the validity of weighted automata - perso.telecom-paristech.fr ·...
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The validity of weighted automata
Jacques Sakarovitch
CNRS / Universite Denis-Diderot and Telecom ParisTech
Joint work with
Sylvain Lombardy
LaBRI, CNRS / Universite de Bordeaux / Institut Polytechnique de Bordeaux
CIAA, 31 July 2018, Charlottetown, PEI
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Dedicated to the memory of Zoltan Esik
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First version presented at CIAA 2012 under the title:
The removal of weighted ε-transitions,
in: Proc. CIAA 2012, Lect. Notes in Comput. Sci. n◦ 7381.
Published in
International Journal of Algebra and Computation 23 (2013)
DOI: 10.1142/S0218196713400146
Supported by ANR Project 10-INTB-0203 VAUCANSON 2.
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The automaton model
p qb
a
b
a
b
B1
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The automaton model
p qb
a
b
a
b
B1 L(B1) = A∗bA∗
−→ pb−−→p
a−−→ pb−−→ q −→
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
1−−→ p12b−−−→p
12a−−−→ p
12b−−−→ q
1−−→1−−→ p
12b−−−→q
a−−→ qb−−→ q
1−−→
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
1−−→ p12b−−−→p
12a−−−→ p
12b−−−→ q
1−−→1−−→ p
12b−−−→q
a−−→ qb−−→ q
1−−→
� Weight of a path c : product of the weights of transitions in c
� Weight of a word w : sum of the weights of paths with label w
bab �−→ 1
2+
1
8=
5
8
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
1−−→ p12b−−−→p
12a−−−→ p
12b−−−→ q
1−−→1−−→ p
12b−−−→q
a−−→ qb−−→ q
1−−→
� Weight of a path c : product of the weights of transitions in c
� Weight of a word w : sum of the weights of paths with label w
bab �−→ 1
2+
1
8=
5
8= <0.101>2
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1 C1 ∈ Q〈〈A∗〉〉
1−−→ p12b−−−→p
12a−−−→ p
12b−−−→ q
1−−→1−−→ p
12b−−−→q
a−−→ qb−−→ q
1−−→
� Weight of a path c : product of the weights of transitions in c
� Weight of a word w : sum of the weights of paths with label w
bab �−→ 1
2+
1
8=
5
8C1 : A∗ −→ Q
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1 C1 ∈ Q〈〈A∗〉〉
1−−→ p12b−−−→p
12a−−−→ p
12b−−−→ q
1−−→1−−→ p
12b−−−→q
a−−→ qb−−→ q
1−−→
� Weight of a path c : product of the weights of transitions in c
� Weight of a word w : sum of the weights of paths with label w
C1 =1
2b+
1
4ab+
1
2b a+
3
4b b+
1
8aab+
1
4ab a+
3
8ab b+
1
2b aa+ . . .
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
C1 =⟨I1,E1,T1
⟩=
⟨(1 0
),
(12 a +
12 b
12 b
0 a+ b
),
(01
)⟩
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The weighted automaton model
A = 〈 I ,E ,T 〉 E = adjacency matrix
E p,q =∑
{wl(e) | e transition from p to q}= linear combination of letters in A
E np,q =
∑{wl(c) | c computation from p to q of length n}
E ∗ =∑n∈N
E n
Since E is proper, E ∗ is well-defined
E ∗p,q =
∑{wl(c) | c computation from p to q}
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
C1 =⟨I1,E1,T1
⟩=
⟨(1 0
),
(12 a +
12 b
12 b
0 a+ b
),
(01
)⟩
C1 = I1 · E1∗ · T1
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
C1 =⟨I1,E1,T1
⟩=
⟨(1 0
),
(12 a +
12 b
12 b
0 a+ b
),
(01
)⟩
C1 = I1 · E1∗ · T1
Every K-automaton defines a series in K〈〈A∗〉〉whose coefficients are effectively computable
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
C1 = I1 · E1∗ · T1
Every K-automaton defines a series in K〈〈A∗〉〉whose coefficients are effectively computable
Where is the problem ?
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The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
C1 = I1 · E1∗ · T1
Every K-automaton defines a series in K〈〈A∗〉〉whose coefficients are effectively computable
Where is the problem ?
