1 / 32

Covering problems from a formal language point of view

Covering problems from a formal language point of view. Marcella ANSELMO Maria MADONIA.  X.  X.  X.  X.  X. w. Covering a word. Covering a word w with words in a set X. Covering = concatenations +overlaps Example : X = ab+ba w = abababa. a b a b a b a.

simone-randall
Télécharger la présentation

Covering problems from a formal language point of view

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Covering problemsfroma formal language point of view Marcella ANSELMO Maria MADONIA Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  2. X X X X X w Covering a word Covering a word w with words in a set X Covering = concatenations +overlaps Example: X = ab+ba w = abababa a b a b a b a Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  3. Why study covering ? • Molecular biology: • manipulating DNA molecules (e.g. fragment assembly) • Data compression • Computer-assisted music analysis Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  4. Apostolico, Ehrenfeucht (1993) • Brodal, Pedersen (2000) w is ‘quasiperiodic’ • 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 Literature All algorithmic problems!!! (given w find ‘optimal’ X) Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  5. If X  A*, X cov = set of words ‘covered’ by words in X Formal language point of view Formal language point of view is needed! Madonia, Salemi, Sportelli (1999) [MSS99]: also Xcov = (X, A*), set of z-decompositions over (X, A*) Here: Coverings not simple generalizations of z-decompositions! Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  6. red(w) = canonical representative of the class of w in the free group Def. A covering (over X) of w in A* is  =(w1, …, wn) s.t. 1. n is odd; for any odd i, wi  X for any even i, wi  2. red(w1… wn) = w 3. for any i, red(w1…wi) is prefix of w Formal Definition Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  7.  =(ab,,ba,  , ba, , ab) is a covering of w over X • n is odd; for any odd i, wi  X; • for any even i, wi  * • 2. red(abba  ba ab) = ababab • 3. for any i, red(w1…wi) is prefix of w a b a b a b : Example: X= ab+baw= ababab. Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  8. Concatenation Zig-zag Covering Concatenation, zig-zag, covering submonoid X* z-submonoid X cov-submonoid Xcov cov-submonoid z-submonoid submonoid Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  9. COV2(X) = Start with:  x $, xX or COV2(X) Rules: (, x, $), xX (, x, x3$), x=x1x2, x2x3 X x = ba a b a b $ x = ab b a a b $ a b a $ Splicing systems for Xcov X, finite S, splicing system s.t. L(S) =  Xcov$ Example: X= ab+ba, w=#ababaab$  L(S) a b a b a $ a b a b a a b$ Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  10. 1=(ab,  ,ab,  , ab) 2=(ab,,ba, , ba ,  , ba, , ab) 3=(ab, ,ba, , ab,  , ab) 4=(ab, ,ab, , ba, , ab) 5=(ab, ,ba, , ab, , ba, ,ab) Coding problems [MSS99] How many coverings has a word? • Example: X=ab + ba, w = ababab Xcov • w has many different coverings over X: Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  11. 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 Covering codes [MSS 99] X A* is a covering code if any word in A* has at most one minimal covering (over X). Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  12. Let M  A*, cov-submonoid. cov-G(M) is the minimal X  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’? Cov - freeness We want ‘cov-stability’ = global notion equivalent to cov-freeness. Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  13. stable u,w,uv,vwM implies wM z-stable w, vwM , uv, uZ-p-s(uvw) implies vZ-p-s(uvw) cov-stable? Not always! Example: X = abcd+bcde+cdef+defg Toward a cov-stability definition (I) w, vw, uvx, uyM, for  x<w and  y<vw, implies vx M ? Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  14. Main observation in the classical proof of (stable implies free): • x minimal word with 2 different factorizations: • the last step in a factorization  from the last step • in the other factorization u w Example: X = abc + bcd + cde Toward a cov-stability definition (II) New situation with covering: So we have to study the case v = . Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  15. If v, then vz  M, for some   z w • Moreover vx  M if y v 2. If v=, u and x y then t  M, for some t propersuffix of ux Cov – stability Def. M is cov-stable if w, vw, uvx, uyM, for  x w and  y vw Remark: cov-stable implies stable Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  16. Theorem: M covering submonoid. M is cov-stable M is cov-free Cov-stable iff cov-free Proof: many cases and sub-cases (as in definition!) Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  17. Fact 5: cov –free z-free free Some consequences Fact 1: (cov-free  cov-free)  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. Remark: Covering not simple generalization of z-decomposition! Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  18. Let X  A*, covering code. X is cov-complete if Fact(Xcov). X is cov-maximal if X  X1, covering code X=X1 Remark[MSS99]: X cov-completeX infinite (unless X=A) Fact: X cov-complete X cov-maximal Remark complete cov-complete (not viceversa) maximal cov-maximal (not viceversa) Cov - maximality and cov-completeness Example: X=ab+a +a Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  19. X  A*, regular language covX : w number of minimal coverings of w X X A 1 Counting minimal coverings A, 1DFA recognizing X B, 2FA recognizing Xcov Remark: B counts all coverings of w Xcov Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  20. w 1 1 1 1 1 1 1 1 1 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: Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  21. b a 2 4 Example: X = ab + ba, A : 1 b 3 a 2 2 1 3 1 a b a b C : b a 1 b a 1 3 a b 1 2 3 A 1NFA automaton for covX CS3 = crossing sequences of length  3 and no twice state 1 (cs,a) =cs’ if cs matches cs’ on a C= (CS3, (1), , (1) ) Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  22. 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]) Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  23. 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: • covering codes: measure • special cases: |X| =1, XAk • suggestions … Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  24. x x x x w x x x x w X X X X X w w is‘quasiperiodic’ x is a ‘cover’ of w x ‘covers’ w X Ak ‘set of k-covers’ of w Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  25. a b a b a b Example: X = ab+ba w = ababab Xcov a b a b a b w = ababab (X, A*) Xcov = (ab + ba+ aba + bab)* Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  26. a b a b a b 1: a b a b a b 2: All the steps to the right are needed for covering w: δ1, δ2 are minimal coverings! Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  27. a b a b a b 3: a b a b a b 4: a b a b a b 5: All blue steps are useless for covering w : δ3, δ4, δ5 are not minimal. We count only minimal coverings. Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  28. v u w z-stable w, vwM , uv, uZ-prefix-strict(uvw) v Z -prefix-strict(uvw) v u w Toward a cov-stability definition (I) stableu,w,uv,vwMvM Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  29. Example: X= abcd+bcde+cdef+defg M=Xcov vx a b c d e f g Set u=ab, v=c, w=defg, x=de, y=cd. Therefore w, vw, uvx, uy  M but vx =cde M. • Note vz=cdef  M,   z w . Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  30. Example: X = abc + bcd + cde M=Xcov w x u a b c d e Set u=ab, v= , w=cde, x=cd, y=c. Therefore w, vw, uvx, uy  M but vzM for no  z w. • Note bcd M, bcd proper suffix of ux. Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  31. w u v v y  v  x y w u v v y  v  x y Case 1. vz  M   z w vx  M Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

  32. w u x y Case 2. v x y u t  M, t propersuffix of ux Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia

More Related