1. Dipartimento di Informatica e Applicazioni, Univ. of Salerno, ITALY
“Nuovi risultati sui sistemi splicing lineari finiti” Palermo, 13/15 Febbraio 2003
Paola Bonizzoni, Clelia De Felice, Giancarlo Mauri, Rosalba Zizza
Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. of Milano - Bicocca, ITALY
Dipartimento di Informatica e Applicazioni, Univ. of Salerno, ITALY

2. In the following… Characterize regular languages
Finite linear splicing system: SPA = ( A, I, R) with A, I, R finite sets
Characterize regular languages
generated by finite linear Paun splicing systems
Problem 1
Given L regular,
can we decide whether L  H(FIN,FIN) ?
Problem 2

3. Remark SPA = (A, I, R) Reflexive splicing system
[Handbook 1996]
SPA = (A, I, R)
finite + (reflexive hypothesis on R)
u1 | u2 $ u3 | u4 R  u1 | u2 $ u1 | u2 , u3 | u4 $ u3 | u4 R
Remark
[Handbook 1996]
Finite Paun splicing system,
reflexive and symmetric
Finite Head splicing system

4. Reflexive Paun splicing languages
Main result 1
The characterization of reflexive Paun splicing languages
by means of
finite set of (Schutzenberger) constants C
finite set of factorizations of these constants into 2 words
FINITE UNION OF
Reflexive Paun splicing languages
languages containing constants in C 
languages containing mixed factorizations of constants

5. (and 2) Pixton Pixton mapping of some pairs of constants into a word
languages containing images of constants

6. Head splicing languages
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

7. LINEAR SPLICING DNA Strand 2 DNA Strand 1 restriction enzyme
ligase enzyme
ligase enzyme

8. Paun’s linear splicing operation (1996)
r = u1 | u2 $ u3 | u4 rule
: (x u1u2 y, wu3u4 z)
(x u1 u4 z , wu3 u2 y)
sites x u1 u2 y u3 u4 w z
Pattern recognition x u1 u4 z
cut u2 y u3 w
paste u1 u4 u3 u2 x z w y

9. 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}
(aab , aab) = (aaaab, b)
H(F1, F2) = {L=L(SPA) | SPA = (A,I,R), IF1, R  F2, F1, F2 families in the Chomsky hierarchy}
Known results
[Head, Paun, Pixton, Handbook of Formal Languages, 1996]
H(F1, F2)
{ L | L=L(SPA), I regular, R finite } = Regular
{ L | L=L(SPA), I, R finite sets }  Regular
(aa)*  L(SPA)
(proper subclass)

10. Head 2002 Splicing systems: regular languages and below (DNA8)
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 Splicing systems: regular languages and below (DNA8)

11. TOOLS: Automata Theory
Main Difficulty
Rules for generating... c c v’ v v’ u v z u u’ c v u z
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’
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

12. L=L(A ) , A = (A, Q,, q0 ,F) minimal
Partial results
[Bonizzoni, De Felice, Mauri, Zizza (2002)]
L=L(A ) , A = (A, Q,, q0 ,F) minimal
Marker w[x]
deterministic
[x]
> w
> qF
> q0
>
>
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

13. Reflexive splicing system
[Handbook 1996]
L is a reflexive splicing language  L=L(SPA), SPA reflexive splicing system
Theorem
[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

14. Our result
[Bonizzoni, De Felice, Mauri, Zizza]
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 only one-sided contexts
Extend Head’s result
Alternative, constructive, effective proof for constant languages
Reflexive splicing languages
Decidability property
Marker languages
Contain some constant languages, but also reflexive splicing languages

15. Split-languages
T finite subset of N, {mt | mt is a constant for a regular language L, t  T}
Constant language L(mt) = {x mt y L| x,yA*}
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 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)
L’1 m t’ L’2 = L’1m(j’,1) m(j’,2) L’2
mt’