Covering problems
from
a formal language point of view
Marcella ANSELMO
Maria MADONIA
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Covering a word
Covering a word w with words in a set X
X X X
X
X
w
Covering = concatenations +overlaps
Example: X = ab+ba
w = abababa
a b a b a b a
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Why study covering ?
• Molecular biology:
manipulating DNA molecules (e.g. fragment assembly)
• Data compression
• Computer-assisted music analysis
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Literature
• Apostolico, Ehrenfeucht (1993)
• Brodal, Pedersen (2000)
• Moore, Smyth (1995)
w is ‘quasiperiodic’
x is a ‘cover’ of w
• Iliopulos, Moore, Park (1993)
• Iliopulos, Smyth (1998)
x ‘covers’ w
‘set of k-covers’ of w
• Sim, Iliopulos, Park, Smyth (2001)
(complete references)
p ‘approximated
period’ of w
All algorithmic problems!!!
(given w find ‘optimal’ X)
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Formal language point of view
Formal language point of view is needed!
Madonia, Salemi, Sportelli (1999) [MSS99]:
If X  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!
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Formal Definition
A , alphabet
A , disjoint copy
w  an ... a1 if
If


s.t. a a  aa  


*
w  a1 ... an  A  A .


If X  A* , X  x | x  X .
*
w A A ,
red(w) = canonical representative of the class of w in the free group
(Ex: w  ab b ba ba a ab, red(w)  ababab)
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  A
2. red(w1… wn) = w
3. for any i, red(w1…wi) is prefix of w
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Example: X = ab+ba
w = ababab.
 =(ab, b , ba,  , ba, a , ab) is a covering of w over X
1. n is odd; for any odd i, wi  X;
for any even i, wi  A *
2. red(ab b ba  ba a ab) = ababab
3. for any i, red(w1…wi) is prefix of w
:
a b a b a b
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Concatenation, zig-zag, covering
Concatenation
submonoid
X*
Zig-zag
z-submonoid
X
cov-submonoid Xcov
Covering
cov-submonoid
Ravello 19-21 settembre 2003
z-submonoid
submonoid
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Splicing systems for Xcov
X, finite
S, splicing system s.t. L(S) =  Xcov $
COV2(X) =  x1 x 2 x3 | x1, x 2, x3  A*, x1 x 2, x 2 x3  X 
Start with:  x $, xX or COV2(X)
Rules: (, x, $), xX
(, x, x3$), x=x1x2, x2x3 X
Example: X= ab+ba, w=#ababaab$  L(S)
a b a b $ x = ab
a b a $
Ravello 19-21 settembre 2003
a b a b a $
b a a b $
x = ba
a b a b a a b$
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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 :
1 =(ab,  , ab,  , ab)
2 =(ab, b, ba, , ba ,  , ba, a, ab)
3 =(ab, b , ba, a , ab,  , ab)
4 =(ab, , ab, b, ba, a , ab)
5 =(ab, b , ba, a , ab, b , ba, a ,ab)
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Covering codes [MSS 99]
X  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
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is a covering code
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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)
Question: M cov-free
M ‘cov-stable’?
We want
‘cov-stability’ = global notion equivalent to cov-freeness.
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Toward a cov-stability definition (I)
stable
u,w,uv,vw  M
z-stable
w, vw  M , uv, u  Z-p-s(uvw)
implies v Z-p-s(uvw)
cov-stable?
w, vw, uvx, uy  M, for  x <w
and  y <vw, implies vx  M ?
implies
wM
Not always!
Example:
X = abcd+bcde+cdef+defg
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Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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:
u
w
So we have to study the case v = .
Example: X = abc + bcd + cde
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Cov – stability
Def. M is cov-stable if
w, vw, uvx, uy  M, for  x  w and  y  vw
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
Remark: cov-stable implies stable
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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!)
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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.
Fact 5:
cov –free
z-free
free
Remark: Covering not simple generalization of
z-decomposition!
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Cov - maximality and cov-completeness
Let X  A*, covering code.
X is cov-complete if Fact(Xcov).
X is cov-maximal if X  X1, covering code
Fact:
X cov-complete
Remark [MSS99]:
X cov-complete
Remark complete
maximal
X=X1
X cov-maximal
X infinite (unless X=A)
cov-complete (not viceversa)
cov-maximal (not viceversa)
Example: X=ab+a +a
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Counting minimal coverings
X  A*, regular language
covX : w
number of minimal coverings of w
A, 1DFA recognizing X
X
A
1
X
B, 2FA recognizing Xcov
Remark: B counts all coverings of w Xcov
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Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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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Covering Problems from a Formal Language Point of View
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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) )
a
A:
Example: X = ab + ba,
4
1
b
2
1
C:
2
a
a
b
b
b
2
3
a
3
1
b
a
1
1
3
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a
b
a
3
b
1
2
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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])
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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:
covering codes: measure
special cases: |X| =1, X  Ak
 suggestions …
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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x
w is‘quasiperiodic’
x is a ‘cover’ of w
x
x
x
w
x
x
x
x
x ‘covers’ w
w
X X
‘set of k-covers’
of w
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X
X
X
X  Ak
w
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Example: X = ab+ba
a b a b a b
w = ababab  Xcov
a b a b a b
w = ababab  (X, A*)
Xcov = (ab + ba+ aba + bab)*
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Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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1: a b a b a b
2: a b a b a b
All the steps to the right are needed for covering w:
δ1, δ2 are minimal coverings!
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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3: a b a b a b
4: a b a b a b
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.
Ravello 19-21 settembre 2003
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Toward a cov-stability definition (I)
stable
u,w,uv,vw  M
u
z-stable
v
vM
w
w, vw  M , uv, u  Z-prefix-strict(uvw)
v  Z -prefix-strict(uvw)
u
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v
w
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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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  M but vx =cde  M.
•Note vz=cdef  M,
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zw.
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Example: X = abc + bcd + cde
M=Xcov
w
a b c d e
u
x
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,
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bcd proper suffix of ux.
Covering Problems from a Formal Language Point of View
M. Anselmo - M. Madonia
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Case 1.
u
w
v
v
vz  M
y  v 
zw
x
y
u
w
v
v
vx  M
y  v 
y
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x
Covering Problems from a Formal Language Point of View
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Case 2.
w
u
v
x  y 
u
x
y
t  M, t proper suffix of ux
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