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Paola Bonizzoni, Giancarlo Mauri Dipartimento di Informatica Sistemistica e Comunicazioni,

“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

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Paola Bonizzoni, Giancarlo Mauri Dipartimento di Informatica Sistemistica e Comunicazioni,

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  1. “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

  2. COFIN auditorium after this talk COFIN auditorium working on splicing themes

  3. sites u1 u2 u3 u4 x y w z u1 u4 x z u2 u3 y w u1 u4 u3 u2 y x z w Paun’s linear splicing operation (1996) r = u1| u2 $ u3 | u4 rule : (x u1u2 y, wu3u4z) (x u1 u4z , wu3 u2 y) Pattern recognition cut paste

  4. s| s $ s | t me s s o, pa s t o u2 u1 u3 u4 Example mesto, passo

  5. 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; 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) 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, F1, F2 families in the Chomsky hierarchy}

  6. Problem 1 Problem 2 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) ?

  7. 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 dominolanguages • 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)

  8.  Generative process of the language  Consistency of the model Rules for generating... c c v’ v v’ u v z u u’ c v u z Main Difficulty Model Language

  9. Syntactic Congruence (w.r.t. L) [x] Context of x and x’ x L x’  [ w,z A* wxz  L  wx’z L]  C(x,L) = C(x’,L) L regular  M (L) finite syntactic monoidM(L)= A*/L • 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 TOOLS: Automata Theory • Minimal Automaton

  10. 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

  11. 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

  12. 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|1or r=1|u $ 1|v

  13. [Bonizzoni, De Felice, Mauri, Zizza, DLT03] (and 2) Pixton Main result 1 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

  14. Reflexive Paun splicing languages Reflexive and “transitive” Paun splicing languages The characterization of Head splicing languages Main result 3 Headsplicing languages FINITE UNION OF Head splicing languages languages containing constants in C  languages containing “constrained” mixed factorizations of constants

  15. m(j,1) m(j,2) m(j’,1) m(j’,2) mt’ • TheoremL 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 languageL(mt) = {x mt y L| x,yA*} Lis 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 tL2 = 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 L’1m t’L’2 = L’1m(j’,1) m(j’,2) L’2

  16. CIRCULAR SPLICING restriction enzyme 2 restriction enzyme 1 ligase enzyme

  17. shorter than L  a* generated by a finite circular splicing system L =L 1  { ag | g  G } + finite set subgroup of Zn Decidable property forA (minimal automaton A for L) > qn’ > a a > > > > > > > > > q0 q1 q2 ... ^ All regular languages length of the closed path = p | n’, p | n [Bonizzoni, De Felice, Mauri, Zizza 2002] Result

  18. Theorem X* star language AND fingerprint closed X* generated (by Paun circular splicing) ExampleGROUP CODES Definition Star languages L  A* star language = L closed under the conjugacy relation and L=X*, 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)

  19. 2-splicing (1996) r = u1| u2 $ u3 | u4 rule 2 : (x u1u2 y, wu3u4z) (x u1 u4z , wu3 u2 y) 1-splicing (1996) r = u1| u2 $ u3 | u4 rule 1 : (x u1u2 y, wu3u4z) x u1 u4z H2 (F1, F2)  H1 (F1, F2) [Handbook 1996] = ... ?

  20. Result [Words03] CONSTANT LANGUAGES (2-splicing): Lc, cL, LcL, cLc (LA* regular, c A) [Head 98] H1 (Fin,Fin) Reg H2 (Fin,Fin) L+c*L, L+Lc* L+cLc L+cL+Ld, L+cL+Ld, L+LcL

  21. al prossimo COFIN !

  22. 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]

  23. Example (aa)*b =L(SPA) , I={b, aab} , R={1| b$ 1| aab} (aa b , aab)= (aaaab, b) (aaaa b , aab)= (aaaaaab, b)

  24. c a a a qF q0 b b a a a c Example (reflexive language) c 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

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