Download presentation

Presentation is loading. Please wait.

Published byCody Emery Modified over 4 years ago

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

Similar presentations

OK

Molecular Computing Formal Languages Theory of Codes Combinatorics on Words.

Molecular Computing Formal Languages Theory of Codes Combinatorics on Words.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google