o, pa s t ou2u1 u4u3
L(SPA) = I (I) 2(I) ... = n0 n(I) splicing language
H(F1, F2) = {L=L(SPA) | SPA = (A,I,R), IF1, R F2, F1, F2 families in the Chomsky hierarchy}
Paun’s linear splicing system (1996) SPA = (A, I, R)
A=finite alphabet; I A* initial language; RA*|A*$A*|A* set of rules;
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
{ L | L=L(SPA), I, R finite sets } Regular
{ L | L=L(SPA), I regular, R finite } = Regular
(aa)* L(SPA) (proper subclass)
[Head, Paun, Pixton,Handbook of Formal Languages, 1996]H(F1, F2)
Finite linear splicing system: SPA = ( A, I, R) with A, I, R finite sets
In the following…In the following…
Characterize regular languages generated by finite linear Paun splicing
systemsProblem 1
Problem 2Given 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 LREG, a finite set of triples X, we can decide whether IL 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, aA); we can decide whether LREG, L= L(SPA ), SPA simple splicing system.
Head 1998 (Computing with Bio-Molecules): given LREG, we can decide whether L= L(SPA )
with
“special” one sided-contexts rR: r=u|1 $ v|1 (resp. r=1|u $ 1|v), u|1 $ u|1R (resp. 1|u $ 1|uR)
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)
Model Language
Generative process of the language
Consistency of the model
Main DifficultyMain Difficulty
c
z u v v’ c
u’ u v v’
Rules for generating...
z u c v
TOOLS: Automata TheoryTOOLS: Automata Theory TOOLS: Automata TheoryTOOLS: Automata Theory
Syntactic Congruence (w.r.t. L) [x]
x L x’ Context of x and x’[ w,z A* wxz L wx’z L] C(x,L) = C(x’,L)
syntactic monoid M(L)= A*/ L 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 resultsPartial results[Bonizzoni, De Felice, Mauri, Zizza (2002)]
L=L([x])={y’1x’ y’2 A*|(q0 ,y’1 x’ y’2) F , x’ [x]} finite splicing
language
Marker Language
L=L(A ) , A = (A, Q,, q0 ,F) minimal
[x]
only here
q0
>
>
>>
>
Marker [x]
u1 | u2 $ u3 | u4 R u1 | u2 $ u1 | u2 , u3 | u4 $ u3 | u4 R
SPA = (A, I, R) finite + (reflexive hypothesis on R)
Reflexive splicing system[Handbook 1996]
Finite Head splicing systemFinite Paun splicing system,
reflexive and symmetric
RemarkRemark[Handbook 1996]
We can decide the above property,
but only when ALL rules are either r=u|1 $ v|1 or r=1|u $ 1|v
L is a regular language generated by a reflexive SPA=(A, I, R) , where
rR: 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
[Head, Splicing languages generated by one-sided context (1998)]TheoremTheorem
Reflexive splicing system [Handbook 1996]
L is a reflexive splicing language L=L(SPA), SPA reflexive splicing system
Main result 1
The characterization of reflexive Paun splicing languages
structure described by means of
• finite set of (Schutzenberger) constants C
• finite set of factorizations of these constants into 2 words
Reflexive Paun splicing
languages
languages containing constants in C
languages containing mixed factorizations of constants
FINITE UNION OF
(and 2) Pixton
mapping of some pairs of constants into a word
Pixton
languages containing images of constants
[Bonizzoni, De Felice, Mauri, Zizza, DLT03]
Main result 3 The characterization of Head splicing languages
Head splicing languages
languages containing constants in C
languages containing “constrained” mixed factorizations of constants
FINITE UNION OF
Head splicing
languages
Reflexive Paun splicing languages
Reflexive and “transitive” Paun splicing languages
T finite subset of N, {mt | mt is a constant for a regular language L,
t T}
L is a split language L = X t T L(mt) (j,j’)L(j,j’)
Finite set, s.t. no word in X has mt as a factor
Union of constant languages
m(j’,1) m(j’,2) L’1 m t’ L’2 = L’1m(j’,1) m(j’,2) L’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
Constant language L(mt) = {x mt y L| x,yA*}
m(j,1) m(j,2)
mt
mt’
Theorem L is a regular reflexive splicing language L is a split-language.
CIRCULAR SPLICING
restriction enzyme 1
restriction enzyme 2
ligase enzyme
ResultResult [Bonizzoni, De Felice, Mauri, Zizza 2002]
L a* generated by a
finite circular splicing systemL =L 1 { ag | g G } +
subgroup of Znfinite set
shorter than
length of the closed path = p | n’, p | n
qn’
q0 q1 q2 ...a a
(minimal automaton A for
L)
All regular languages
> >> > > > > > >
> >
^
Decidable property for A
Star languages
L A* star language = L closed under the conjugacy relation and L=X*, X regular
Definition
Fingerprint closed languages
For any cycle c, L contains the Fingerprint of c
(“suitable” finite crossing of the closed path labelled with c)
Definition
X* star language AND fingerprint closed
X* generated (by Paun circular splicing)
Theorem
Example GROUP CODES
2 : (x u1u2 y, wu3u4 z)r = u1 | u2 $ u3 | u4 rule
(x u1 u4 z , wu3 u2 y)
2-splicing (1996)
1 : (x u1u2 y, wu3u4 z)r = u1 | u2 $ u3 | u4 rule
x u1 u4 z
1-splicing (1996)
= ... ?
H2 (F1, F2) H1 (F1, F2) [Handbook 1996]
Result [Words03]
H2 (Fin,Fin)
H1 (Fin,Fin) Reg
L+cL+Ld, L+cL+Ld, L+LcL L+cLc
L+c*L, L+Lc*
CONSTANT LANGUAGES (2-splicing): Lc, cL, LcL, cLc (LA* regular, c A) [Head 98]
al prossimo
COFIN !
Outline of the talk (and of the research steps…)
Let us recall the splicing
operation Let us manage splicing languages
Let us understand the “crux” of splicing languages
Let us construct reflexive splicing languages [DLT03]
1-splicing vs. 2-splicing: separating results [R.Z. & Sergey Verlan,
WORDS03]
Let us recall our results on circular splicing
(aa)*b =L(SPA) , I={b, aab} , R={1| b$ 1| aab}
Example
(aa b , aab) = (aaaab, b)
(aaaa b , aab) = (aaaaaab, b)
Example (reflexive language)
c c
a a
c
a
q0
c
qFa a a
b b
aac*a =L(SPA) , I={aaa, aaca} , R={c| 1$ 1|c}
caa c*ac =L(SPA) , I={caaac, aaacac} , R={caac| 1$ caa|1}
CONSTANT LANGUAGES!
aac*a + caac*ac NOT (FINITE UNION OF) CONSTANT LANGUAGES!
aac*a + caac*ac + bb + ab + bac*a
REFLEXIVE LANGUAGE