deciding emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf ·...
TRANSCRIPT
![Page 1: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/1.jpg)
Deciding Emptiness of min-automata
Szymon Toruńczykjoint work with
Mikołaj Bojańczyk
LSV Cachan / University of Warsaw
Wednesday, November 18, 2009
![Page 2: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/2.jpg)
“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
![Page 3: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/3.jpg)
“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
![Page 4: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/4.jpg)
“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
![Page 5: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/5.jpg)
“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
![Page 6: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/6.jpg)
“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
![Page 7: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/7.jpg)
“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
![Page 8: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/8.jpg)
“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
![Page 9: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/9.jpg)
“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
![Page 10: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/10.jpg)
“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
![Page 11: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/11.jpg)
“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
![Page 12: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/12.jpg)
“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
![Page 13: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/13.jpg)
“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
![Page 14: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/14.jpg)
“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
![Page 15: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/15.jpg)
“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
![Page 16: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/16.jpg)
“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
![Page 17: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/17.jpg)
“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
![Page 18: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/18.jpg)
“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
![Page 19: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/19.jpg)
“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
![Page 20: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/20.jpg)
“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
![Page 21: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/21.jpg)
“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
![Page 22: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/22.jpg)
Max-automata
Logic
Wednesday, November 18, 2009
![Page 23: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/23.jpg)
Extension of WMSO by the quanti$er
Max-automata
Logic
Wednesday, November 18, 2009
![Page 24: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/24.jpg)
Extension of WMSO by the quanti$erUX φ(X)
„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”
which says
Max-automata
Logic
Wednesday, November 18, 2009
![Page 25: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/25.jpg)
Extension of WMSO by the quanti$er
Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}
UX φ(X)
„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”
which says
Max-automata
Logic
Wednesday, November 18, 2009
![Page 26: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/26.jpg)
Extension of WMSO by the quanti$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
![Page 27: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/27.jpg)
Extension of WMSO by the quanti$er
Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}
UX “X is a block of a’s”
UX φ(X)
„there exist arbitrarily large ( "nite) sets X, satisfying φ(X)”
which says
Max-automata Min-automata
Logic
Wednesday, November 18, 2009
![Page 28: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/28.jpg)
Extension of WMSO by the quanti$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
![Page 29: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/29.jpg)
Extension of WMSO by the quanti$er
Language: {an1 b an2 b an3 b... : n1 n2 n3... is unbounded}
UX “X is a block of a’s”
UX φ(X) which says
Max-automata Min-automata
RX φ(X)
„there exist in"nitely many sets X of same size, satisfying φ(X)”
Logic
Wednesday, November 18, 2009
![Page 30: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/30.jpg)
Extension of WMSO by the quanti$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
![Page 31: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/31.jpg)
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
![Page 32: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/32.jpg)
max-automata min-automata
WMSO + U WMSO + R
Wednesday, November 18, 2009
![Page 33: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/33.jpg)
eorem. WMSO+U has the same expressive power as deterministic max-automata.
max-automata min-automata
WMSO + U WMSO + R
Wednesday, November 18, 2009
![Page 34: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/34.jpg)
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
![Page 35: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/35.jpg)
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
![Page 36: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/36.jpg)
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
![Page 37: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/37.jpg)
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
![Page 38: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/38.jpg)
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
![Page 39: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/39.jpg)
Emptiness of min-automata
Wednesday, November 18, 2009
![Page 40: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/40.jpg)
eorem. Emptiness of min-automata is decidable.
Emptiness of min-automata
Wednesday, November 18, 2009
![Page 41: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/41.jpg)
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
![Page 42: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/42.jpg)
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
![Page 43: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/43.jpg)
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
![Page 44: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/44.jpg)
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
![Page 45: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/45.jpg)
Simplyfying the min-automata model
Wednesday, November 18, 2009
![Page 46: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/46.jpg)
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
Wednesday, November 18, 2009
![Page 47: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/47.jpg)
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
![Page 48: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/48.jpg)
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
![Page 49: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/49.jpg)
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
![Page 50: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/50.jpg)
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
![Page 51: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/51.jpg)
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
"·
#
$1 ! !! 0 !! ! 0
%
&
Wednesday, November 18, 2009
![Page 52: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/52.jpg)
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$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
"·
#
$1 ! !! 0 !! ! 0
%
& =!