We want to be able to deal with weighted automata
where transitions might be labelled by the empty word
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The need for a richer model: eg, the concatenation product
p qa
a
r s
b
b
![Page 19: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/19.jpg)
The need for a richer model: eg, the concatenation product
p qa
a
r s
b
b
pa
a
s
b
b
![Page 20: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/20.jpg)
The need for a richer model: eg, the concatenation product
p qa
a
r s
b
b
pa
a
s
b
b
![Page 21: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/21.jpg)
The need for a richer model: eg, the concatenation product
p qa
a
r s
b
b
pa
a
s
b
b
p qa
a
r s
b
b
ε
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The need for a richer model: eg, the concatenation product
p qa
a
r s
b
b
pa
a
s
b
b
p qa
a
r s
b
b
ε
p qa
a
r s
b
b
b
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A basic result in (classical) automata theory
Theorem (Folk–Lore)
Every ε-NFA is equivalent to an NFA
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A basic result in (classical) automata theory
Theorem (Folk–Lore)
Every ε-NFA is equivalent to an NFA
Usefulness of ε-transitions:
Preliminary step for many constructions on NFA’s:
� Product and star of position (Glushkov, standard) automata
� Thompson construction
� Construction of the universal automaton
� Computation of the image of a transducer
� ...
May correspond to the structure of the computations
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A basic result in (classical) automata theory
Theorem (Folk–Lore)
Every ε-NFA is equivalent to an NFA
Usefulness of ε-transitions:
Preliminary step for many constructions on NFA’s:
� Product and star of position (Glushkov, standard) automata
� Thompson construction
� Construction of the universal automaton
� Computation of the image of a transducer
� ...
May correspond to the structure of the computations
Removal of ε-transitions is implemented in all automata software
![Page 26: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/26.jpg)
A basic result in (classical) automata theory
p rqε a
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A basic result in (classical) automata theory
p rqε a
p rq
a
a
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A basic result in (classical) automata theory
p rqε a
p rq
a
a
p rq
a
a
![Page 29: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/29.jpg)
A basic result in (classical) automata theory
p rqε a
p rq
a
a
p rq
a
a
p r
a
![Page 30: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/30.jpg)
A basic result in (classical) automata theory
p rqε a
p rqa
ε
ε
p rq
a
a
p rq
a
a
p r
a
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A basic result in (classical) automata theory
p rqε a
p rqa
ε
ε
p rq
a
ap rq
aε
ε
p rq
a
a
p r
a
![Page 32: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/32.jpg)
A basic result in (classical) automata theory
p rqε a
p rqa
ε
ε
p rq
a
ap rq
aε
ε
p rq
a
ap r
aε
p r
a
![Page 33: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/33.jpg)
A basic result in (classical) automata theory
p rqε a
p rqa
ε
ε
p rq
a
ap rq
aε
ε
p rq
a
ap r
aε
p r
a
p ra
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A basic result in (classical) automata theory
Theorem (Folk–Lore)
Every ε-NFA is equivalent to an NFA
A proof
A = 〈 I ,E ,T 〉 E transition matrix of AEntries of E = subsets of A ∪ {ε}
L(A) = I · E ∗ · TE = E 0 + Ep
L(A) = I · (E 0 + E p)∗ · T = I · (E ∗
0 · Ep)∗ · E ∗
0 · TA = 〈 I ,E ,T 〉 equivalent to B =
⟨I ,E ∗
0 · Ep,E∗0 · T
⟩
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A basic result in (classical) automata theory
Theorem (Folk–Lore)
Every ε-NFA is equivalent to an NFA
A proof
A = 〈 I ,E ,T 〉 E transition matrix of AEntries of E = subsets of A ∪ {ε}
L(A) = I · E ∗ · TE = E 0 + Ep
L(A) = I · (E 0 + E p)∗ · T = I · (E ∗
0 · Ep)∗ · E ∗
0 · TA = 〈 I ,E ,T 〉 equivalent to B =
⟨I ,E ∗
0 · Ep,E∗0 · T
⟩
One proof = several algorithms for computing E ∗0 or E ∗
0 · Ep
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Automata and expressions
E2 = (a∗ + b∗)∗
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Automata and expressions
E2 = (a∗ + b∗)∗
a
b
The Thompson automaton of E2
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Automata and expressions
E2 = (a∗ + b∗)∗
a
b
The Thompson automaton of E2
Theorem (Folk–Lore ?)
