Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf ·...
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Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk LSV Cachan / University of Warsaw Wednesday, November 18, 2009
Transcript of Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf ·...
Deciding Emptiness of min-automata
Szymon Toruńczykjoint work with
Mikołaj Bojańczyk
LSV Cachan / University of Warsaw
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
200
cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
200
300
cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
200
300
030
cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
200
300
030
130
cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
200
300
030
130
010
cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
200
300
030
130
010
110
cdz
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
200
300
030
130
010
110
cdz
210
Wednesday, November 18, 2009
“c tends to ∞”
liminf(c) = ∞
deterministic automata with counterstransitions invoke counter operations:
c:=min(d,e)
c:=c+1
acceptance condition is a boolean combination of:
Min-automata
Example. L = {an1 b an2 b an3 b...: n1,n2... does not converge to ∞}Min-automaton has one state and three counters: c,d,z-when reading a, do c:=c+1-when reading b, do d:=min(c,c); c:=min(z,z)
Acceptance condition: ¬ liminf(c) = ∞ ∧¬ liminf(d) = ∞
a a a b a b a a b...000
100
200
300
030
130
010
110
cdz
210
020
Wednesday, November 18, 2009
Max-automata
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$er
Max-automata
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$erUX φ(X)
„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”
which says
Max-automata
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$er
Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}
UX φ(X)
„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”
which says
Max-automata
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$er
Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}
UX “X is a block of a’s”
UX φ(X)
„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”
which says
Max-automata
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$er
Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}
UX “X is a block of a’s”
UX φ(X)
„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”
which says
Max-automata Min-automata
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$er
Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}
UX “X is a block of a’s”
UX φ(X)
„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”
which says
Max-automata Min-automata
RX φ(X)
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$er
Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}
UX “X is a block of a’s”
UX φ(X) which says
Max-automata Min-automata
RX φ(X)
„there exist in"nitely many sets X of same size, satisfying φ(X)”
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$er
UX “X is a block of a’s”
UX φ(X) which says
Max-automata Min-automata
RX φ(X)
„there exist in"nitely many sets X of same size, satisfying φ(X)”
Language: {an1 b an2 b an3 b... : n1 n2 n3... converges to ∞}
Logic
Wednesday, November 18, 2009
Extension of WMSO by the quanti$erUX φ(X)
which says
Max-automata Min-automata
RX φ(X)
„there exist in"nitely many sets X of same size, satisfying φ(X)”
Language: {an1 b an2 b an3 b... : n1 n2 n3... converges to ∞}
¬RX “X is a block of a’s”
Logic
Wednesday, November 18, 2009
max-automata min-automata
WMSO + U WMSO + R
Wednesday, November 18, 2009
eorem. WMSO+U has the same expressive power as deterministic max-automata.
max-automata min-automata
WMSO + U WMSO + R
Wednesday, November 18, 2009
eorem. WMSO+U has the same expressive power as deterministic max-automata.
max-automata min-automata
eorem. WMSO+R has the same expressive power as deterministic min-automata.
WMSO + U WMSO + R
Wednesday, November 18, 2009
WMSO + U + R
eorem. WMSO+U has the same expressive power as deterministic max-automata.
max-automata min-automata
eorem. WMSO+R has the same expressive power as deterministic min-automata.
What if we allow both U and R?
Wednesday, November 18, 2009
WMSO + U + R
eorem. WMSO+U has the same expressive power as deterministic max-automata.
max-automata min-automata
eorem. WMSO+R has the same expressive power as deterministic min-automata.
eorem. WMSO+U+R has the same expressive power as boolean combinations of min- and max-automata.
Wednesday, November 18, 2009
WMSO + U + R
eorem. WMSO+U has the same expressive power as deterministic max-automata.
eorem. WMSO+R has the same expressive power as deterministic min-automata.
eorem. WMSO+U+R has the same expressive power as boolean combinations of min- and max-automata.
Boolean combinations of min- & max-automata
Wednesday, November 18, 2009
WMSO + U + R
eorem. WMSO+U has the same expressive power as deterministic max-automata.
eorem. WMSO+R has the same expressive power as deterministic min-automata.
eorem. WMSO+U+R has the same expressive power as boolean combinations of min- and max-automata.
Boolean combinations of min- & max-automata
Equivalently: Nesting the quanti$ers U and R does not contribute anything to the expressive power of WMSO.
Wednesday, November 18, 2009
Emptiness of min-automata
Wednesday, November 18, 2009
eorem. Emptiness of min-automata is decidable.
Emptiness of min-automata
Wednesday, November 18, 2009
eorem. Emptiness of min-automata is decidable.