c + 1 d z )
Wednesday, November 18, 2009
![Page 53: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/53.jpg)
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$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
"·
#
$1 ! !! 0 !! ! 0
%
& =!
c + 1 d z )
Wednesday, November 18, 2009
![Page 54: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/54.jpg)
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$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
"·
#
$! 0 !! ! !0 ! 0
%
&
=!
c + 1 d z )
Wednesday, November 18, 2009
![Page 55: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/55.jpg)
Simplyfying the min-automata model• Instructions c:=0, c:=d can be implemented into the model, as in the example
• Can introduce the unde$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
![Page 56: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/56.jpg)
eorem. Min-automata are equivalent to min-automata in matrix form, with one state.
Wednesday, November 18, 2009
![Page 57: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/57.jpg)
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
![Page 58: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/58.jpg)
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
![Page 59: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/59.jpg)
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
![Page 60: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/60.jpg)
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
![Page 61: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/61.jpg)
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
![Page 62: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/62.jpg)
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
![Page 63: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/63.jpg)
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
![Page 64: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/64.jpg)
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
![Page 65: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/65.jpg)
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
![Page 66: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/66.jpg)
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
![Page 67: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/67.jpg)
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
![Page 68: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/68.jpg)
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
![Page 69: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/69.jpg)
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
![Page 70: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/70.jpg)
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
![Page 71: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/71.jpg)
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
![Page 72: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/72.jpg)
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
![Page 73: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/73.jpg)
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
![Page 74: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/74.jpg)
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
![Page 75: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/75.jpg)
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
![Page 76: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/76.jpg)
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
![Page 77: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/77.jpg)
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
![Page 78: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/78.jpg)
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
![Page 79: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/79.jpg)
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
![Page 80: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/80.jpg)
Wednesday, November 18, 2009
![Page 81: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/81.jpg)
Example of a min-automaton A.
Wednesday, November 18, 2009
![Page 82: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/82.jpg)
Example of a min-automaton A.
0 ⊤ ⊤
⊤ 1 ⊤
⊤ ⊤ 0
1 1 ⊤
⊤ ⊤ 0
⊤ ⊤ 0a: b:
Wednesday, November 18, 2009
![Page 83: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/83.jpg)
Example of a min-automaton A.
0 ⊤ ⊤
⊤ 1 ⊤
⊤ ⊤ 0
1 1 ⊤
⊤ ⊤ 0
⊤ ⊤ 0a: b:
2 2 1
⊤ ⊤ 0
⊤ ⊤ 0abb:
Wednesday, November 18, 2009
![Page 84: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/84.jpg)
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
![Page 85: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/85.jpg)
e tropical semiring
Wednesday, November 18, 2009
![Page 86: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/86.jpg)
e tropical semiring
T = {0,1,2,..., ∞, ⊤}
Wednesday, November 18, 2009
![Page 87: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/87.jpg)
e tropical semiring
T = {0,1,2,..., ∞, ⊤}ordered by 0<1<2<...<∞<⊤
with operations +, min
where ⊤+ x = x +⊤=⊤
Wednesday, November 18, 2009
![Page 88: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/88.jpg)
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
![Page 89: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/89.jpg)
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
![Page 90: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/90.jpg)
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
![Page 91: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/91.jpg)
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
![Page 92: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/92.jpg)
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
![Page 93: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/93.jpg)
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
![Page 94: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/94.jpg)
Pro$nite monoid
Wednesday, November 18, 2009
![Page 95: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/95.jpg)
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
Wednesday, November 18, 2009
![Page 96: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/96.jpg)
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
Wednesday, November 18, 2009
![Page 97: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/97.jpg)
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
M3T0 M3T1 M3T2 M3T3
Wednesday, November 18, 2009
![Page 98: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/98.jpg)
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
Wednesday, November 18, 2009
![Page 99: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/99.jpg)
Pro$nite monoid
3 ∞ 10 ∞ 12 ∞ ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.
Wednesday, November 18, 2009
![Page 100: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/100.jpg)
Pro$nite monoid
∞ ∞ 10 ∞ 12 ∞ ∞
3 ∞ 10 ∞ 12 ∞ ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
..
