# Covering Problems from a formal language point of view by pptfiles

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```									                Covering problems
from
a formal language point of view
Marcella ANSELMO

Ravello 19-21 settembre 2003   Covering Problems from a Formal Language Point of View   1
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

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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)
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)
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Formal language point of view

Formal language point of view is needed!

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!
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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
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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

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Concatenation, zig-zag, covering

Concatenation                                                    submonoid                    X*

Zig-zag                                                     z-submonoid                    X­

Covering                                                   cov-submonoid Xcov

cov-submonoid                      z-submonoid                           submonoid

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Splicing systems for Xcov
X, finite
S, splicing system s.t. L(S) = # Xcov \$
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 \$
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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)
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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

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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.

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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

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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 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
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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

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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!)

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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!
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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
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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

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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 1
1

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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

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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

•Further problems:
ücovering codes: measure
üspecial cases: |X| =1, X Í Ak
ü suggestions …

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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

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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)*

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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!

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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.

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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

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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.

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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.

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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
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Case 2.

u                                  w
v=l
çx ç> çy ç
u¹l                                                          x
y

t Î M, t proper suffix of ux

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