1st proof. min-automata are a special case of ωBS-automata (Bojańczyk, Colcombet [06]), so emptiness is decidable. is gives bad complexity, however.
Emptiness of min-automata
Wednesday, November 18, 2009
eorem. Emptiness of min-automata is decidable.
1st proof. min-automata are a special case of ωBS-automata (Bojańczyk, Colcombet [06]), so emptiness is decidable. is gives bad complexity, however.
Emptiness of min-automata
2nd proof. Reduction to the limitedness problem for distance-automata. Gives PSPACE algorithm, which is optimal.
Wednesday, November 18, 2009
eorem. Emptiness of min-automata is decidable.
1st proof. min-automata are a special case of ωBS-automata (Bojańczyk, Colcombet [06]), so emptiness is decidable. is gives bad complexity, however.
Emptiness of min-automata
2nd proof. Reduction to the limitedness problem for distance-automata. Gives PSPACE algorithm, which is optimal.
eorem. Emptiness of max-automata is decidable.
Wednesday, November 18, 2009
eorem. Emptiness of min-automata is decidable.
1st proof. min-automata are a special case of ωBS-automata (Bojańczyk, Colcombet [06]), so emptiness is decidable. is gives bad complexity, however.
Emptiness of min-automata
2nd proof. Reduction to the limitedness problem for distance-automata. Gives PSPACE algorithm, which is optimal.
eorem. Emptiness of max-automata is decidable.
eorem. Emptiness of a boolean combination of min- and max-automata is decidable.
Wednesday, November 18, 2009
Simplyfying the min-automata model
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞
• Can introduce matrix operations on counters, which stems from the semiring structure on {0,1,2,..., ∞, ⊤}, where min with respect to 0<1<2<...< ∞<⊤ is addition and + is multiplication
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞
• Can introduce matrix operations on counters, which stems from the semiring structure on {0,1,2,..., ∞, ⊤}, where min with respect to 0<1<2<...< ∞<⊤ is addition and + is multiplication
In Example 1, c:=c+1 can be written as:
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞
• Can introduce matrix operations on counters, which stems from the semiring structure on {0,1,2,..., ∞, ⊤}, where min with respect to 0<1<2<...< ∞<⊤ is addition and + is multiplication
In Example 1, c:=c+1 can be written as:
!c d z
":=
!c d z
"·
#
$1 ! !! 0 !! ! 0
%
&
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞
• Can introduce matrix operations on counters, which stems from the semiring structure on {0,1,2,..., ∞, ⊤}, where min with respect to 0<1<2<...< ∞<⊤ is addition and + is multiplication
In Example 1, c:=c+1 can be written as:
!c d z
":=
!c d z
"·
#
$1 ! !! 0 !! ! 0
%
& =!
c + 1 d z )
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞
• Can introduce matrix operations on counters, which stems from the semiring structure on {0,1,2,..., ∞, ⊤}, where min with respect to 0<1<2<...< ∞<⊤ is addition and + is multiplication
In Example 1, c:=c+1 can be written as:
d:=min(c,c); c:=min(z,z) can be written as:
!c d z
":=
!c d z
"·
#
$1 ! !! 0 !! ! 0
%
& =!
c + 1 d z )
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞
• Can introduce matrix operations on counters, which stems from the semiring structure on {0,1,2,..., ∞, ⊤}, where min with respect to 0<1<2<...< ∞<⊤ is addition and + is multiplication
In Example 1, c:=c+1 can be written as:
d:=min(c,c); c:=min(z,z) can be written as:
!c d z
":=
!c d z
"·
#
$1 ! !! 0 !! ! 0
%
&
!c d z
":=
!c d z
"·
#
$! 0 !! ! !0 ! 0
%
&
=!
c + 1 d z )
Wednesday, November 18, 2009
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$ned counter values ⊤ this can be eliminated by storing in the states the info about which counters are de$ned
• Can introduce the counter value ∞ the difference between ⊤ and ∞ is that lim ∞ = ∞ while lim ⊤≠ ∞
• Can introduce matrix operations on counters, which stems from the semiring structure on {0,1,2,..., ∞, ⊤}, where min with respect to 0<1<2<...< ∞<⊤ is addition and + is multiplication
In Example 1, c:=c+1 can be written as:
d:=min(c,c); c:=min(z,z) can be written as:
!c d z
":=
!c d z
"·
#
$1 ! !! 0 !! ! 0
%
&
!c d z
":=
!c d z
"·
#
$! 0 !! ! !0 ! 0
%
&
=!
c + 1 d z )
=!
z c z )
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c.
