“Towards a characterization of regular
languages generated by finite splicing
systems: where are we?”
Ravello, 19-21 Settembre 2003
Paola Bonizzoni, Giancarlo Mauri
Clelia De Felice, Rosalba Zizza
Dipartimento di Informatica Sistemistica e
Comunicazioni,
Dipartimento di Informatica e Applicazioni,
Univ. of Milano - Bicocca, ITALY
Univ. of Salerno, ITALY
COFIN
auditorium
after this talk
COFIN
auditorium working on
splicing themes
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
y
Example
s| s $ s | t
me s s o, pa s t o
u1 u2
u3 u4
mesto, passo
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
H(F1, F2) = {L=L(SPA) | SPA = (A,I,R), IF1, R  F2,
H(F1, F2)
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
I\R
FIN
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
Model
Language
 Generative process of the language
 Consistency of the model
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]

C(x,L) = C(x’,L)
Context of x and x’
L regular  M (L) finite
syntactic monoid M(L)= A*/ L
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 [x]
[x]
>
q0
>
>
>
>
only here
L=L([x])={y’1x’ y’2 A*|(q0 ,y’1 x’ y’2) F , x’  [x]} finite splicing language
Marker Language
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
Main result 1
(and 2)
[Bonizzoni, De Felice, Mauri, Zizza, DLT03]
Pixton
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
mapping of some pairs of constants into a word
Pixton
Reflexive Paun
splicing languages
FINITE UNION OF
languages containing constants in C

languages containing mixed factorizations of constants
languages containing images of constants
Main result 3
The characterization of Head splicing languages
Reflexive
Paun splicing
languages
Reflexive and
“transitive” Paun
splicing languages
Head splicing
languages
FINITE UNION OF
Head splicing
languages
languages containing constants in C

languages containing “constrained” mixed factorizations of
constants
Theorem L is a regular reflexive splicing language 
T finite subset of N,
Constant language
L is a split-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
CIRCULAR SPLICING
restriction
enzyme 2
restriction
enzyme 1
ligase enzyme
Result
[Bonizzoni, De Felice, Mauri, Zizza 2002]
shorter than
L  a* generated by a
L =L 1  { ag | g  G } +
finite circular splicing system
finite set
(minimal automaton
A
for L)
q0
All regular
languages
>
q1
a
>
q2
>
...
>
> > >
Zn
Decidable property
for A
>
a
subgroup of
>
>
qn’
>
^
length of the closed path = p | n’, p | n
Definition
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)
Theorem
X* star language AND fingerprint closed 
X* generated (by Paun circular splicing)
Example GROUP CODES
2-splicing (1996)
2 : (x u1u2 y, wu3u4 z)
r = u1 | u2 $ u3 | u4 rule
(x u1 u4 z , wu3 u2 y)
1-splicing (1996)
1 : (x u1u2 y, wu3u4 z)
r = u1 | u2 $ u3 | u4 rule
x u1 u4 z
H2 (F1, F2)  H1 (F1, F2)
= ... ?
[Handbook 1996]
Result [Words03]
CONSTANT LANGUAGES (2-splicing): Lc, cL, LcL, cLc (LA* regular, c A) [Head 98]
H1 (Fin,Fin)
H2 (Fin,Fin)
L+c*L,
L+Lc*
L+cLc
L+cL+Ld, L+cL+Ld, L+LcL
Reg
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]
Let us recall our results on circular splicing
1-splicing vs. 2-splicing: separating results [R.Z. & Sergey Verlan, WORDS03]
Example
(aa)*b =L(SPA) , I={b, aab} , R={1| b$ 1| aab}
(aa b , aab) = (aaaab, b)
(aaaa b , aab) = (aaaaaab, b)
Example (reflexive language)
c
q0
a
a
b
c
qF
a
b
c
a
a
a
c
CONSTANT LANGUAGES!
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}
aac*a + caac*ac
NOT (FINITE UNION OF) CONSTANT LANGUAGES!
aac*a + caac*ac + bb + ab + bac*a
REFLEXIVE LANGUAGE