The closure of the Thompson automaton of Eyields the position automaton of E
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A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
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A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p rq2ε 3a
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A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
q
6a
3a
![Page 42: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/42.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
q
6a
3a
![Page 43: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/43.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
![Page 44: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/44.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
q
2ε
ε
![Page 45: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/45.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
q
2ε
2ε
![Page 46: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/46.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
q
2ε
2ε
![Page 47: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/47.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
2ε
2
![Page 48: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/48.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
2ε
2
1−−→ pa−−→ r
1−−→ ,
![Page 49: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/49.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
2ε
2
1−−→ pa−−→ r
1−−→ ,1−−→ p
2 ε−−−→ pa−−→ r
1−−→ ,
![Page 50: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/50.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
2ε
2
1−−→ pa−−→ r
1−−→ ,1−−→ p
2 ε−−−→ pa−−→ r
1−−→ ,
1−−→ p2 ε−−−→ p
2 ε−−−→ pa−−→ r
1−−→ , . . .
![Page 51: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/51.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
2ε
2
1−−→ pa−−→ r
1−−→ ,1−−→ p
2 ε−−−→ pa−−→ r
1−−→ ,
1−−→ p2 ε−−−→ p
2 ε−−−→ pa−−→ r
1−−→ , . . .
a �−→ 1 + 2 + 4 + · · ·
![Page 52: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/52.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
2ε
2
1−−→ pa−−→ r
1−−→ ,1−−→ p
2 ε−−−→ pa−−→ r
1−−→ ,
1−−→ p2 ε−−−→ p
2 ε−−−→ pa−−→ r
1−−→ , . . .
a �−→ 1 + 2 + 4 + · · · undefined
![Page 53: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/53.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
ε
1
![Page 54: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/54.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
ε
1
1−−→ pa−−→ r
1−−→ ,1−−→ p
ε−−→ pa−−→ r
1−−→ ,
1−−→ pε−−→ p
ε−−→ pa−−→ r
1−−→ , . . .
a �−→ 1 + 1 + 1 + · · · undefined
![Page 55: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/55.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
12 ε
12
![Page 56: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/56.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
12 ε
12
1−−→ pa−−→ r
1−−→ ,1−−→ p
12ε−−−→ p
a−−→ r1−−→ ,
1−−→ p12ε−−−→ p
12ε−−−→ p
a−−→ r1−−→ , . . .
![Page 57: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/57.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
12 ε
12
1−−→ pa−−→ r
1−−→ ,1−−→ p
12ε−−−→ p
a−−→ r1−−→ ,
1−−→ p12ε−−−→ p
12ε−−−→ p
a−−→ r1−−→ , . . .
a �−→ 1 + 12 +
14 + · · · undefined?
![Page 58: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/58.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
12 ε
12
1−−→ pa−−→ r
1−−→ ,1−−→ p
12ε−−−→ p
a−−→ r1−−→ ,
1−−→ p12ε−−−→ p
12ε−−−→ p
a−−→ r1−−→ , . . .
a �−→ 1 + 12 +
14 + · · · undefined? defined?
![Page 59: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/59.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
12 ε
12
1−−→ pa−−→ r
1−−→ ,1−−→ p
12ε−−−→ p
a−−→ r1−−→ ,
1−−→ p12ε−−−→ p
12ε−−−→ p
a−−→ r1−−→ , . . .
a �−→ 1 + 12 +
14 + · · · undefined? defined?
![Page 60: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/60.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
12 ε
12
1−−→ pa−−→ r
1−−→ ,1−−→ p
12ε−−−→ p
a−−→ r1−−→ ,
1−−→ p12ε−−−→ p
12ε−−−→ p
a−−→ r1−−→ , . . .
a �−→ 1 + 12 +
14 + · · · undefined? defined?
![Page 61: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/61.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p r2a
1
1−−→ pa−−→ r
1−−→ ,1−−→ p
12ε−−−→ p
a−−→ r1−−→ ,
1−−→ p12ε−−−→ p
12ε−−−→ p
a−−→ r1−−→ , . . .
a �−→ 1 + 12 +
14 + · · · undefined? defined?
![Page 62: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/62.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p r
k εa
1
![Page 63: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/63.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p r
k εa
1
![Page 64: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/64.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p rk∗a
k∗
if k∗ =∞∑n=0
kn is defined in K
![Page 65: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/65.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
certainly not !
![Page 66: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/66.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
certainly not !
New questions
Which ε-WFAs have a well-defined behaviour?
![Page 67: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/67.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
certainly not !
New questions
Which ε-WFAs have a well-defined behaviour?
