Post on 24-Jul-2020
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
Dedicated to the memory of Zoltan Esik
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.
The automaton model
p qb
a
b
a
b
B1
The automaton model
p qb
a
b
a
b
B1 L(B1) = A∗bA∗
−→ pb−−→p
a−−→ pb−−→ q −→
The weighted automaton model
p q
12 b
12 a
12 b
a
b
C1
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−−→
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
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
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
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+ . . .
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
)⟩
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}
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
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
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 ?
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
The need for a richer model: eg, the concatenation product
p qa
a
r s
b
b
The need for a richer model: eg, the concatenation product
p qa
a
r s
b
b
pa
a
s
b
b
The need for a richer model: eg, the concatenation product
p qa
a
r s
b
b
pa
a
s
b
b
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
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
A basic result in (classical) automata theory
Theorem (Folk–Lore)
Every ε-NFA is equivalent to an NFA
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
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
A basic result in (classical) automata theory
p rqε a
A basic result in (classical) automata theory
p rqε a
p rq
a
a
A basic result in (classical) automata theory
p rqε a
p rq
a
a
p rq
a
a
A basic result in (classical) automata theory
p rqε a
p rq
a
a
p rq
a
a
p r
a
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
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
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
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
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
⟩
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
Automata and expressions
E2 = (a∗ + b∗)∗
Automata and expressions
E2 = (a∗ + b∗)∗
a
b
The Thompson automaton of E2
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
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p rq2ε 3a
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
q
6a
3a
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
q
6a
3a
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
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
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
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
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
2ε
2
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−−→ ,
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−−→ ,
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 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 + · · ·
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
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
p r
2
6a
p ra
ε
1
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
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
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 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?
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?
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?
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?
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?
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
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
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
A basic question in weighted automata theory
QuestionIs every ε-WFA is equivalent to a WFA?
certainly not !
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?
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 )?
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?
Behaviour of weighted automata
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
Behaviour of weighted automata
A = 〈K,A,Q, I ,E ,T 〉 possibly with ε-transitions
u ∈ A∗
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
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
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!
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∗
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
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
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
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
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
Behaviour of weighted automata
First solution
behaviour well-defined ⇐⇒ acyclic
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)
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
Behaviour of weighted automata
A not acyclic ⇒ weight of u in A may be an infinite sum.
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
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
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
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)
Behaviour of weighted automata
Second point of view (more analytical)
Infinite sums are given a meaning via a topology on K
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
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∗〉〉
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
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
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
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
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
Problems in computing the behaviour of a weighted automaton
−11
−1
1
A1
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
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
Problems in computing the behaviour of a weighted automaton
A2A2
−121
2
12
−12
Problems in computing the behaviour of a weighted automaton
A2A2
(−12
)∗= 2
3
−121
2
12
−12
Problems in computing the behaviour of a weighted automaton
A2A223
13
12
−12
Problems in computing the behaviour of a weighted automaton
A2A2
(−12
)∗= 2
323
13
12
−12
Problems in computing the behaviour of a weighted automaton
A2A223
13
13
Problems in computing the behaviour of a weighted automaton
A2A223
13
13
Problems in computing the behaviour of a weighted automaton
A2A223
19
Problems in computing the behaviour of a weighted automaton
A2A2
(19
)∗= 9
823
19
Problems in computing the behaviour of a weighted automaton
A2A234
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
Problems in computing the behaviour of a weighted automaton
p
1
A3
Problems in computing the behaviour of a weighted automaton
p
1
A3 (1)∗ = undefined
N natural integers A3 not defined
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
Problems in computing the behaviour of a weighted automaton
p
+∞
A3
N natural integers A3 not defined
N N ∪ +∞ compact topology A3 defined
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
Problems in computing the behaviour of a weighted automaton
p q
1+∞
1
1
A4
N N ∪ +∞ compact topology A4 defined
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
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
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
A chicken and egg problem
automaton algorithm
A A
A chicken and egg problem
automaton algorithm
A A
valid ? success ?
A chicken and egg problem
automaton algorithm
A A
valid ? success ?
valid
A chicken and egg problem
automaton algorithm
A A
valid ? success ?
valid =⇒ success
A chicken and egg problem
automaton algorithm
A A
valid ? success ?
valid =⇒ success
success
A chicken and egg problem
automaton algorithm
A A
valid ? success ?
valid =⇒ success
valid?⇐= success
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 ∗
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
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
A new definition of validity for weighted automata
Validity implies the well-definition of behaviour
A new definition of validity for weighted automata
Validity implies the well-definition of behaviour
The notion of validity settles the previous examples
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 .
A new definition of validity for weighted automata
TheoremA is valid iff the behaviour of every covering of A is well-defined
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
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.
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
Deciding validity
DefinitionK topological, ordered, positive, star-domain downward closed
(TOP SDDC)
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
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
Deciding validity
DefinitionIf A is a Q- or R-automaton,
then abs(A) is a Q+- or R+-automaton
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.
Automata and expressions validity
‘Kleene’ theorem
Automata ⇐⇒ Expressions
A ⇐⇒ E
Weighted automata ⇐⇒ Weighted expressions
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
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
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∗)∗
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):
Conclusion
Conclusion
� Semiring structure is weak, topology does not help so much.
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.
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
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.
Conclusion (2)
Conclusion (2)
� Apart the trivial cases, and the TOP SDDC case,decision of validity is never granted, and is to be established.
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.
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.
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.
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!
Hidden parts
� The removal algorithm itself
Hidden parts
� The removal algorithm itself:
� Termination issues (weighted versus Boolean cases)� Complexity issues
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
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
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
DefinitionK topological: K regular Hausdorff ⊕, ⊗ continuous
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 .
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
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
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
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
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
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
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
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
Hidden parts
� The removal algorithm itself
� Details on the topology we put semirings
� Validity of automata and covering
� ‘Infinitary’ axioms : strong, star-strong semirings
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
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
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
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
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
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
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)
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
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
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):
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)