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Covering problems from a formal language point of view Marcella ANSELMO Maria MADONIA Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 1 M. Anselmo - M. Madonia Covering a word Covering a word w with words in a set X ÎX ÎX ÎX ÎX ÎX w Covering = concatenations +overlaps Example: X = ab+ba w = abababa a b a b a b a Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 2 M. Anselmo - M. Madonia Why study covering ? • Molecular biology: manipulating DNA molecules (e.g. fragment assembly) • Data compression • Computer-assisted music analysis Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 3 M. Anselmo - M. Madonia Literature • Apostolico, Ehrenfeucht (1993) w is ‘quasiperiodic’ • Brodal, Pedersen (2000) • Moore, Smyth (1995) x is a ‘cover’ of w • Iliopulos, Moore, Park (1993) x ‘covers’ w • Iliopulos, Smyth (1998) ‘set of k-covers’ of w • Sim, Iliopulos, Park, Smyth (2001) p ‘approximated (complete references) period’ of w All algorithmic problems!!! (given w find ‘optimal’ X) Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 4 M. Anselmo - M. Madonia Formal language point of view Formal language point of view is needed! Madonia, Salemi, Sportelli (1999) [MSS99]: If X Í A*, X cov = set of words ‘covered’ by words in X also Xcov = (X, A*) set of z-decompositions over (X, A*) , Here: Coverings not simple generalizations of z-decompositions! Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 5 M. Anselmo - M. Madonia Formal Definition red(w) = canonical representative of the class of w in the free group Def. A covering (over X) of w in A* is d =(w1, …, wn) s.t. 1. n is odd; for any odd i, wi Î X for any even i, wi Î 2. red(w1… wn) = w 3. for any i, red(w1…wi) is prefix of w Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 6 M. Anselmo - M. Madonia Example: X = ab+ba w = ababab. d =(ab, , ba, l , ba, , ab) is a covering of w over X • n is odd; for any odd i, wi Î X; for any even i, wi Î * 2. red(ab ba l ba ab) = ababab 3. for any i, red(w1…wi) is prefix of w d: a b a b a b Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 7 M. Anselmo - M. Madonia Concatenation, zig-zag, covering Concatenation submonoid X* Zig-zag z-submonoid X Covering cov-submonoid Xcov cov-submonoid z-submonoid submonoid Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 8 M. Anselmo - M. Madonia Splicing systems for Xcov X, finite S, splicing system s.t. L(S) = # Xcov $ COV2(X) = Start with: # x $, xÎX or COV2(X) Rules: (l, x, $), xÎX (#, x, x3$), x=x1x2, x2x3 ÎX Example: X= ab+ba, w=#ababaab$ Î L(S) #a b a b $ x = ab x = ba #a b a b a $ #a b a $ #a b a b a a b$ #b a a b $ Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 9 M. Anselmo - M. Madonia Coding problems [MSS99] How many coverings has a word? Example: X=ab + ba, w = ababab Î X cov • w has many different coverings over X : d1 =(ab, l , ab, l , ab) d2 =(ab, , ba, l, ba , l , ba, , ab) d3 =(ab, , ba, , ab, l , ab) d4 =(ab, l, ab, , ba, , ab) d5 =(ab, , ba, , ab, , ba, ,ab) Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 10 M. Anselmo - M. Madonia Covering codes [MSS 99] X Í A* is a covering code if any word in A* has at most one minimal covering (over X). Example: X = ab + ba is not a covering code (remember δ1, δ2) Example: X = aabab + abb is a covering code Example: X= ab+a + a is a covering code Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 11 M. Anselmo - M. Madonia Cov - freeness Let M Í A*, cov-submonoid. cov-G(M) is the minimal X Í A* such that M= Xcov. M is cov-free if cov-G(M) is a covering code. Fact: M free M stable (well-known) M z-free M z-stable (known) Question: M cov-free M ‘cov-stable’? We want ‘cov-stability’ = global notion equivalent to cov-freeness. Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 12 M. Anselmo - M. Madonia Toward a cov-stability definition (I) stable u,w,uv,vw Î M implies wÎM z-stable w, vw Î M , uv, u Î Z-p-s(uvw) implies vÎ Z-p-s(uvw) cov-stable? w, vw, uvx, uy Î M, for l£ x <w and l£ y <vw, implies vx Î M ? Not always! Example: X = abcd+bcde+cdef+defg Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 13 M. Anselmo - M. Madonia Toward a cov-stability definition (II) Main observation in the classical proof of (stable implies free): • x minimal word with 2 different factorizations: the last step in a factorization ¹ from the last step in the other factorization New situation with covering: u w So we have to study the case v = l. Example: X = abc + bcd + cde Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 14 M. Anselmo - M. Madonia Cov – stability Def. M is cov-stable if w, vw, uvx, uy Î M, for l£ x < w and l£ y < vw • If v ¹ l, then vz Î M, for some l £ z < w Moreover vx Î M if çy ç< çv ç 2. If v = l, u ¹ l and çx ç> çy ç then t Î M, for some t proper suffix of ux Remark: cov-stable implies stable Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 15 M. Anselmo - M. Madonia Cov-stable iff cov-free Theorem: M covering submonoid. M is cov-stable M is cov-free Proof: many cases and sub-cases (as in definition!) Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 16 M. Anselmo - M. Madonia Some consequences Fact 1: (cov-free Ç cov-free) ¹ cov-free Fact 2: cov-free implies free (not viceversa) Fact 3: cov-free implies very pure (not viceversa) Fact 4: M covering submonoid, X= cov-G(M). M cov-free implies X* free. Fact 5: cov –free z-free free Remark: Covering not simple generalization of z-decomposition! Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 17 M. Anselmo - M. Madonia Cov - maximality and cov-completeness Let X Í A*, covering code. X is cov-complete if Fact(Xcov). X is cov-maximal if X Í X1, covering code X=X1 Fact: X cov-complete X cov-maximal Remark [MSS99]: X cov-complete X infinite (unless X=A) Remark complete cov-complete (not viceversa) maximal cov-maximal (not viceversa) Example: X=ab+a +a Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 18 M. Anselmo - M. Madonia Counting minimal coverings X Í A*, regular language covX : w number of minimal coverings of w X A, 1DFA recognizing X A 1 X B, 2FA recognizing Xcov Remark: B counts all coverings of wÎ Xcov Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 19 M. Anselmo - M. Madonia Remark on minimal coverings Remark: In minimal coverings, no 2 steps to the left under the same occurrence of a letter Crossing sequences in B for minimal coverings of w: w 1 1 1 1 1 1 1 1 1 Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 20 M. Anselmo - M. Madonia A 1NFA automaton for covX CS3 = crossing sequences of length £ 3 and no twice state 1 d(cs,a) =cs’ if cs matches cs’ on a C = (CS3, (1), d, (1) ) a 2 b Example: X = ab + ba, A: 1 4 b 3 a 2 2 3 1 a b 1 a b C: b a 1 1 a b a b 1 3 2 3 Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 21 M. Anselmo - M. Madonia Some remarks • Language recognized by C = X cov • X regular implies X cov regular • Behaviour of C is covX • X regular implies covX rational • X covering code iff C unambiguous (decidable) (different proof in [MSS99]) Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 22 M. Anselmo - M. Madonia Conclusions and future works • Formal language point of view is needed • Covering not generalization of zig-zag (or z-decomposition): many new problems and results •Further problems: ücovering codes: measure üspecial cases: |X| =1, X Í Ak ü suggestions … Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 23 M. Anselmo - M. Madonia x x x x w is‘quasiperiodic’ x is a ‘cover’ of w w x x x x x ‘covers’ w w ÎX ÎX ÎX ÎX ÎX ‘set of k-covers’ X Í Ak of w w Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 24 M. Anselmo - M. Madonia Example: X = ab+ba a b a b a b w = ababab Î Xcov a b a b a b w = ababab Î (X, A*) Xcov = (ab + ba+ aba + bab)* Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 25 M. Anselmo - M. Madonia d 1: a b a b a b d 2: a b a b a b All the steps to the right are needed for covering w: δ1, δ2 are minimal coverings! Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 26 M. Anselmo - M. Madonia d 3: a b a b a b d 4: a b a b a b d 5: a b a b a b All blue steps are useless for covering w : δ3, δ4, δ5 are not minimal. We count only minimal coverings. Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 27 M. Anselmo - M. Madonia Toward a cov-stability definition (I) stable u,w,uv,vw Î M vÎM u v w z-stable w, vw Î M , uv, u Î Z-prefix-strict(uvw) v Î Z -prefix-strict(uvw) u v w Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 28 M. Anselmo - M. Madonia Example: X= abcd+bcde+cdef+defg M=Xcov a b c d e f g vx Set u=ab, v=c, w=defg, x=de, y=cd. Therefore w, vw, uvx, uy Î M but vx =cde Ï M. •Note vz=cdef Î M, l£z<w. Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 29 M. Anselmo - M. Madonia Example: X = abc + bcd + cde M=Xcov w a b c d e u x Set u=ab, v=l , w=cde, x=cd, y=c. Therefore w, vw, uvx, uy Î M but vz Ï M for no l £ z < w. • Note bcdÎ M, bcd proper suffix of ux. Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 30 M. Anselmo - M. Madonia Case 1. u v w v¹l vz Î M çy ç³ çv ç l£z<w x y u v w v¹l vx Î M çy ç< çv ç y x Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 31 M. Anselmo - M. Madonia Case 2. u w v=l çx ç> çy ç u¹l x y t Î M, t proper suffix of ux Ravello 19-21 settembre 2003 Covering Problems from a Formal Language Point of View 32 M. Anselmo - M. Madonia