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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 COFIN auditorium working on splicing themes after this talk

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: (x u 1 u 2 y, wu 3 u 4 z) r = u 1 | u 2 $ u 3 | u 4 rule (x u 1 u 4 z, wu 3 u 2 y) Pauns linear splicing operation (1996) cut paste Pattern recognition xy wz sites u1u1 u2u2 u3u3 u4u4 x w z y u1u1 u2u2 u3u3 u4u4 x u1u1 z u4u4 w u3u3 u2u2 y

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Example mesto, passo s| s $ s | t me s s o, pa s t o u2u2 u1u1 u4u4 u3u3

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L(S PA ) = I (I) 2 (I)... = n 0 n (I) splicing language H(F 1, F 2 ) = {L=L(S PA ) | S PA = (A,I,R), I F 1, R F 2, F 1, F 2 families in the Chomsky hierarchy} Pauns linear splicing system (1996)S PA = (A, I, R) A=finite alphabet; I A* initial language; R A*|A*$A*|A* set of rules; { L | L=L(S PA ), I, R finite sets } Regular { L | L=L(S PA ), I regular, R finite } = Regular (aa)* L( S PA ) (proper subclass) [Head, Paun, Pixton, Handbook of Formal Languages, 1996] H(F 1, F 2 )

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Finite linear splicing system: S PA = ( A, I, R) with A, I, R finite sets In the following… Characterize regular languages generated by finite linear Paun splicing systems Problem 1 Problem 2 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.): SLT=languages 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(S H ) Kim 1997 (Cocoon97): given L REG, a finite set of triples X, we can decide whether I L s.t. L= L(S H ) 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(S PA ), S PA simple splicing system. Head 1998 (Computing with Bio-Molecules): given L REG, we can decide whether L= L(S PA ) 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)

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ModelLanguage Generative process of the language Consistency of the model Main Difficulty c z u v v c u u v v Rules for generating... z u cv

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TOOLS: Automata Theory Syntactic Congruence (w.r.t. L) [x] x L x Context of x and x [ w,z A* wxz L wxz L] C(x,L) = C(x,L) 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)=C l (w,L) C r (w,L) Left context Right context

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Partial results [Bonizzoni, De Felice, Mauri, Zizza (2002)] L=L([x])={y 1 x y 2 A*| (q 0,y 1 x y 2 ) F, x [x]} finite splicing language Marker Language L=L( A ), A = (A, Q,, q 0,F) minimal [x] only here q0q0 > > > > > Marker [x]

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u 1 | u 2 $ u 3 | u 4 R u 1 | u 2 $ u 1 | u 2, u 3 | u 4 $ u 3 | u 4 R S PA = (A, I, R) finite + (reflexive hypothesis on R) Reflexive splicing system [Handbook 1996] Finite Head splicing system Finite Paun splicing system, reflexive and symmetricRemark [Handbook 1996]

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We can decide the above property, but only when ALL rules are either r=u|1 $ v|1 or r=1|u $ 1|v L is a regular language generated by a reflexive S PA =(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 [Head, Splicing languages generated by one-sided context (1998)]Theorem Reflexive splicing system [Handbook 1996] L is a reflexive splicing language L=L(S PA ), S PA reflexive splicing system

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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 Reflexive Paun splicing languages languages containing constants in C languages containing mixed factorizations of constants FINITE UNION OF (and 2) Pixton mapping of some pairs of constants into a word Pixton languages containing images of constants [Bonizzoni, De Felice, Mauri, Zizza, DLT03]

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Main result 3 The characterization of Head splicing languages Head splicing languages languages containing constants in C languages containing constrained mixed factorizations of constants FINITE UNION OF Head splicing languages Reflexive Paun splicing languages Reflexive and transitive Paun splicing languages

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T finite subset of N, {m t | m t is a constant for a regular language L, t T} L is a split language L = X t T L( m t ) (j,j) L (j,j) Finite set, s.t. no word in X has m t as a factor Union of constant languages m (j,1) m (j,2) L 1 m t L 2 = L 1 m (j,1) m (j,2) L 2 L 1 m (j,1) m (j,2) L 2 Constant language L(m t ) = {x m t y L| x,y A*} m (j,1) m (j,2) mtmt mtmt Theorem L is a regular reflexive splicing language L is a split-language.

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CIRCULAR SPLICING restriction enzyme 1 restriction enzyme 2 ligase enzyme

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Result [Bonizzoni, De Felice, Mauri, Zizza 2002] L a* generated by a finite circular splicing system L =L 1 { a g | g G } + subgroup of Z n finite set shorter than length of the closed path = p | n, p | n qnqn q0q0 q1q1 q2q2... aa (minimal automaton A for L) All regular languages >>>>>>>>> > > ^ Decidable property for A

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Star languages L A* star language = L closed under the conjugacy relation and L=X*, X regular Definition Fingerprint closed languages For any cycle c, L contains the Fingerprint of c (suitable finite crossing of the closed path labelled with c) Definition X* star language AND fingerprint closed X* generated (by Paun circular splicing) Theorem Example GROUP CODES

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2 : (x u 1 u 2 y, wu 3 u 4 z) r = u 1 | u 2 $ u 3 | u 4 rule (x u 1 u 4 z, wu 3 u 2 y) 2-splicing (1996) 1 : (x u 1 u 2 y, wu 3 u 4 z) r = u 1 | u 2 $ u 3 | u 4 rule x u 1 u 4 z 1-splicing (1996) =... ? H 2 (F 1, F 2 ) H 1 (F 1, F 2 ) [Handbook 1996]

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Result [Words03] H 2 (Fin,Fin) H 1 (Fin,Fin) Reg L+cL+Ld, L+cL+Ld, L+LcL L+cLc L+c*L, L+Lc* CONSTANT LANGUAGES (2-splicing): Lc, cL, LcL, cLc (LA* regular, c A) [Head 98]

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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] 1-splicing vs. 2-splicing: separating results [R.Z. & Sergey Verlan, WORDS03] Let us recall our results on circular splicing

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(aa)*b =L(S PA ), I={b, aab}, R={1| b$ 1| aab} Example ( aa b, aab ) = (aaaab, b) ( aaaa b, aab ) = (aaaaaab, b)

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Example (reflexive language) c c aa c a q0q0 c qFqF aaa bb aac*a =L(S PA ), I={aaa, aaca}, R={c| 1$ 1|c} caa c*ac =L(S PA ), I={caaac, aaacac}, R={caac| 1$ caa|1} CONSTANT LANGUAGES! aac*a + caac*ac NOT (FINITE UNION OF) CONSTANT LANGUAGES! aac*a + caac*ac + bb + ab + bac*a REFLEXIVE LANGUAGE

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