Dipartimento di Informatica e Applicazioni, Univ. of Salerno, ITALY “Nuovi risultati sui sistemi splicing lineari finiti” Palermo, 13/15 Febbraio 2003 Paola Bonizzoni, Clelia De Felice, Giancarlo Mauri, Rosalba Zizza Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. of Milano - Bicocca, ITALY Dipartimento di Informatica e Applicazioni, Univ. of Salerno, ITALY

In the following… Characterize regular languages Finite linear splicing system: SPA = ( A, I, R) with A, I, R finite sets Characterize regular languages generated by finite linear Paun splicing systems Problem 1 Given L regular, can we decide whether L  H(FIN,FIN) ? Problem 2

Remark SPA = (A, I, R) 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

Reflexive Paun splicing languages Main result 1 The characterization of reflexive Paun splicing languages by means of finite set of (Schutzenberger) constants C finite set of factorizations of these constants into 2 words FINITE UNION OF Reflexive Paun splicing languages languages containing constants in C  languages containing mixed factorizations of constants

(and 2) Pixton Pixton mapping of some pairs of constants into a word languages containing images of constants

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

LINEAR SPLICING DNA Strand 2 DNA Strand 1 restriction enzyme ligase enzyme ligase enzyme

Paun’s linear splicing operation (1996) r = u1 | u2 $ u3 | u4 rule : (x u1u2 y, wu3u4 z) (x u1 u4 z , wu3 u2 y) sites x u1 u2 y u3 u4 w z Pattern recognition x u1 u4 z cut u2 y u3 w paste u1 u4 u3 u2 x z w y

Paun’s linear splicing system (1996) SPA = (A, I, R) A=finite alphabet; I A* initial language; RA*|A*$A*|A* set of rules; L(SPA) = I  (I)  2(I)  ... = n0 n(I) splicing language Example (aa)*b =L(SPA) , I={b, aab} , R={1| b$ 1| aab} (aab , aab) = (aaaab, b) H(F1, F2) = {L=L(SPA) | SPA = (A,I,R), IF1, R  F2, F1, F2 families in the Chomsky hierarchy} 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)

Head 2002 Splicing systems: regular languages and below (DNA8) 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 domino languages 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)

TOOLS: Automata Theory Main Difficulty Rules for generating... c c v’ v v’ u v z u u’ c v u z TOOLS: Automata Theory Syntactic Congruence (w.r.t. L) [x] x L x’  [ w,z A* wxz  L  wx’z L]  C(x,L) = C(x’,L) Context of x and x’ 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)=Cl (w,L)  Cr (w,L) Left context Right context

L=L(A ) , A = (A, Q,, q0 ,F) minimal Partial results [Bonizzoni, De Felice, Mauri, Zizza (2002)] L=L(A ) , A = (A, Q,, q0 ,F) minimal Marker w[x] deterministic [x] > w > qF > q0 > > only here > L(w[x])={y’1wx’ y’2 L|(q0 ,y’1 w x’ y’2)=qF, x’  [x]} finite splicing language Marker Language Note that we can ERASE Locally reversible Hypotheses, - qF  F

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

Our result [Bonizzoni, De Felice, Mauri, Zizza] Lemma L is a regular reflexive splicing language  finite splicing system SPA=(A, I, R) s.t. L=L(SPA) and each site is a constant for L Theorem L is a regular reflexive splicing language  L is a split-language. Not only one-sided contexts Extend Head’s result Alternative, constructive, effective proof for constant languages Reflexive splicing languages Decidability property Marker languages Contain some constant languages, but also reflexive splicing languages

Split-languages T finite subset of N, {mt | mt is a constant for a regular language L, t  T} Constant language L(mt) = {x mt y L| x,yA*} L is 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 m(j,1) m(j,2) L1m t L2 = 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 m(j’,1) m(j’,2) L’1 m t’ L’2 = L’1m(j’,1) m(j’,2) L’2 mt’