Automata on linear orderings
Olivier Carton
LIAFA
Université Paris Diderot
Orléans 2009
Olivier Carton (LIAFA)
Automata on orderings
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1
Introduction
2
Orderings and words
3
Automata
Automata
Examples
4
Results
Kleene’s theorem
Complementation
Logic
5
Questions
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Automata on orderings
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Infinite computations
There are automata on
infinite words,
bi-infinite words,
transfinite words,
b
b
b
trees
a
a
a
a
b
a
a
a
a
a
a
a
Olivier Carton (LIAFA)
,
Automata on orderings
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Words
word: function from an ordering J (its length) to an alphabet A
Example (aj )j∈J
abaab
(ab)ω
(ab)−ω
(ab)ω c
((ab)ω c)ω
(a−ω bω )ω
ω
aω
(
r 7→ a if r = n/2m
r 7→ b if r ∈ Q \ {n/2m | m ≥ 0}
(
x 7→ a if x ∈ Q
x 7→ b if x ∈ R \ Q
Olivier Carton (LIAFA)
Automata on orderings
length J
5
ω
−ω
ω+1
ω2 = ω · ω
ζ ·ω
ωω
η
λ
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Operations on orderings and words
Mirror image: −J backwards linear ordering
Sum of orderings and concatenation of words
◮
J + K places K to the right of J
J
◮
K
if x and y are of length J and K, xy is of length J + K.
Generalized
sum and concatenation
X
Kj generalized sum
◮
j∈J
···
j ′ j ′′
• •
···
J
X
Kj
Kj ′′
j∈J
Y
X
If each word xj is of length Kj ,
xj is of length
Kj
Kj
◮
j
•
Kj ′
j∈J
Olivier Carton (LIAFA)
Automata on orderings
j∈J
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Scattered orderings
dense ordering: ∀i < j ∃k
i < k < j.
scattered ordering: no dense subordering.
Theorem (Hausdorff)
The class of countable scattered orderings is
S
α∈O
Vα where
V0 = {0, 1},
S
P
Vα = { j∈J Kj | J ∈ {ω, −ω} and Kj ∈ β<α Vβ }.
Example
(ω + (−ω)) · −ω = · · · (• • • · · · · · · • ••) (• • • · · · · · · • ••)
X
X
X X
1+
1
=
• • ··· ··· • • =
i∈−ω
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i∈−ω
Automata on orderings
j∈ω
j∈−ω
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Cuts of a linear ordering
A cut of a linear ordering J is a pair (K, L) of intervals such that
K ∪L=J
∀k ∈ K, ∀l ∈ L k < l.
————
| {z } | ————
| {z }
K
J
L
Jˆ linear ordering of the cuts
Examples
J =3
J =ω
J =ζ +ζ
Olivier Carton (LIAFA)
J ∪ Jˆ = | • | • | • |
J ∪ Jˆ = | • | • | • | · · · |
J ∪ Jˆ = · · · | • | • | • | · · · · · · | • | • | • | · · · Automata on orderings
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Consecutive cuts
... •
•
j
•
•
c−
j
•
•
• ...
J ∪ Jˆ
c+
j
Fact
+
Each pair of consecutive cuts is of the form (c−
j , cj ).
Olivier Carton (LIAFA)
Automata on orderings
Orléans 2009
8 / 16
Automata on linear orderings
states - initial states
A = (Q, A, E, I, F )
transitions - final states
E ⊆Q×A×Q
successor transitions
∪ P(Q) × Q
left limit transitions
∪ Q × P(Q)
right limit transitions
Example
b
b
1 → {2}
1
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2
a
3
Automata on orderings
{3} → 1
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Paths
A path labeled by a word x = (aj )j∈J is a sequence of states
γ = (qc )c∈Jˆ such that
a
If | a | , then p → q ∈ E,
p q
If · · · | , then P → q ∈ E,
P q
If | · · · , then q → P ∈ E.
qP
Example
b
b
1 → {2}
1
2
a
{3} → 1
3
(b−ω abω )2 ∈ L(A) but (b−ω abω )ω 6∈ L(A)
···b | b | a | b | b··· ···b | b | a | b | b··· 1 {2}
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2
2
3
3
{3} 1 {2}
2
Automata on orderings
2
3
3
{3} 1
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10 / 16
Precise definitions
For any cut c of J, define the sets
lim γ = {q | ∀c′ < c ∃c′ < k < c
q = qk },
lim
γ = {q | ∀c < c′ ∃c < k < c′
+
q = qk }.
c−
c
◮
◮
◮
aj
+
For any consecutive cuts c−
→ qc+j is a successor
j and cj , qc−
j
transition,
For any cut c with no predecessor, limc− γ → qc is a left limit
transition
For any cut c with no successor, qc → limc+ γ is a right limit
transition
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Automata on orderings
Orléans 2009
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Another example
0
2
a
b
{0, 1, 2, 3, 4} → 0
{0, 1, 2, 3, 4} → 2
1
{0, 1, 2, 3, 4} → 4
0 → {0, 1, 2, 3, 4}
3
1 → {0, 1, 2, 3, 4}
4
Word
Path
Olivier Carton (LIAFA)
3 → {0, 1, 2, 3, 4}
(
r 7→ a
r 7→ b
c 7→ 0
c 7→ 1
c 7→ 2
c 7→ 3
c 7→ 4
if r = n/2m
if r ∈ Q \ {n/2m | m ≥ 0}
if
if
if
if
if
c = c−
r and
c = c+
r and
c = c−
r and
c = c+
r and
c∈R\Q
Automata on orderings
r
r
r
r
= n/2m
= n/2m
6= n/2m
6= n/2m
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12 / 16
Kleene theorem
Rational expressions
∅ empty set and a ∈ A letter
Rational operations
◮
◮
◮
◮
◮
union, product, and star iteration
ω and −ω iterations
ordinal and backwards ordinal iteration
iteration on all linear orderings
shuffle iteration for non-scattered orderings
Theorem (C.-Bruyère, C.-Bès)
A set of words on linear orderings is rational iff it is accepted by some
finite automaton.
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Automata on orderings
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Complementation
Theorem
For scattered and countable orderings, the complement of a rational set
is also rational
Büchi has already pointed out that complementation does not
hold for non-countable ordinals (greater than ω1 ).
→ Only countable orderings are considered.
The set of all scattered orderings is not rational while its
complement is rational.
Determinization is not possible. Some rational sets like (A−ω )−ω
cannot be accepted by a deterministic automaton.
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Automata on orderings
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Logic
Theorem
For scattered and countable orderings, MSO is equivalent to automata.
Theorem
For all orderings, MSO is more expressive than automata.
Olivier Carton (LIAFA)
Automata on orderings
Orléans 2009
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