How to compute the behaviour of an ε-WFA (when it is well-defined )?
![Page 68: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/68.jpg)
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
certainly not !
New questions
Which ε-WFAs have a well-defined behaviour?
How to compute the behaviour of an ε-WFA (when it is well-defined )?
How to decide if the behaviour of an ε-WFA is well-defined?
![Page 69: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/69.jpg)
Behaviour of weighted automata
![Page 70: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/70.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
![Page 71: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/71.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗
![Page 72: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/72.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A
![Page 73: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/73.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A<A , u> sum of weights of computations labelled by u in A
![Page 74: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/74.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A<A , u> sum of weights of computations labelled by u in A
if it is defined!
![Page 75: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/75.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A<A , u> sum of weights of computations labelled by u in A
if it is defined!
A is defined if <A , u> is defined ∀u ∈ A∗
![Page 76: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/76.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A<A , u> sum of weights of computations labelled by u in A
if it is defined!
A is defined if <A , u> is defined ∀u ∈ A∗
Trivial case
![Page 77: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/77.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A<A , u> sum of weights of computations labelled by u in A
if it is defined!
A is defined if <A , u> is defined ∀u ∈ A∗
Trivial case
Every u in A∗ is the label of a finite number of paths
![Page 78: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/78.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A<A , u> sum of weights of computations labelled by u in A
if it is defined!
A is defined if <A , u> is defined ∀u ∈ A∗
Trivial case
Every u in A∗ is the label of a finite number of paths
⇑no ε-transitions in A
![Page 79: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/79.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A<A , u> sum of weights of computations labelled by u in A
if it is defined!
A is defined if <A , u> is defined ∀u ∈ A∗
Trivial case
Every u in A∗ is the label of a finite number of paths
no circuits of ε-transitions in A
![Page 80: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/80.jpg)
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗ possibly infinitely many paths labelled by u in A<A , u> sum of weights of computations labelled by u in A
if it is defined!
A is defined if <A , u> is defined ∀u ∈ A∗
Trivial case
Every u in A∗ is the label of a finite number of paths
no circuits of ε-transitions in A
acyclic K-automata
![Page 81: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/81.jpg)
Behaviour of weighted automata
First solution
behaviour well-defined ⇐⇒ acyclic
![Page 82: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/82.jpg)
Behaviour of weighted automata
First solution
behaviour well-defined ⇐⇒ acyclic
Legitimate, as far as the behaviours of the automata are concerned
(Kuich–Salomaa 86, Berstel–Reutenauer 84-88;11)
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Behaviour of weighted automata
First solution
behaviour well-defined ⇐⇒ acyclic
Legitimate, as far as the behaviours of the automata are concerned
(Kuich–Salomaa 86, Berstel–Reutenauer 84-88;11)
p ra
12 ε
12
not valid
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Behaviour of weighted automata
A not acyclic ⇒ weight of u in A may be an infinite sum.
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Behaviour of weighted automata
A not acyclic ⇒ weight of u in A may be an infinite sum.
Second family of solutions
Accepting the idea of infinite sums
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Behaviour of weighted automata
A not acyclic ⇒ weight of u in A may be an infinite sum.
Second family of solutions
Accepting the idea of infinite sums
First point of view (algebraico-logic)
� Definition of a new operator for infinite sums∑
I
� Setting axioms on∑
I
such that the star of a matrix be meaningful
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Behaviour of weighted automata
A not acyclic ⇒ weight of u in A may be an infinite sum.
Second family of solutions
Accepting the idea of infinite sums
First point of view (algebraico-logic)
� Definition of a new operator for infinite sums∑
I
� Setting axioms on∑
I
such that the star of a matrix be meaningful
Less a definition on automata than conditions on K
for all K-automata have well-defined behaviour
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Behaviour of weighted automata
A not acyclic ⇒ weight of u in A may be an infinite sum.
