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Developments on linear and circular splicing
Paola Bonizzoni, Clelia De Felice, Giancarlo
Mauri, Rosalba Zizza Dipartimento di Informatica
Sistemistica e Comunicazioni, Univ. di Milano -
Bicocca ITALY Dipartimento di Informatica e
Applicazioni, Univ. di Salerno, ITALY
Bibliography
Circular splicing and regularity (submitted, 2001)
Developments on circular splicing (WORDS01,
Palermo 2001)
On the power of linear and circular splicing
(submitted 2002)
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CIRCULAR SPLICING
Problem 1
Structure of circular regular languages (regular
languages closed under conjugacy relation)
Problem 2
Characterize circular regular languages generated
by finite circular splicing
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Circular languages
(Formal) languages closed under conjugation
Regular
Regular
Circular (Paun) splicing systems
SCPA (A, I, R)
R? A A A A rules
A finite alphabet, I? A initial language,
r u1 u2 u3 u4 ? R
hu1u2 ,
ku3u4
? A
u2 hu1
u4ku3
u2 hu1 u4ku3
In the literature...
Other definitions, other models, additional
hypothesis (on R)
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Contributions
-Reg ? C(Fin, Fin)
Words99, DNA6, Words01, submitted
X, X finite set (X closed under conj.) or X
regular group code
X, X closed under conj. and fingerprint
closed
cyclic and weak cyclic languages
Computational power of (finite) Pixtons systems
(no additional hyp.)
dna6
C(SCH ) ? C(SCPA ) ? C(SCPI ) ? Reg
new!
((A2) ? (A3)) ? Reg \ C(SCPI )

• All known examples of regular circular splicing
  languages
• ? F (a class of languages Pixton generated)

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The case of one-letter alphabet
(Each language on a is closed under conjugacy
relation)
Characterization
L ? a is CPA generated
L L 1 ? (aG )

• L 1 is a finite set

• ? n G is a set of representatives of the
  elements in a subgroup G of Zn

• max m am ?? L 1 lt n min ag ag ? G
  min aG

(extended to uniform languages J ? N, L AJ
? j ? J Aj w? A wj)
Complexity description / minimal splicing system
L ? a CPA generated by I L1 ? aG and R
an 1 1 an
Example
L a 3 , a 4 ? a 6 , a 14, a 16
Ia 3 , a 4 , a 6 , a 14, a 16 Ra6 1
1 a6
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Given L ? a , we CAN NOT DECIDE whether L is
generated by a circular (Paun) splicing
system (Rices theorem)
Theorem Given L ? a , regular , we decide
whether L is generated by a finite circular
(Paun) splicing system
The proof is quite technical... via automata
(frying-pan shape) properties
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Linear (iterated) splicing systems
(A finite alphabet, I? A initial language)
Pauns definition
SPA (A, I, R)
R? A A A A rules
r u1 u2 u3 u4 ? R
? A ,
x u1u2 y,
wu3u4 z
x u1 u4 z , wu3 u2 y
A known result
Fin ? H(Fin, Fin) ? Reg
Head Paun Pixton 1996-
Result P. B. , C. Ferretti, G. M., R.Z.,
IPL 01
Strict inclusion among the three definitions of
(finite) splicing
Problem (HEAD) Can we decide whether a regular
language is generated by a finite splicing system?
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Splicing languages defined by markers M
M w x wx x ? x where q ? Q
?(q , m), m ? M is defined 1 and x
finite or ?x ? x s.t. x cycle
Existence of a (right) marker for L decidible
Trim automaton for L exist y1,y2 s.t. y1 m y2 ?
L
L(M)y ? L yy1 m y2 , y1 ? y1, y1
?y1, m ? M L(S)
x
w