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Formal Languages Theory of Codes Combinatorics on words Molecular Computing

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Formal Languages Molecular Computing Theory of Codes Combinatorics on words THESIS On the power of classes of splicing systems Dottoranda: Rosalba Zizza (XIII ciclo) Supervisori: Prof. Giancarlo Mauri Prof.ssa Clelia De Felice (Univ. di Salerno)

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Formal Language Theory and DNA : an analysis of the generative capacity of specific recombinant behaviors SPLICING Modelli non convenzionali di calcolo Tom Head 1987 (Bull. of Math. Biology) LINEARE CIRCOLARE

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Una motivazione generale per lo splicing Gearchia di Chomsky Splicing systems REmT CSLBA CFPDA REGDFA F 1, F 2 {FIN, RE, CS, CF,REG} H(F 1, F 2 ) ; C(F 1, F 2 ) 1) Processo generativo del linguaggio 2) Prove di consistenza del sistema splicing

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LINEAR SPLICING restriction enzyme 1 ligase enzyme restriction enzyme 2

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Pauns definition Linear 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 * r = u 1 | u 2 $ u 3 | u 4 R x u 1, u 2 ywu 3, u 4 zx u 1 u 4 z, wu 3 u 2 y Definitions Splicing language L(S PA ), H(F1, F2) Some known results Some known results [Head; Paun; Pixton; 1996-] Fin H(Fin, Fin) Reg Fin H(Fin, Reg) Re H(Reg, Fin) = Reg

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Problem (HEAD) Can we decide whether a regular language is generated by a finite splicing system? [P. Bonizzoni, R.Z., Tech Rep DSI, submitted] L(S H ) L(S PA ) L(S PI ) [P. Bonizzoni, C. Ferretti, G. Mauri, R.Z., Grammar Systems 2000, IPL 01] Comparing the three definitions of (finite) splicing

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Theorem L regular language 0-generated L generated by finite splicing Monoide sintattico: Rappresentazione di L attraverso classi di congruenza Proprieta delle classi di congruenza... regole splicing

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CIRCULAR SPLICING restriction enzyme 1 restriction enzyme 2 ligase enzyme

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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 ~ L = { ~ w | w L} (circularization of L) CL C {w A*| ~ w C}= Lin(C) (Full linearization of C) (A linearization of C, i.e. ~ L =C)

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Il nostro approccio... Linguaggi Circolari Linguaggi Formali chiusi sotto coniugazione Regolari

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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 In the literature... Other models, additional hypothesis (on R) Other definitions of circular splicing (Head, Pixton) Splicing language C(SC PA )

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

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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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Contributions Reg ~ Fingerprint closed star languages X*, X regular group code Cir (X*) X finite cyclic languages weak cyclic, altri esempi ~ (a*ba*)* [P. Bonizzoni, C. De Felice, G. Mauri, R.Z., Words99, DNA6 (2000), submitted] -Reg ~ C(Fin, Fin) -Comparing the three def. of circular splicing systems C(SC H ) C(SC PA ) C(SC PI )

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Consistence easily follows!!! then the circular language generated by SC PA is ~ X* The unique problem is the generation of all words of the language L A* star language = L regular, closed under conjugacy relation, L=X*, with X regular Proposition Why studying star languages? Given SC PA = (A,I,R), if I ~ X*, ~ X* star language

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Proposition Theorem X* star language AND fingerprint closed ~ X* generated (by splicing) X regular group code. For any automaton A and for any cycle c in A, c X*. (X* is fingerprint closed) X* star language, X finite set ~ X* generated (by splicing)

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The case of one-letter alphabet Each language on a* is closed under conjugacy relation Theorem 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 Uniform languages characterization J N, L = A J = j J A j = {w A * | |w|=j}

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Complexity description / minimal splicing system Theorem L a* generated by a finite circular (Paun) system, then L is generated by ({a}, I, R) with I = L 1 a G R= { a n | 1 $ 1 | a n } Examples 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 } L = { a 3, a 4, a 5, a 7 } {a 8, a 9, a 10, a 12, a 13, a 14, a 15 } + I={} R={ } I={a 3, a 4, a 5, a 7, a 8, a 9, a 10, a 12, a 13, a 14, a 15 } R={ a 8 | 1 $ 1 | a 8 }

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Given L a*, we CAN NOT DECIDE whether L is generated by a circular (Paun) splicing system (Rices theorem) Problem: Problem: Given L a*, regular, can we decide whether L is generated by a circular (Paun) splicing system? Probably YES !!!

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L = { a 3, a 4 } { a 6, a 14, a 16 } + Sketch: G = {0, 2, 4} subgroup of Z 6 |F l |=1, F l ={q n } p | n : n p { a 3, a 4, a 6 } a 11 a 12 G ={6, 14, 16 }

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Computational power of Pixtons systems Computational power of Pixtons systems SC PI = (A, I, R) A ~ (, ; ), (, ; ) R ~ h h ~ h, ~ h h Pixtons definition R A* A* A* rules h C(SC H ) C(SC PA ) C(SC PI ) ~ Reg Remind Pixton recombinant process ~ ((A 2 )* (A 3 )*) ~ Reg \ C(SC PI )

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F Class of circular regular languages generated by Pixton X* generated by regular group codes F All known examples of regular splicing languages F ~ A* \ a+ = ~ (a*ba*)* (star free language) ~ (aa)*b ~ {w A* | h,k N : |w| a =2h+1, |w| b =2k+1} ~ (aa)*a, ~ (ab)*a ~ (ab)*b

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(Linear splicing) Inclusion results (Circular splicing) Characterization of regular (finite) splicing languages Fingerprint closed star languages, cyclic languages, weak cyclic languages, unary languages (CODES) Pixton systems (subclasses or regular languages) (Linear splicing) Problems on descriptional complexity (Formal languages) Characterization of circular regular languages Unary Languages: linear splicing vs. circular splicing (Circular splicing) Pixton systems

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