Second family of solutions
Accepting the idea of infinite sums
First point of view (algebraico-logic)
� Definition of a new operator for infinite sums∑
I
� Setting axioms on∑
I
such that the star of a matrix be meaningful
Less a definition on automata than conditions on K
for all K-automata have well-defined behaviour
Works of Bloom, Esik, Kuich (90’s –)based on the axiomatisation described by Conway (72)
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Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
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Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
Topology allows to define summable families in K
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Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
Topology on K defines a topology on K〈〈A∗〉〉
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Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
Topology on K defines a topology on K〈〈A∗〉〉
Third solution (Lombardy, S. 03 –)
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
PA set of all paths in A
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Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
Topology on K defines a topology on K〈〈A∗〉〉
Third solution (Lombardy, S. 03 –)
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
PA set of all paths in AA well-defined ⇐⇒ WL
(PA
)summable
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Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
Topology on K defines a topology on K〈〈A∗〉〉
Third solution (Lombardy, S. 03 –)
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
PA set of all paths in AA well-defined ⇐⇒ ∀p, q ∈ Q WL
(PA(p, q)
)summable
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Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
Topology on K defines a topology on K〈〈A∗〉〉
Third solution (Lombardy, S. 03 –)
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
PA set of all paths in AA well-defined ⇐⇒ ∀p, q ∈ Q WL
(PA(p, q)
)summable
� Yields a consistent theory
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Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
Topology on K defines a topology on K〈〈A∗〉〉
Third solution (Lombardy, S. 03 –)
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
PA set of all paths in AA well-defined ⇐⇒ ∀p, q ∈ Q WL
(PA(p, q)
)summable
� Yields a consistent theory
� Two pitfalls for effectivity� effective computation of a summable family may not be possible� effective computation may give values to non summable families
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Problems in computing the behaviour of a weighted automaton
−11
−1
1
A1
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Problems in computing the behaviour of a weighted automaton
−11
−1
1
A1
A1 =⟨I1,E1,T1
⟩=
⟨(1 0
),(
1 1−1 −1
),(10
)⟩
A1 = I1 · E1∗ · T1
E12 = 0 =⇒ E1
∗ =(
2 1−1 0
)=⇒ A1 = 2
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Problems in computing the behaviour of a weighted automaton
−11
−1
1
A1
A1 =⟨I1,E1,T1
⟩=
⟨(1 0
),(
1 1−1 −1
),(10
)⟩
A1 = I1 · E1∗ · T1
E12 = 0 =⇒ E1
∗ =(
2 1−1 0
)=⇒ A1 = 2
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Problems in computing the behaviour of a weighted automaton
A2A2
−121
2
12
−12
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Problems in computing the behaviour of a weighted automaton
A2A2
(−12
)∗= 2
3
−121
2
12
−12
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Problems in computing the behaviour of a weighted automaton
A2A223
13
12
−12
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Problems in computing the behaviour of a weighted automaton
A2A2
(−12
)∗= 2
323
13
12
−12
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Problems in computing the behaviour of a weighted automaton
A2A223
13
13
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Problems in computing the behaviour of a weighted automaton
A2A223
13
13
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Problems in computing the behaviour of a weighted automaton
A2A223
19
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Problems in computing the behaviour of a weighted automaton
A2A2
(19
)∗= 9
823
19
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Problems in computing the behaviour of a weighted automaton
A2A234
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Problems in computing the behaviour of a weighted automaton
A2A234
A2 =⟨I2,E2,T2
⟩=
⟨(1 0
),(−1
212
12 −1
2
),(10
)⟩
A2 = I2 · E2∗ · T2
E23 = E2 =⇒ E2
∗ undefined =⇒ A2 undefined
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Problems in computing the behaviour of a weighted automaton
p
1
A3
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Problems in computing the behaviour of a weighted automaton
p
1
A3 (1)∗ = undefined
N natural integers A3 not defined
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Problems in computing the behaviour of a weighted automaton
p
1
A3 (1)∗ = +∞
N natural integers A3 not defined
N N ∪ +∞ compact topology A3 defined
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Problems in computing the behaviour of a weighted automaton
p
+∞
A3
N natural integers A3 not defined
N N ∪ +∞ compact topology A3 defined
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Problems in computing the behaviour of a weighted automaton
p
1
A3 (1)∗ = undefined
N natural integers A3 not defined
N N ∪ +∞ compact topology A3 defined
N∞ N ∪ +∞ discrete topology A3 not defined
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Problems in computing the behaviour of a weighted automaton
p q
1+∞
1
1
A4
N N ∪ +∞ compact topology A4 defined
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Problems in computing the behaviour of a weighted automaton
p q
1+∞
1
1
A4
N N ∪ +∞ compact topology A4 defined
N∞ N ∪ +∞ discrete topology A4 defined
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Problems in computing the behaviour of a weighted automaton
p q
1+∞
1
1
A4
N N ∪ +∞ compact topology A4 defined
N∞ N ∪ +∞ discrete topology A4 defined
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Problems in computing the behaviour of a weighted automaton
p q
1+∞
1
1
A4 (1)∗ = undefined
N N ∪ +∞ compact topology A4 defined
N∞ N ∪ +∞ discrete topology A4 defined
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A chicken and egg problem
automaton algorithm
A A
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A chicken and egg problem
automaton algorithm
A A
valid ? success ?
