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DNA and splicing (circular) Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. di Milano - Bicocca ITALY Dipartimento di Informatica e Applicazioni, Univ. di Salerno, ITALY Paola Bonizzoni, Clelia De Felice, Giancarlo Mauri, Rosalba Zizza Circular splicing, definitions State of the art Our contributions Works in progress

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Before Adleman experiment (1994)... Tom Head 1987 (Bull. of Math. Biology) Formal Language Theory and DNA : an analysis of the generative capacity of specific recombinant behaviors SPLICING Unconventional models of computation

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SPLICING LINEAR CIRCULAR

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CIRCULAR SPLICING restriction enzyme 1 restriction enzyme 2 ligase enzymes

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Circular languages: definitions and examples Conjugacy relation on A* w, w A*, w ~ w w=xy, w = yx Example abaa, baaa, aaab,aaba are conjugate A ~ = A* ~ = set of all circular words ~ w = [w] ~, w A* Circular language C A ~ set of equivalence classes A* A* ~ L Cir(L) = { ~ w | w L} (circularization of L) C L C {w A*| ~ w C}= Lin(C) (Full linearization of C) (A linearization of C, i.e. Cir(L)=C )

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FA ~ ={ C A ~ | L A*, Cir(L) = C, L FA, FA Chomsky hierarchy} Definition: Theorem [Head, Paun, Pixton] C Reg Lin (C) Reg C Reg ~ Lin (C) Reg

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Pauns definition Circular splicing systems (A= finite alphabet, I A ~ initial language) SC PA = (A, I, R) R A* | A* $ A* | A* rules ~hu1u2,~hu1u2, ~ku3u4~ku3u4 A ~ r = u 1 | u 2 $ u 3 | u 4 R u 2 hu 1 u4ku3u4ku3 ~ u 2 hu 1 u 4 ku 3 Definition I and closed under the application of the rules in R A circular splicing language C(SC PA ) (i.e. a circular language generated by a splicing system SC PA ) is the smallest circular language containing

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Other definitions of splicing systems Heads definitionSC H = (A, I, T) T A* A* A* triples A ~ (p, x, q ), ( u,x,v) T vkux ~ hpx vkux q ~ hpxq, ~ kuxv q hpx (A= finite alphabet, I A ~ initial language) SC PI = (A, I, R) A ~ (, ; ), (, ; ) R ~ h h ~ h, ~ h h Pixtons definition R A* A* A* rules h

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Problem: Theorem [ Paun96] Characterize FA ~ C(Fin, Fin) C(Reg, Fin) class of circular languages C= C(SC PA ) generated by SC PA with I and R both finite sets. F {Reg ~, CF ~, RE ~ } R +add. hyp. (symmetry, reflexivity, self-splicing) Theorem [Pixton95-96] R Fin+add. hyp. (symmetry, reflexivity) C(F, Fin) F F {Reg ~, CF ~, RE ~ } C(F, Reg) FC(Reg ~, Fin) Reg ~,

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Circular finite splicing languages and Chomsky hierarchy CS ~ CF ~ Reg ~ ~ ((aa)*b) ~ (aa)* ~ (a n b n ) I= ~ aa ~ 1, R={aa | 1 $ 1 | aa} I= ~ ab ~ 1, R={a | b $ b | a}

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Our contributions Reg ~ Fingerprint closed star languages X*, X regular group code Cir (X*) X finite cyclic languages weak cyclic, other examples ~ (a*ba*)* Reg ~ C(Fin, Fin)

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Our contributions (continued) Comparing the three definitions of splicing systems C(SC H ) C(SC PA ) C(SC PI ) ~ (a*ba*)*, ~ ((aa)*b) =... ?

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Star languages L A* is star language if L is regular, closed under conjugacy relation and L=X*, with X regular Proposition: SC PA =(A,I,R), I Cir(X*) C(SC PA ) Cir (X*) Consistence easily follows!!! Examples (b*(ab*a)*)* = X* (a*ba*)* = X* X=b ab*a X= a*ba* Definition

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Fingerprint closed languages Definition For any cycle c, L contains the Fingerprints of c Fingerprint of a cycle c n c L power of the cycle, where the internal cycles are crossed a finite number of times c=(x(y(zz) j y) i x) n c i n y, j n x c q0q0 x x y y z z q0q0

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Fingerprint closed star languages C(Fin,Fin) Theorem I=Cir({successful path containing fingerprint of cycles}) R={1 | 1 $ 1 | ƒ | ƒ fingerprint of cycle c, for any cycle c} Star languages not fingerprint closed (a*ba*)* but not generated!!! Star languages fingerprint closed X*, X regular group code X finite, Cir(X*) Sketch Take SC PA = (A, I, R) with (for example X=b ab*a) (for example X=A d )

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Not Star Languages in C(Fin, Fin) new! Definition Cyclic(z) ={( ~ (z* p)) | p Pref (Lin( ~ z))} Example Cyclic(abc)= ~ (abc)*a ~ (abc)*ab ~ (abc)*b ~ (abc)*bc ~ (abc)*c ~ (abc)*ca z = abc A* Lin ( ~ z) =Lin ( ~ abc) ={abc, bca,cab} Pref(Lin ( ~ z)) =Pref(Lin ( ~ abc)) =Pref({abc, bca,cba}) = {a, ab, b, bc, c, ca} Cyclic Languages

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Theorem Cyclic(z) C(Fin,Fin) The proof is quite technical... Example (continued) Cyclic (abc) is generated by SC PA = (A,I,R) where I,R are defined as follows I={ ~ ((abc) i p | 0 i 3, p Pref(Lin( ~ (abc))) } R={z ab | z $ z | ca z, z ab | z $ z b | c z, z ca | z $ z $ bc z, z a | z $ z | b z, z b | z $ z $ c z, z c | z $ z | a z } For any z, |z|>2, z unbordered word, then i.e. z uA* A*u

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Other circular regular splicing languages ~ (abc)*a ~ (abc)*ab ~ (abc)*b ~ (abc)*bc ~ (abc)*c ~ (abc)*ca Cyclic(abc) ~ (abc)*ac weak cyclic languages Cyclic (abca).... bordered word...

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Works in progress Characterize Reg ~ C(Fin, Fin) Characterize FA ~ C(Fin, Fin) C(SC PI ) = Star languages Additional hypothesis r= u 1 | u 2 $ u 3 | u 4 in R Reflexive: r = u 1 | u 2 $ u 1 | u 2 Symmetric: r = u 3 | u 4 $ u 1 | u 2 Self-splicing: From ~ xu 1 u 2 yu 3 u 4, with r,r as above, generates ~ u 4 xu 1, ~ u 2 yu 3.

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