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

Ravello 19-21 settembre 2003

Covering Problems from a Formal Language Point of ViewM. Anselmo - M. Madonia

1

Covering problemsfrom

a formal language point of view

Marcella ANSELMOMaria MADONIA



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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



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Why study covering ?

• Molecular biology: manipulating DNA molecules (e.g. fragment assembly)

• Data compression

• Computer-assisted music analysis



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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)



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Formal language point of view

alsoXcov = (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]:



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Formal Definition

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) = w3. for any i, red(w1…wi) is prefix of w

*A

.if...*

11 AAa...awaaw nn .| ,If * XxxXAX

) , :(Ex abababred(w) aba ba ba b ab w

aaaaA

A

s.t.copydisjoint,

alphabet,

red(w) = canonical representative of the class of w in the free group

,If*

AAw



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=(ab, , ba, , ba, , ab) is a covering of w over X b a

a b a b a b :

Example: X = ab+ba w = ababab.

1. n is odd; for any odd i, wi X; for any even i, wi *

2. red(ab ba ba ab) = ababab3. for any i, red(w1…wi) is prefix of w

A

b a



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Concatenation, zig-zag, covering

Xcovcov-submonoid

Xz-submonoid

X*submonoid

Covering

Zig-zag

Concatenation

cov-submonoid z-submonoid submonoid



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Example: X= ab+ba, w=#ababaab$ L(S)

Splicing systems for Xcov

a b a b a $b a a b $

x = baa b a b a a b$

X, finiteS, splicing system s.t. L(S) = Xcov $

Start with: x $, xX or COV2(X) Rules: (, x, $), xX

(, x, x3$), x=x1x2, x2x3 X

COV2(X) = XxxxxAxxxxxx 3221321321 ,*,,,|

a b a b $a b a $

x = ab



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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) b a

5 =(ab, , ba, , ab, , ba, ,ab) b b a a

3 =(ab, , ba, , ab, , ab) b a

1 =(ab, , ab, , ab)

2 =(ab, , ba, , ba , , ba, , ab) b a



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Covering codes [MSS 99]

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

X A* is a covering code if any word in A* has at most one minimal covering (over X).



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Cov - freeness

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)

We want ‘cov-stability’ = global notion equivalent to cov-freeness.

Question: M cov-free M ‘cov-stable’?



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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



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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



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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



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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



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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!



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Cov - maximality and cov-completeness

Fact: X cov-complete X cov-maximal

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-complete X infinite (unless X=A)

Example: X=ab+a +a

Remark complete cov-complete (not viceversa) maximal cov-maximal (not viceversa)



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B, 2FA recognizing Xcov

A, 1DFA recognizing X

Counting minimal coverings

X A*, regular language

covX : w number of minimal coverings of w

Remark: B counts all coverings of w Xcov

X1A

X



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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 11



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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) )

1

2

3

4a

a

b

bExample: X = ab + ba, A :

C :1

3

2 31

12

13

21

a

a

a

aa

b

bb

b

b



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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])



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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: measurespecial cases: |X| =1, X Ak

suggestions …

•Further problems:



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w

x xx x

w

x xx x

w

X XX X XX Ak

w is‘quasiperiodic’x is a ‘cover’ of w

‘set of k-covers’ of w

x ‘covers’ w



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Example: X = ab+ba

a b a b a b

a b a b a b

Xcov = (ab + ba+ aba + bab)*

w = ababab Xcov

w = ababab (X, A*)



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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!



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a b a b a b

a b a b a b4:

a b a b a b 5:

3:

All blue steps are useless for covering w :δ3, δ4, δ5 are not minimal.We count only minimal coverings.



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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)



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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+defg M=Xcov

but vx =cde M.



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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 + cde M=Xcov

u x

w

but vz M for no z w.



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Case 1.

u v w

yx

vz M

z w

u v w

xy

v

y v

v

y v vx M



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Case 2.

v

x y

u

t M, t proper suffix of ux

u w

yx

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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