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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
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COFIN auditorium
after this talk
COFIN auditorium working on splicing themes
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Pauns linear splicing operation (1996)
r u1 u2 u3 u4 rule
? (x u1u2 y, wu3u4 z)
(x u1 u4 z , wu3 u2 y)
Pattern recognition
cut
paste
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Example
mesto, passo
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L(SPA) I ? ?(I) ? ?2(I) ? ... ?n?0 ?n(I)
splicing language
H(F1, F2) LL(SPA) SPA (A,I,R), I?F1, R ?
F2, F1, F2 families in the Chomsky
hierarchy
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In the following
Finite linear splicing system SPA ( A, I, R)
with A, I, R finite sets
Characterize regular languages generated by
finite linear Paun splicing systems
Given L regular, can we decide whether L ?
H(FIN,FIN) ?
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Computational power of splicing languages and
regular languages a short survey

• Head 1987 (Bull. Math. Biol.) SLTlanguages
  generated by Null Context splicing systems
  (triples (1,x,1))
• Gatterdam 1992 (SIAM J. of Comp.) specific
  finite Heads splicing systems
• Culik, Harju 1992 (Discr. App. Math.) (Heads)
  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 a1 a1, 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
  ru1 v1 (resp. r1u 1v), u1 u1?R
  (resp. 1u 1u?R)
• Head 1998 (Discr. Appl. Math.) SLThierarchy 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)
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Main Difficulty
Model
Language
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TOOLS Automata Theory

• Minimal Automaton

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(No Transcript)
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Finite Paun splicing system, reflexive and
symmetric
Finite Head splicing system
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• We can decide the above property,
• but only when ALL rules are
  either ru1 v1 or r1u 1v

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Main result 1
Bonizzoni, De Felice, Mauri, Zizza, DLT03
(and 2)
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
FINITE UNION OF
Reflexive Paun splicing languages
languages containing constants in C
?
languages containing mixed factorizations of
constants
languages containing images of constants
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Main result 3
The characterization of Head splicing languages
Head splicing languages
FINITE UNION OF
Head splicing languages
languages containing constants in C
?
languages containing constrained mixed
factorizations of constants
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• Theorem L is a regular reflexive splicing
  language ? L is a split-language.

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
L1m t L2 L1 m(j,1) m(j,2) L2
L1 m(j,1) m(j,2) L2 ? L1m(j,1) m(j,2) L2
L1 m t L2 L1m(j,1) m(j,2) L2
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CIRCULAR SPLICING
restriction enzyme 2
restriction enzyme 1
ligase enzyme
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Result
Bonizzoni, De Felice, Mauri, Zizza 2002
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Star languages
Definition
L ? A star language L closed under the
conjugacy relation and LX, X regular
Fingerprint closed languages
Definition
For any cycle c, L contains the Fingerprint of
c (suitable finite crossing of the closed path
labelled with c)
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... ?
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Result Words03
CONSTANT LANGUAGES (2-splicing) Lc, cL, LcL, cLc
(L?A regular, c ?A) Head 98
H1 (Fin,Fin)
Reg
H2 (Fin,Fin)
LcL, LLc
LcLc
LcLLd, LcLLd, LLcL
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al prossimo COFIN !
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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

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Example
(aa)b L(SPA) , Ib, aab , R1 b 1 aab
?(aa b , aab) (aaaab, b)
?(aaaa b , aab) (aaaaaab, b)
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Example (reflexive language)
c
c
CONSTANT LANGUAGES!
aaca L(SPA) , Iaaa, aaca , Rc 1 1c
caa cac L(SPA) , Icaaac, aaacac ,
Rcaac 1 caa1
aaca caacac
NOT (FINITE UNION OF) CONSTANT LANGUAGES!
aaca caacac bb ab baca
REFLEXIVE LANGUAGE