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
1⊤
c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
1⊤
⊤2
c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
1⊤
⊤2
2⊤
c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
1⊤
⊤2
2⊤
⊤2
c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
1⊤
⊤2
2⊤
⊤2
2⊤
c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
1⊤
⊤2
2⊤
⊤2
2⊤
⊤3
c0
c1
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
1⊤
⊤2
2⊤
⊤2
2⊤
⊤3
c0
c1
3⊤
Wednesday, November 18, 2009
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.Proof. We eliminate states as in the following example.
Example. Min-automaton which counts a’s on odd positions.States: q0, q1, one counter c. Transitions:
-saw a in state q0 – go to q1; c:=c+1-saw a in state q1 – go to q0
-saw b in state q0 – go to q1
-saw b in state q1 – go to q0
Min-automaton in matrix form with one state and two counters: c0, c1.e initial counter valuation is (c0, c1)=(0, ⊤).
a :!
c0 c1
":=
!c0 c1
"·#! 01 !
$.
b :!
c0 c1
":=
!c0 c1
"·#! 00 !
$.
a a a b b b a a b...0⊤
⊤1
1⊤
⊤2
2⊤
⊤2
2⊤
⊤3
c0
c1
3⊤
⊤3
Wednesday, November 18, 2009
Wednesday, November 18, 2009
Example of a min-automaton A.
Wednesday, November 18, 2009
Example of a min-automaton A.
0 ⊤ ⊤
⊤ 1 ⊤
⊤ ⊤ 0
1 1 ⊤
⊤ ⊤ 0
⊤ ⊤ 0a: b:
Wednesday, November 18, 2009
Example of a min-automaton A.
0 ⊤ ⊤
⊤ 1 ⊤
⊤ ⊤ 0
1 1 ⊤
⊤ ⊤ 0
⊤ ⊤ 0a: b:
2 2 1
⊤ ⊤ 0
⊤ ⊤ 0abb:
Wednesday, November 18, 2009
Example of a min-automaton A.
0 ⊤ ⊤
⊤ 1 ⊤
⊤ ⊤ 0
1 1 ⊤
⊤ ⊤ 0
⊤ ⊤ 0a: b:
2 2 1
⊤ ⊤ 0
⊤ ⊤ 0abb:
0 ⊤ ⊤Initial valuation:
Acceptance condition: ¬ liminf(c3) = ∞
Wednesday, November 18, 2009
e tropical semiring
Wednesday, November 18, 2009
e tropical semiring
T = {0,1,2,..., ∞, ⊤}
Wednesday, November 18, 2009
e tropical semiring
T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤
with operations +, min
where ⊤+ x = x +⊤=⊤
Wednesday, November 18, 2009
e tropical semiring
T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤
with operations +, min
Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞
where ⊤+ x = x +⊤=⊤
Wednesday, November 18, 2009
e tropical semiring
T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤
with operations +, min
Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞
πn : T → Tn
maps n+1, n+2, ... to ∞
is a homomorphism of semirings
where ⊤+ x = x +⊤=⊤
Wednesday, November 18, 2009
e tropical semiring
T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤
with operations +, min
Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞
πn : T → Tn
maps n+1, n+2, ... to ∞
is a homomorphism of semirings
where ⊤+ x = x +⊤=⊤
πn : Tm → Tn for m>n
Wednesday, November 18, 2009
e tropical semiring
T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤
with operations +, min
Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞
πn : T → Tn
maps n+1, n+2, ... to ∞
is a homomorphism of semirings
where ⊤+ x = x +⊤=⊤
πn : Tm → Tn for m>n
Wednesday, November 18, 2009
e tropical semiring
T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤
with operations +, min
Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞
πn : T → Tn
maps n+1, n+2, ... to ∞
is a homomorphism of semirings
where ⊤+ x = x +⊤=⊤
πn : Tm → Tn for m>n
MkT – k by k matrices over Twith matrix multiplication
MkTn – k by k matrices over Tnwith matrix multiplication
πn : MkT → MkTn
maps n+1, n+2, ... to ∞
is a homomorphism of semirings
πn : Tm → Tn for m>n
Wednesday, November 18, 2009
e tropical semiring
T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤
with operations +, min
Tn = {0,1,2,...,n, ∞, ⊤} where n+1=∞
πn : T → Tn
maps n+1, n+2, ... to ∞
is a homomorphism of semirings
where ⊤+ x = x +⊤=⊤
πn : Tm → Tn for m>n
MkT – k by k matrices over Twith matrix multiplication
MkTn – k by k matrices over Tnwith matrix multiplication
πn : MkT → MkTn
maps n+1, n+2, ... to ∞
is a homomorphism of semirings
πn : Tm → Tn for m>n
3 ∞ 10 ∞ 12 ∞ ∞
3 32 10 11 12 7 ∞
∞ ∞ 10 ∞ 1∞ ∞ ∞
π6 π2
Wednesday, November 18, 2009
Pro$nite monoid
Wednesday, November 18, 2009
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
Wednesday, November 18, 2009
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
Wednesday, November 18, 2009
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
M3T0 M3T1 M3T2 M3T3
Wednesday, November 18, 2009
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
Wednesday, November 18, 2009
Pro$nite monoid
3 ∞ 10 ∞ 12 ∞ ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.
