Approximate comparison of distance automata
Thomas Colcombet
Laure Daviaud
LIAFA, CNRS
Highlights of logic, games and automata, Paris 2013
T.Colcombet, L.Daviaud (LIAFA)
Comparison of distance automata
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Distance automata
Distance automaton: Non deterministic nite automaton for which each transition is also labelled by a non-negative integer called the weight of the transition.
(A, Q , I , T , E ) with E ⊆ (Q × A × N × Q ))
b: 1
q1
a: 0
q2
a,b: 0
b: 0
q3
b: 0
a: 1
T.Colcombet, L.Daviaud (LIAFA)
q4
a,b: 0
Comparison of distance automata
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Distance automata
Distance automaton: Non deterministic nite automaton for which each transition is also labelled by a non-negative integer called the weight of the transition.
(A, Q , I , T , E ) with E ⊆ (Q × A × N × Q ))
b: 1
Weight of a run:
sum of the weights of the transitions
q1
a: 0
q2
a,b: 0
b: 0
q3
b: 0
a: 1
T.Colcombet, L.Daviaud (LIAFA)
q4
a,b: 0
Comparison of distance automata
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Distance automata
Distance automaton: Non deterministic nite automaton for which each transition is also labelled by a non-negative integer called the weight of the transition.
(A, Q , I , T , E ) with E ⊆ (Q × A × N × Q ))
b: 1
Weight of a run:
sum of the weights of the transitions
q1
Computed function:
a: 0
q2
a,b: 0
b: 0
q3
A∗
w
b: 0
a: 1
T.Colcombet, L.Daviaud (LIAFA)
q4
→
7
→
N ∪ {+∞}
minimum of the weights of
the runs labelled by w going
from an initial state
to a nal state
(+∞ if no such run)
a,b: 0
Comparison of distance automata
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Distance automata
b: 1
q
1
a: 0
b: 0
q
2
a,b: 0
T.Colcombet, L.Daviaud (LIAFA)
q
3
a: 1
b: 0
q
4
a,b: 0
Comparison of distance automata
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Distance automata
b: 1
q
1
a: 0
b: 0
q
2
a,b: 0
an ban b · · · bank
0
T.Colcombet, L.Daviaud (LIAFA)
1
q
3
a: 1
b: 0
q
4
a,b: 0
7→ min(n0 , n1 , . . . , nk , k )
Comparison of distance automata
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Decision problems on comparison
f, g
A∗ → N ∪ {+∞}
words w , f (w ) 6 g (w )
computed by distance automata :
f
6g
T.Colcombet, L.Daviaud (LIAFA)
if for all
Comparison of distance automata
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Decision problems on comparison
f, g
A∗ → N ∪ {+∞}
words w , f (w ) 6 g (w )
computed by distance automata :
f
6g
if for all
Undecidable [Krob, 92]
Decidable [Colcombet, 09]
Given
Is there a polynomial
f, g
computed by
f
distance automata,
is
f
6g
?
6P ◦g
P
s.t
?
(context of cost functions)
Generalisation of results by
Hashiguchi, Leung and Simon
T.Colcombet, L.Daviaud (LIAFA)
Comparison of distance automata
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Theorem of ane domination
Proposition
Given
f, g
computed by distance automata, the two assertions are
equivalent:
1
There is a polynomial
2
There is an integer
P
s.t
a s.t f
f
6 P ◦ g.
6 ag + a.
Theorem
f , g computed by distance automata, one can decide if there is an
integer a s.t f 6 ag + a.
Given
T.Colcombet, L.Daviaud (LIAFA)
Comparison of distance automata
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Theorem of approximate comparison
Input :
f ,g
N
computed by distance automata and
ε>0
(1 + ε)g
g
f
=⇒
YES
=⇒
NO
=⇒
YES or NO
A∗
N
(1 + ε)g
g
f
A∗
N
(1 + ε)g
g
f
A∗
Theorem: Existence of an algorithm having this behaviour.
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Comparison of distance automata
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Conclusion and further questions
[Krob, 92]
f 6g ?
Undecidable
↓
[Colcombet, 09]
Is there a polynomial P s.t
f 6P ◦g ?
Decidable
Algorithm of approximate
m
comparison
Decidable
Is there an integer
EXPSPACE
(problem PSPACE-hard)
f
6 ag + a
a s.t
?
Next steps
Capture other kinds of asymptotic behaviours
Case of max-+ automata
T.Colcombet, L.Daviaud (LIAFA)
Comparison of distance automata
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