“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 Circular splicing and regularity (submitted, 2001) Developments on circular splicing (WORDS01, Palermo 2001) On the power of linear and circular splicing (submitted 2002) Bibliography:

Problem 1 Problem 2 Characterize circular regular languages generated by finite circular splicing Structure of circular regular languages (regular languages closed under conjugacy relation) CIRCULAR SPLICING

Circular languages(Formal) languages closed under conjugation Regular Circular (Paun) splicing systems SC PA = (A, I, R) R  A* | A* $ A* | A* rules ~hu1u2,~hu1u2, ~ku3u4~ku3u4  A~ A~ r = u 1 | u 2 $ u 3 | u 4  R u 2 hu 1 u4ku3u4ku3 ~ u 2 hu 1 u 4 ku 3 A= finite alphabet, I  A ~ initial language, In the literature... Other definitions, other models, additional hypothesis (on R)

Contributions [Words99, DNA6, Words01, submitted] -Reg ~  C(Fin, Fin) C(SC H )  C(SC PA )  C(SC PI )  ~ Reg ~ ((A 2 )*  (A 3 )*)  ~ Reg \ C(SC PI ) Computational power of (finite) Pixton’s systems (no additional hyp.) dna6 new! All known examples of regular circular splicing languages  F (a class of languages Pixton generated) ~ 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

The case of one-letter alphabet ( Each language on a* is closed under conjugacy relation) L  a* is CPA generatedL =L 1  (a G ) + L 1 is a finite set  n : G is a set of representatives of the elements in a subgroup G’ of Z n max{ m | a m  L 1 } < n = min{ a g | a g  G } = min a G L  a* CPA generated by I = L 1  a G and R= { a n | 1 $ 1 | a n } Example L = { a 3, a 4 }  { a 6, a 14, a 16 } + I={} R={ } I={a 3, a 4, a 6, a 14, a 16 } R={ a 6 | 1 $ 1 | a 6 } Complexity description / minimal splicing system Characterization (extended to uniform languages: J  N, L = A J =  j  J A j = {w  A * | |w|=j})

Given L  a*, we CAN NOT DECIDE whether L is generated by a circular (Paun) splicing system (Rice’s 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

Paun’s definition Linear (iterated) splicing systems (A= finite alphabet, I  A * initial language) S PA = (A, I, R) R  A* | A* $ A* | A* rules x u 1 u 2 y,wu3u4 zwu3u4 z  A*, A*, r = u 1 | u 2 $ u 3 | u 4  R x u 1 u 4 z, wu 3 u 2 y A known result: Fin  H(Fin, Fin)  Reg Problem (HEAD): Can we decide whether a regular language is generated by a finite splicing system? [Head; Paun; Pixton; 1996-] Result: Result: [P. B., C. Ferretti, G. M., R.Z., IPL ‘01] Strict inclusion among the three definitions of (finite) splicing

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 y 1,y 2 s.t. y 1 m y 2  L L(M)={ y  L | y=y’ 1 m y’ 2, y’ 1  [y 1 ], y’ 1  [y 1 ], m  M } = L(S) w [x]