Wednesday, November 18, 2009
Pro$nite monoid
∞ ∞ 10 ∞ 12 ∞ ∞
3 ∞ 10 ∞ 12 ∞ ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
..
Wednesday, November 18, 2009
Pro$nite monoid
∞ ∞ 10 ∞ 1∞ ∞ ∞
∞ ∞ 10 ∞ 12 ∞ ∞
3 ∞ 10 ∞ 12 ∞ ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
...
Wednesday, November 18, 2009
Pro$nite monoid
∞ ∞ ∞0 ∞ ∞∞ ∞ ∞
∞ ∞ 10 ∞ 1∞ ∞ ∞
∞ ∞ 10 ∞ 12 ∞ ∞
3 ∞ 10 ∞ 12 ∞ ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
....
Wednesday, November 18, 2009
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
....
Wednesday, November 18, 2009
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
-approximation:0
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
∞ ∞ ∞0 ∞ ∞∞ ∞ ∞
Wednesday, November 18, 2009
Pro$nite monoid
-approximation:1
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
∞ ∞ 10 ∞ 1∞ ∞ ∞
Wednesday, November 18, 2009
Pro$nite monoid
-approximation:2
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
∞ ∞ 10 ∞ 12 ∞ ∞
Wednesday, November 18, 2009
Pro$nite monoid
-approximation:3
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
3 ∞ 10 ∞ 12 ∞ ∞
Wednesday, November 18, 2009
Pro$nite monoid
-approximation:3 ∞ 10 ∞ 12 7 ∞
7
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
-approximation:3 32 10 11 12 7 ∞
1000
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings
multiplication
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings
3 8 43 8 45 18 3
=
multiplication
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings
3 8 43 8 45 18 3
=
is continuous
multiplication
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings
3 8 43 8 45 18 3
=
is continuous
...................
multiplication
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplication
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplication
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplicationaddition
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplicationaddition
addition
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplicationaddition
addition
idempotent power (ω)
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplicationaddition
addition
idempotent power (ω)
ω-power
·
.... . . .........................
Wednesday, November 18, 2009
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplicationaddition
addition
idempotent power (ω)
ω-power
·
compact space.... . . .........................
Wednesday, November 18, 2009
Wednesday, November 18, 2009
Example of a min-automaton A.
Wednesday, November 18, 2009
Example of a min-automaton A.
0 ⊤ ⊤
⊤ 1 ⊤
⊤ ⊤ 0
1 1 ⊤
⊤ ⊤ 0
⊤ ⊤ 0a: b:
∞ ∞ 1⊤ ⊤ 0
⊤ ⊤ 0bω:
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
.x x x x x x x x x x x x x x x x x x x x x x x x x
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
.x x x x x x x x x x x x x x x x x x x x x x x x x
colored by elements of a compact space
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
.x x x x x x x x x x x x x x x x x x x x x x x x x . . .. . . . . .
..colored by elements of a compact space
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
.x x x x x x x x x x x x x x x x x x x x x x x x x . . .. . . . . .
..
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
.x x x x x x x x x x x x x x x x x x x x x x x x x . . .. . . . . .
...
. .
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
.x x x x x x x x x x x x x x x x x x x x x x x x x . . .. . . . . .
...
. .
.
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . ..
. .
.x x x x x x x x x x x x x x x
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . ..
. .
.x x x x x x x x x x x x x x x
. .
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . ..
. .
.x x x x x x x x x x x x x x x
. .
.
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . ..
. .
.x x x x x x x x x x x x x x x
. .
.x x
.
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . ..
. .
.x x x x x x x x x x x x x x x
. .
.x x
..
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . ..
. .
.x x x x x x x x x x x x x x x
. .
.x x
.. .
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . ..
. .
.x x x x x x x x x x x x x x x
. .
.x x
.. .
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . .. .
.x x x x x x x x x x x x x x x
.
.x x .x
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . .. .
.x x x x x x x x x x x x x x x
.
.x x .x
lim
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . .. .
.x x x x x x x x x x x x x x x
.