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A chicken and egg problem
automaton algorithm
A A
valid ? success ?
valid
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A chicken and egg problem
automaton algorithm
A A
valid ? success ?
valid =⇒ success
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A chicken and egg problem
automaton algorithm
A A
valid ? success ?
valid =⇒ success
success
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A chicken and egg problem
automaton algorithm
A A
valid ? success ?
valid =⇒ success
valid?⇐= success
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A new definition of validity for weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
E ∗ free monoid generated by E
PA set of paths in A (local) rational subset of E ∗
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A new definition of validity for weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
E ∗ free monoid generated by E
PA set of paths in A (local) rational subset of E ∗
DefinitionR rational family of paths of A R ∈ RatE ∗ ∧ R ⊆ PA
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A new definition of validity for weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
E ∗ free monoid generated by E
PA set of paths in A (local) rational subset of E ∗
DefinitionR rational family of paths of A R ∈ RatE ∗ ∧ R ⊆ PA
DefinitionA is valid iff
∀R rational family of paths of A, WL(R) is summable
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A new definition of validity for weighted automata
Validity implies the well-definition of behaviour
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A new definition of validity for weighted automata
Validity implies the well-definition of behaviour
The notion of validity settles the previous examples
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A new definition of validity for weighted automata
Validity implies the well-definition of behaviour
The notion of validity settles the previous examples
RemarkIf every subfamily of a summable family in K is summable,
then validity is equivalent to the well-definition of behaviour
Eg. R , C (and N , Z , N ).
If every rational subfamily of a summable family in K is summable,then validity is equivalent to the well-definition of behaviour
Eg. Q .
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A new definition of validity for weighted automata
TheoremA is valid iff the behaviour of every covering of A is well-defined
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A new definition of validity for weighted automata
TheoremA is valid iff the behaviour of every covering of A is well-defined
TheoremIf A is valid, then ‘every’ removal algorithm on A is successful
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A new definition of validity for weighted automata
TheoremA is valid iff the behaviour of every covering of A is well-defined
TheoremIf A is valid, then ‘every’ removal algorithm on A is successful
Nota BeneWe do not know yet how to decide whether
a Q - or an R -automaton is valid.
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Deciding validity
Straightforward cases
� Non starable semirings (eg. N, Z)A valid ⇐⇒ A acyclic
� Complete topological semirings (eg. N ) every A valid
� Rationally additive semirings (eg. RatA∗ ) every A valid
� Locally closed commutative semirings every A valid
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Deciding validity
DefinitionK topological, ordered, positive, star-domain downward closed
(TOP SDDC)
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Deciding validity
DefinitionK topological, ordered, positive, star-domain downward closed
(TOP SDDC)
N, N , Q+, R+, Zmin, RatA∗,... are TOP SDDC
N∞, (binary) positive decimals,... are not TOP SDDC
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Deciding validity
DefinitionK topological, ordered, positive, star-domain downward closed
(TOP SDDC)
N, N , Q+, R+, Zmin, RatA∗,... are TOP SDDC
N∞, (binary) positive decimals,... are not TOP SDDC
TheoremK topological, ordered, positive, star-domain downward closedA K-automaton is valid if and only if
the ε-removal algorithm succeeds
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Deciding validity
DefinitionIf A is a Q- or R-automaton,
then abs(A) is a Q+- or R+-automaton
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Deciding validity
DefinitionIf A is a Q- or R-automaton,
then abs(A) is a Q+- or R+-automaton
TheoremA Q- or R-automaton A is valid if and only if abs(A) is valid.
