“New results on finite H-systems” Budapest, 29/30 November 2002 Jointly with Paola Bonizzoni, Clelia De Felice, Giancarlo Mauri Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. of Milano - Bicocca, ITALY Dipartimento di Informatica e Applicazioni, Univ. of Salerno, ITALY Partially supported by: - MIUR Project “Formal Languages and Automata: Theory and Applications” - 60% Project “Linguaggi Formali e Modelli di Calcolo” - the contribution of EU Commission under The 5th Framework Programme, Project MolCoNet (IST ) Rosalba Zizza

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

 : (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) Paun’s linear splicing operation (1996) xy wz x w z cut paste y sites Pattern recognition u1u1 u2u2 u3u3 u4u4 u1u1 u2u2 u3u3 u4u4 x u1u1 z u4u4 w u3u3 u2u2 z

(aa)*b =L(S PA ), I={b, aab}, R={1| b$ 1| aab} Example  ( aab, aab ) = (aaaab, b) 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} Paun’s linear splicing system (1996)S PA = (A, I, R) A=finite alphabet; I  A* initial language; R  A*|A*$A*|A* set of rules; [Head, Paun, Pixton, Handbook of Formal Languages, 1996] { L | L=L(S PA ), I, R finite sets }  Regular (aa)*  L( S PA ) { L | L=L(S PA ), I regular, R finite } = Regular (proper subclass) H(F 1, F 2 )

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

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(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)

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

L(w[x])={y’ 1 wx’ y’ 2  L|  (q 0,y’ 1 w x’ y’ 2 )=q F, x’  [x]} finite splicing language w [x] Partial results L=L( A ), A = (A, Q, , q 0,F) minimal only here [Bonizzoni, De Felice, Mauri, Zizza (2002)] q0q0 qFqF > > > > > > deterministic Marker Language Marker w[x] Note that we can -ERASE Locally reversible Hypotheses, -- q F  F

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 Reflexive splicing system S PA = (A, I, R) finite + (reflexive hypothesis on R) [Handbook 1996] Remark Finite Head splicing system Finite Paun splicing system, reflexive and symmetric [Handbook 1996]

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

Marker languages Lemma L is a regular reflexive splicing language   finite splicing system S PA =(A, I, R) s.t. L=L( S PA ) and each site is a constant for L Theorem L is a regular reflexive splicing language  L is a split-language. [Bonizzoni, De Felice, Mauri, Zizza, submitted (2002)] Our result Extend Head’s result Alternative, constructive, effective proof for constant languages Reflexive splicing languages Decidability property Contain some constant languages, but also reflexive splicing languages Not all one-sided contexts

Split-languages 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) 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 t L 2 = L 1 m (j,1) m (j,2) 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*} mtmt m t’