1. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 1 Covering problems from a formal language point of view Marcella ANSELMO Maria MADONIA

2. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 2 Covering a word Covering a word w with words in a set X w Covering = concatenations +overlaps Example: X = ab+ba w = abababa a b a b a b a X X X X X

3. Ravello 19-21 settembre 2003Covering 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

4. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 4 Literature 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 All algorithmic problems!!! (given w find optimal X)

5. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 5 Formal language point of view also X cov = (X, A * ), set of z-decompositions over (X, A * ) Here: Coverings not simple generalizations of z-decompositions! If X A *, X cov = set of words covered by words in X Formal language point of view is needed! Madonia, Salemi, Sportelli (1999) [MSS99]:

6. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 6 Formal Definition Def. A covering (over X) of w in A * is =(w 1, …, w n ) s.t. 1. n is odd; for any odd i, w i X for any even i, w i 2. red(w 1 … w n ) = w 3. for any i, red(w 1 …w i ) is prefix of w red(w) = canonical representative of the class of w in the free group

7. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 7 =(ab,, ba,, ba,, ab) is a covering of w over X a b a b a b : Example: X = ab+ba w = ababab. 1.n is odd; for any odd i, w i X; for any even i, w i * 2. red( ab ba ba ab ) = ababab 3. for any i, red(w 1 …w i ) is prefix of w

8. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 8 Concatenation, zig-zag, covering X cov cov-submonoid X z-submonoid X*X* submonoid Covering Zig-zag Concatenation cov-submonoidz-submonoidsubmonoid

9. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 9 Example: X= ab+ba, w=#ababaab$ L(S) Splicing systems for X cov a b a b a $ b a a b $ x = ba a b a b a a b$ X, finite S, splicing system s.t. L(S) = X cov $ Start with: x $, x X or COV 2 (X) Rules: (, x, $), x X (, x, x 3 $), x=x 1 x 2, x 2 x 3 X COV 2 (X) = a b a b $ a b a $ x = ab

10. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 10 Coding problems [MSS99] How many coverings has a word? Example: X=ab + ba, w = ababab X cov w has many different coverings over X : 4 =(ab,, ab,, ba,, ab) 5 =(ab,, ba,, ab,, ba,,ab) 3 =(ab,, ba,, ab,, ab) 1 =(ab,, ab,, ab) 2 =(ab,, ba,, ba,, ba,, ab)

11. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 11 Covering codes [MSS 99] Example: X = ab + ba is not a covering code (remember δ 1, δ 2 ) Example: X = aabab + abb is a covering code Example: X= ab + a + a is a covering code X A * is a covering code if any word in A * has at most one minimal covering (over X).

12. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 12 Cov - freeness Let M A *, cov-submonoid. cov-G(M) is the minimal X A * such that M= X cov. 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) We want cov-stability = global notion equivalent to cov-freeness. Question: M cov-freeM cov-stable?

13. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 13 w, vw M, uv, u Z-p-s(uvw) implies v Z-p-s(uvw) Toward a cov-stability definition (I) cov-stable? z-stable stable u,w,uv,vw M implies w M w, vw, uvx, uy M, for x <w and y <vw, implies vx M ? Not always! Example: X = abcd+bcde+cdef+defg

14. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 14 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 from the last step in the other factorization New situation with covering: So we have to study the case v =. Example: X = abc + bcd + cde u w

15. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 15 Cov – stability Def. M is cov-stable if w, vw, uvx, uy M, for x w and y vw Remark: cov-stable implies stable 1.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 proper suffix of ux

16. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 16 Cov-stable iff cov-free Proof: many cases and sub-cases (as in definition!) Theorem: M covering submonoid. M is cov-stable M is cov-free

17. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 17 Some consequences Fact 1: (cov-free cov-free) cov-free Fact 5: cov –free z-free 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!

18. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 18 Cov - maximality and cov-completeness Fact: X cov-complete X cov-maximal Let X A *, covering code. X is cov-complete if Fact(X cov ). X is cov-maximal if X X 1, covering code X=X 1 Remark [MSS99] : X cov-complete X infinite (unless X=A) Example: X=ab + a +a Remark complete cov-complete (not viceversa) maximal cov-maximal (not viceversa)

19. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 19 B, 2FA recognizing X cov A, 1DFA recognizing X Counting minimal coverings X A*, regular language cov X : w number of minimal coverings of w Remark: B counts all coverings of w X cov X 1 A X

20. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 20 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

21. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 21 A 1NFA automaton for cov X CS 3 = crossing sequences of length 3 and no twice state 1 (cs,a) =cs if cs matches cs on a C = (CS 3, (1),, (1) ) 1 2 3 4 a a b b Example: X = ab + ba, A : C : 1 3 2 3131 1212 1313 2121 a a a a a b b b b b

22. Ravello 19-21 settembre 2003Covering 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 cov X X regular implies cov X rational X covering code iff C unambiguous (decidable) (different proof in [MSS99])

23. Ravello 19-21 settembre 2003Covering 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 covering codes: measure special cases: |X| =1, X A k suggestions … Further problems:

24. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 24 w xxxx w xxxx w X X X X X X A k w isquasiperiodic x is a cover of w set of k-covers of w x covers w

25. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 25 Example: X = ab+ba a b a b a b X cov = (ab + ba+ aba + bab) * w = ababab X cov w = ababab (X, A * )

26. Ravello 19-21 settembre 2003Covering 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: δ 1, δ 2 are minimal coverings!

27. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 27 a b a b a b 4 : a b a b a b 5 : 3 : All blue steps are useless for covering w : δ 3, δ 4, δ 5 are not minimal. We count only minimal coverings.

28. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 28 Toward a cov-stability definition (I) stable u,w,uv,vw M v M u v w u v w z-stable w, vw M, uv, u Z-prefix-strict(uvw) v Z -prefix-strict(uvw)

29. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 29 a b c d e f g Set u=ab, v=c, w=defg, x=de, y=cd. Note vz=cdef M, z w. Therefore w, vw, uvx, uy M vx Example: X= abcd+bcde+cdef+defgM=X cov but vx =cde M.

30. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 30 a b c d e Set u=ab, v=, w=cde, x=cd, y=c. Therefore w, vw, uvx, uy M Note bcd M, bcd proper suffix of ux. Example: X = abc + bcd + cdeM=X cov u x w but vz M for no z w.

31. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 31 Case 1. u v w y x vz M z w u v w x y v y v v y v vx M

32. Ravello 19-21 settembre 2003Covering Problems from a Formal Language Point of View M. Anselmo - M. Madonia 32 Case 2. v x y u t M, t proper suffix of ux u w y x