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Automata and expressions validity
‘Kleene’ theorem
Automata ⇐⇒ Expressions
A ⇐⇒ E
Weighted automata ⇐⇒ Weighted expressions
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Automata and expressions validity
‘Kleene’ theorem
Automata ⇐⇒ Expressions
A ⇐⇒ E
Weighted automata ⇐⇒ Weighted expressions
Validity of expressions
E valid ⇐⇒ c(E) well-defined
c(E) computed by a bottom-up traversal of the syntactic tree of E
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Automata and expressions validity
Valid A yields valid E
Valid E yields valid A with Glushkov construction
Valid E may yield non valid A with Thompson construction
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Automata and expressions validity
Valid A yields valid E
Valid E yields valid A with Glushkov construction
Valid E may yield non valid A with Thompson construction
a |1
b |1
1 11
1 11
1 1
1
11
1
1
1
1
−1
1
The Thompson automaton of (a∗ + {−1}b∗)∗
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
� Links with the ‘axiomatic’ approach (Bloom–Esik–Kuich)
� References to previous work (on removal algorithms):
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Conclusion
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Conclusion
� Semiring structure is weak, topology does not help so much.
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Conclusion
� Semiring structure is weak, topology does not help so much.
� This weakness imposes a restricted definition of validity,in order to guarantee success of validity algorithms.
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Conclusion
� Semiring structure is weak, topology does not help so much.
� This weakness imposes a restricted definition of validity,in order to guarantee success of validity algorithms.
� Axiomatic approach does not allowto deal wit most common numerical semirings: Zmin, Q
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Conclusion
� Semiring structure is weak, topology does not help so much.
� This weakness imposes a restricted definition of validity,in order to guarantee success of validity algorithms.
� Axiomatic approach does not allowto deal wit most common numerical semirings: Zmin, Q
� On ‘usual’ semirings,the new definition of validity coincides with the former one.
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Conclusion (2)
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Conclusion (2)
� Apart the trivial cases, and the TOP SDDC case,decision of validity is never granted, and is to be established.
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Conclusion (2)
� Apart the trivial cases, and the TOP SDDC case,decision of validity is never granted, and is to be established.
� On ‘usual’ semirings, validity is decidable.
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Conclusion (2)
� Apart the trivial cases, and the TOP SDDC case,decision of validity is never granted, and is to be established.
� On ‘usual’ semirings, validity is decidable.
� The new definition of validityfills the ‘effectivity gap’ left open by the former one.
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Conclusion (2)
� Apart the trivial cases, and the TOP SDDC case,decision of validity is never granted, and is to be established.
� On ‘usual’ semirings, validity is decidable.
� The new definition of validityfills the ‘effectivity gap’ left open by the former one.
� The algorithms implemented in Awaliare given a theoretical framework.
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Conclusion (2)
� Apart the trivial cases, and the TOP SDDC case,decision of validity is never granted, and is to be established.
� On ‘usual’ semirings, validity is decidable.
� The new definition of validityfills the ‘effectivity gap’ left open by the former one.
� The algorithms implemented in Awaliare given a theoretical framework.
All’s well, that ends well!
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Hidden parts
� The removal algorithm itself
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Hidden parts
� The removal algorithm itself:
� Termination issues (weighted versus Boolean cases)� Complexity issues
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Hidden parts
� The removal algorithm itself:
� Termination issues (weighted versus Boolean cases)� Complexity issues
(e) (f )
Boolean ε-removal procedure does not terminate
if newly created ε-transitions are stored in a stack
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Hidden parts
� The removal algorithm itself:
� Termination issues (weighted versus Boolean cases)� Complexity issues
(1)
(2)
(3)
(4)
(5)(6)
(7)(8)
weighted ε-removal procedure does not terminate
if newly created ε-transitions are stored in a queue
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
DefinitionK topological: K regular Hausdorff ⊕, ⊗ continuous
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
DefinitionK topological: K regular Hausdorff ⊕, ⊗ continuous
Definition{ti}i∈I summable of sum t :
∀V ∈ N(t) , ∃JV finite , JV ⊂ I , ∀L finite , JV ⊆ L ⊂ I∑i∈L
ti ∈ V .
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
DefinitionK topological: K regular Hausdorff ⊕, ⊗ continuous
Definition{ti}i∈I summable of sum t :
∀V ∈ N(t) , ∃JV finite , JV ⊂ I , ∀L finite , JV ⊆ L ⊂ I∑i∈L
ti ∈ V .