Wednesday, November 18, 2009
![Page 101: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/101.jpg)
Pro$nite monoid
∞ ∞ 10 ∞ 1∞ ∞ ∞
∞ ∞ 10 ∞ 12 ∞ ∞
3 ∞ 10 ∞ 12 ∞ ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
...
Wednesday, November 18, 2009
![Page 102: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/102.jpg)
Pro$nite monoid
∞ ∞ ∞0 ∞ ∞∞ ∞ ∞
∞ ∞ 10 ∞ 1∞ ∞ ∞
∞ ∞ 10 ∞ 12 ∞ ∞
3 ∞ 10 ∞ 12 ∞ ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
....
Wednesday, November 18, 2009
![Page 103: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/103.jpg)
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
....
Wednesday, November 18, 2009
![Page 104: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/104.jpg)
Pro$nite monoid
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 105: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/105.jpg)
Pro$nite monoid
-approximation:0
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
∞ ∞ ∞0 ∞ ∞∞ ∞ ∞
Wednesday, November 18, 2009
![Page 106: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/106.jpg)
Pro$nite monoid
-approximation:1
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
∞ ∞ 10 ∞ 1∞ ∞ ∞
Wednesday, November 18, 2009
![Page 107: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/107.jpg)
Pro$nite monoid
-approximation:2
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
∞ ∞ 10 ∞ 12 ∞ ∞
Wednesday, November 18, 2009
![Page 108: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/108.jpg)
Pro$nite monoid
-approximation:3
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
3 ∞ 10 ∞ 12 ∞ ∞
Wednesday, November 18, 2009
![Page 109: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/109.jpg)
Pro$nite monoid
-approximation:3 ∞ 10 ∞ 12 7 ∞
7
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 110: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/110.jpg)
Pro$nite monoid
-approximation:3 32 10 11 12 7 ∞
1000
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 111: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/111.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
S0 S1 S2 S3 S7 S32 S1000
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 112: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/112.jpg)
Pro$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
![Page 113: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/113.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 114: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/114.jpg)
Pro$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
![Page 115: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/115.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 116: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/116.jpg)
Pro$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
![Page 117: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/117.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 118: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/118.jpg)
Pro$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
![Page 119: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/119.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish them
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 120: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/120.jpg)
Pro$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
![Page 121: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/121.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings
multiplication
.... . . .........................
Wednesday, November 18, 2009
![Page 122: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/122.jpg)
Pro$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
![Page 123: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/123.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings
3 8 43 8 45 18 3
=
is continuous
multiplication
·
.... . . .........................
Wednesday, November 18, 2009
![Page 124: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/124.jpg)
Pro$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
![Page 125: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/125.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
e n-approximation of x·y is the product of their n-approximations.we again obtain a sequence consistent with the mappings
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplication
·
.... . . .........................
Wednesday, November 18, 2009
![Page 126: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/126.jpg)
Pro$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
![Page 127: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/127.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplicationaddition
·
.... . . .........................
Wednesday, November 18, 2009
![Page 128: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/128.jpg)
Pro$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
![Page 129: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/129.jpg)
Pro$nite monoid
3 32 10 11 12 7 ∞
Metric Two elements are close if only an approx. with high threshold can distinguish themMultiplication
S0 S1 S2 S3 S7 S32 S1000
. . . . . . .........................
3 50 10 11 12 7 ∞
}d = 2-32
preserve
. . . . . . .........................
3 8 43 8 45 18 3
=
is continuous
.....................................
multiplicationaddition
addition
idempotent power (ω)
·
.... . . .........................
Wednesday, November 18, 2009
![Page 130: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/130.jpg)
Pro$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
![Page 131: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/131.jpg)
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
![Page 132: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/132.jpg)
Wednesday, November 18, 2009
![Page 133: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/133.jpg)
Example of a min-automaton A.
Wednesday, November 18, 2009
![Page 134: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/134.jpg)
Example of a min-automaton A.