.x x .
x x x x x x x x x x x x x x x x x x x x x x x x x
x
lim
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . .. .
.x x x x x x x x x x x x x x x
.
.x x .
x x x x x x x x x x x x x x x x x x x x x x x x x .
x
lim
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . .. .
.x x x x x x x x x x x x x x x
.
.x x .
x x x x x x x x x x x x x x x x x x x x x x x x x .
x
lim
d( , )<1/Nlim
N
Wednesday, November 18, 2009
Ramsey eoremfor compact spaces
x x x x x x x x x x x x x x x x x x x x x x x x x . .. .
.x x x x x x x x x x x x x x x
.
.x x .
x x x x x x x x x x x x x x x x x x x x x x x x x ..
x
lim
d( , )<1/Nlim d( , )<1/N’lim
N N'
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x
x x x x x x x x x x x x x x x x x x x x x x x x x
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x
x x x x x x x x x x x x x x x x x x x x x x x x x
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . . . . . ..
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . . . . . ..induced transition matrix
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . . .. . . . . .
..
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lim
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lim
Counter c does not converge to ∞ iff exists a counter d such that lim[d,d]=0 and lim[d,c]< ∞.
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lim
Which limits lim are possible?
Counter c does not converge to ∞ iff exists a counter d such that lim[d,d]=0 and lim[d,c]< ∞.
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lim
Which limits lim are possible?
Counter c does not converge to ∞ iff exists a counter d such that lim[d,d]=0 and lim[d,c]< ∞.
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b
Which limits lim are possible?
Counter c does not converge to ∞ iff exists a counter d such that lim[d,d]=0 and lim[d,c]< ∞.
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b
Which limits lim are possible?
Counter c does not converge to ∞ iff exists a counter d such that lim[d,d]=0 and lim[d,c]< ∞.
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+Which limits lim are possible?
Counter c does not converge to ∞ iff exists a counter d such that lim[d,d]=0 and lim[d,c]< ∞.
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+__________
∈Which limits lim are possible?
Counter c does not converge to ∞ iff exists a counter d such that lim[d,d]=0 and lim[d,c]< ∞.
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+__________
∈Which limits lim are possible?
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+__________
∈
eorem.
Which limits lim are possible?
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+__________
∈
eorem. a b ( , )+__________
Which limits lim are possible?
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+__________
∈
eorem. =a b ( , )+__________
Which limits lim are possible?
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+__________
∈
eorem. a b=a b ( , )+__________
Which limits lim are possible?
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+__________
∈
eorem. a b ( , )+,ω=a b ( , )+__________
Which limits lim are possible?
Wednesday, November 18, 2009
Emptiness of min-automata
a b b a a b a a b a b b a a a b b a b a b a x x x .x x x x x x x x x x x x x x x x x x x x x x x x x . ..
lim
d( , )<1/19)lim
. ..
.d( , )<1/10lim
d( , )<1/15lima b ( , )+__________
∈
eorem. a b ( , )+,ω=if , are matrices over the (min,+)-semiring.a b
a b ( , )+__________
Which limits lim are possible?
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # )
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#
• s# s# = s#
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#
• s# s# = s#
• e# e = e# if e is idempotent
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#
• s# s# = s#
• e# e = e# if e is idempotent
e = e# if e is idempotent
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#
• s# s# = s#
• e# e = e# if e is idempotent
e = e# if e is idempotent
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#
• s# s# = s#
• e# e = e# if e is idempotent
e = e# if e is idempotent
Example 1 (in$nite) ({0,1,2,..., ∞}, +, ω ), 0ω =0, 1ω = 2ω = ...= ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#
• s# s# = s#
• e# e = e# if e is idempotent
e = e# if e is idempotent
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Example 1 (in$nite) ({0,1,2,..., ∞}, +, ω ), 0ω =0, 1ω = 2ω = ...= ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...• (s t)# s = s (t s)#
• s# s# = s#
• e# e = e# if e is idempotent
e = e# if e is idempotent
1,2,3... → 1Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Example 1 (in$nite) ({0,1,2,..., ∞}, +, ω ), 0ω =0, 1ω = 2ω = ...= ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
1 1 1 10 0 0 1
1 1 1 1
1 1
1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
1 1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
1 1 1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
01 1 1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
01 1 1 1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
01 1 1 1
1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
0 01 1 1 1
1 1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
0 01 1 1 1
1 1
∞
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
eorem. For any $nite stabilization semigroup S and word w ∈ S+ there exists a factorization tree over w of height ≤ 9|S|2.
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
Wednesday, November 18, 2009
ank you for your attention!
Wednesday, November 18, 2009