Lemma (Associativity)
{ti}i∈I summable of sum t ,I =
⋃j∈J Kj ∀j ∈ J {ti}i∈Kj
summable of sum sj ,then {sj}j∈J summable of sum t
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
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Validity of automata and covering
y
A5
S ⊂ N2×2, x =(1 00 1
)= 1S, y =
(0 11 0
), x+y = ∞S =
(1 11 1
)
S equipped with the discrete topology
0S, y , and ∞S starable x = y2 x not starable
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Validity of automata and covering
y
A5
S ⊂ N2×2, x =(1 00 1
)= 1S, y =
(0 11 0
), x+y = ∞S =
(1 11 1
)
S equipped with the discrete topology
0S, y , and ∞S starable x = y2 x not starable
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Validity of automata and covering
∞S
A5
S ⊂ N2×2, x =(1 00 1
)= 1S, y =
(0 11 0
), x+y = ∞S =
(1 11 1
)
S equipped with the discrete topology
0S, y , and ∞S starable x = y2 x not starable
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Validity of automata and covering
y
A5 p q B5
y
y
S ⊂ N2×2, x =(1 00 1
)= 1S, y =
(0 11 0
), x+y = ∞S =
(1 11 1
)
S equipped with the discrete topology
0S, y , and ∞S starable x = y2 x not starable
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Validity of automata and covering
y
A5 p B5
∞S
x
S ⊂ N2×2, x =(1 00 1
)= 1S, y =
(0 11 0
), x+y = ∞S =
(1 11 1
)
S equipped with the discrete topology
0S, y , and ∞S starable x = y2 x not starable
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Validity of automata and covering
y
A5 p B5
∞S
x
S ⊂ N2×2, x =(1 00 1
)= 1S, y =
(0 11 0
), x+y = ∞S =
(1 11 1
)
S equipped with the discrete topology
0S, y , and ∞S starable x = y2 x not starable
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
DefinitionA topological semiring is a strong semiring
if the product of two summable families is a summable family
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
DefinitionA topological semiring is a strong semiring
if the product of two summable families is a summable family
TheoremK strong semiring s ∈ K〈〈A∗〉〉 starable iff s0 ∈ K starable
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Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
DefinitionA topological semiring is a strong semiring
if the product of two summable families is a summable family
TheoremK strong semiring s ∈ K〈〈A∗〉〉 starable iff s0 ∈ K starable
Proposition (Madore 18)
There exist (semi)rings K that are not strong
![Page 175: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/175.jpg)
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
DefinitionA topological semiring is a star-strong semiring ifthe star of a summable family, whose sum is starable, is summable
![Page 176: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/176.jpg)
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
DefinitionA topological semiring is a star-strong semiring ifthe star of a summable family, whose sum is starable, is summable
Proposition
A strong semiring K is starable and star-strong iffevery rational family of K is summable
![Page 177: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/177.jpg)
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
DefinitionA topological semiring is a star-strong semiring ifthe star of a summable family, whose sum is starable, is summable
Proposition
A strong semiring K is starable and star-strong iffevery rational family of K is summable
Conjecture
A starable strong semiring star-strong
![Page 178: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/178.jpg)
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
� Links with the ‘axiomatic’ approach (Bloom–Esik–Kuich)
![Page 179: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/179.jpg)
Hidden parts
� The removal algorithm itself:
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
� Links with the ‘axiomatic’ approach (Bloom–Esik–Kuich):
TheoremA starable star-strong semiring is an iteration semiring
![Page 180: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/180.jpg)
Group identities
i 0
1
2
x0
1K
x1
1K
x21K
i
x0 + x1 + x2
0
1
2
0, 0
1, 0
0, 1
1, 1
2, 11, 2
2, 2
0, 2
2, 0
1K
x01K
1K
x1
x2
1K
1Kx0
1Kx2
x1
1K
1K
1K
x0
x1
x2
0
1
2
x0
x1
x2x2
x0
x1
x1
x2
x0
ε-removalε-removal
covering
![Page 181: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/181.jpg)
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
� Links with the ‘axiomatic’ approach (Bloom–Esik–Kuich)
� References to previous work (on removal algorithms):
![Page 182: The validity of weighted automata - perso.telecom-paristech.fr · FirstversionpresentedatCIAA2012underthetitle: Theremovalofweighted ε-transitions, in: Proc. CIAA2012,Lect. NotesinComput.Sci](https://reader034.vdocuments.pub/reader034/viewer/2022050304/5f6cc61049fb787f9d625a11/html5/thumbnails/182.jpg)
Hidden parts
� The removal algorithm itself:
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
� Links with the ‘axiomatic’ approach (Bloom–Esik–Kuich):
� References to previous work (on removal algorithms):
� locally closed srgs (Esik–Kuich), k-closed srgs (Mohri)� links with other algorithms:
shortest-distance algorithm (Mohri),state-elimination method (Hanneforth–Higueira)