0 ⊤ ⊤
⊤ 1 ⊤
⊤ ⊤ 0
1 1 ⊤
⊤ ⊤ 0
⊤ ⊤ 0a: b:
∞ ∞ 1⊤ ⊤ 0
⊤ ⊤ 0bω:
Wednesday, November 18, 2009
![Page 135: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/135.jpg)
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
![Page 136: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/136.jpg)
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
![Page 137: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/137.jpg)
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
![Page 138: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/138.jpg)
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
![Page 139: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/139.jpg)
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
![Page 140: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/140.jpg)
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
![Page 141: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/141.jpg)
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
![Page 142: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/142.jpg)
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
![Page 143: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/143.jpg)
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
![Page 144: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/144.jpg)
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
![Page 145: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/145.jpg)
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
![Page 146: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/146.jpg)
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
![Page 147: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/147.jpg)
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
![Page 148: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/148.jpg)
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
![Page 149: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/149.jpg)
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
![Page 150: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/150.jpg)
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
![Page 151: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/151.jpg)
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
![Page 152: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/152.jpg)
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
![Page 153: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/153.jpg)
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
![Page 154: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/154.jpg)
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
![Page 155: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/155.jpg)
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
![Page 156: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/156.jpg)
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
![Page 157: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/157.jpg)
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
![Page 158: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/158.jpg)
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
![Page 159: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/159.jpg)
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
![Page 160: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/160.jpg)
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
![Page 161: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/161.jpg)
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
![Page 162: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/162.jpg)
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
![Page 163: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/163.jpg)
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
![Page 164: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/164.jpg)
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
![Page 165: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/165.jpg)
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
![Page 166: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/166.jpg)
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
![Page 167: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/167.jpg)
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
![Page 168: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/168.jpg)
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
![Page 169: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/169.jpg)
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
![Page 170: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/170.jpg)
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
![Page 171: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/171.jpg)
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
![Page 172: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/172.jpg)
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
![Page 173: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/173.jpg)
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
![Page 174: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/174.jpg)
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
![Page 175: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/175.jpg)
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
![Page 176: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/176.jpg)
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
![Page 177: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/177.jpg)
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
![Page 178: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/178.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Wednesday, November 18, 2009
![Page 179: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/179.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
Wednesday, November 18, 2009
![Page 180: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/180.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # )
Wednesday, November 18, 2009
![Page 181: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/181.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
Wednesday, November 18, 2009
![Page 182: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/182.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
Wednesday, November 18, 2009
![Page 183: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/183.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
semigroup with stabilization
( S , · , # ){
• s # = (s n )# for n=1,2,3,...
Wednesday, November 18, 2009
![Page 184: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/184.jpg)
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
![Page 185: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/185.jpg)
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
![Page 186: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/186.jpg)
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
![Page 187: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/187.jpg)
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
![Page 188: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/188.jpg)
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
![Page 189: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/189.jpg)
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
![Page 190: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/190.jpg)
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
![Page 191: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/191.jpg)
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
![Page 192: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/192.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 193: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/193.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 194: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/194.jpg)
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 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 195: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/195.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 196: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/196.jpg)
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 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 197: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/197.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 198: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/198.jpg)
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 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 199: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/199.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
1 1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 200: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/200.jpg)
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 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 201: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/201.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
01 1 1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 202: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/202.jpg)
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 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 203: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/203.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
01 1 1 1
1
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 204: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/204.jpg)
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 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 205: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/205.jpg)
Simon’s Factorization eoremfor semigroups with stabilization
Factorization tree of word w ∈ S+
Use the two rules to construct tree:binary rule idempotent rule
s t
st
e ee e
e#
10 0 0 0 0 00001 1 1 1 10
0 01 1 1 1
1 1
∞
Example 2 ($nite) ({0,1, ∞}, +, # ), 1+1=1, 0# =0, 1# = ∞# = ∞
Wednesday, November 18, 2009
![Page 206: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/206.jpg)
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
![Page 207: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/207.jpg)
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
![Page 208: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/208.jpg)
Wednesday, November 18, 2009
![Page 209: Deciding Emptiness of min-automata - mimuw.edu.plszymtor/talks/referat distance workshop.pdf · Deciding Emptiness of min-automata Szymon Toruńczyk joint work with Mikołaj Bojańczyk](https://reader030.vdocuments.us/reader030/viewer/2022041200/5d3f7d4788c993f37c8d451e/html5/thumbnails/209.jpg)
ank you for your attention!
Wednesday, November 18, 2009