Büchi Complementation via Alternating Automata

Automata Theory Seminar
Büchi Complementation via
Alternating Automata
Fabian Reiter
July 16, 2012

Büchi Complementation
BA B
2 Θ(n log n) BA B
BA: Büchi
Automaton
AA A
AA A
AA: Alternating
Automaton
Expensive: If B has n states, B has 2 Θ(n log n) states in
the worst case (Michel 1988, Safra 1988).
Complicated: Direct approaches are rather involved.
Consider indirect approach: detour over alternating automata.
1 / 33

Transition Modes (1)
Existential: some run is accepting
q 0 q 1a q 2a q 3a q 4a q 5a · · ·
q 0 q 1b q 2b q 3b q 4b q 5b · · ·
q 0 q 1c q 2c q 3c q 4c q 5c · · ·
q 0 q 1d q 2d q 3d q 4d q 5d · · ·
q 0 q 1e q 2e q 3e q 4e q 5e · · ·
Universal: every run is accepting
q 0 q 1a q 2a q 3a q 4a q 5a · · ·
q 0 q 1b q 2b q 3b q 4b q 5b · · ·
q 0 q 1c q 2c q 3c q 4c q 5c · · ·
q 0 q 1d q 2d q 3d q 4d q 5d · · ·
q 0 q 1e q 2e q 3e q 4e q 5e · · ·
2 / 33

Transition Modes (2)
Alternating: in some set of runs every run is accepting
q 0 q 1a q 2a q 3a q 4a q 5a · · ·
q 0 q 1b q 2b q 3b q 4b q 5b · · ·
q 0 q 1c q 2c q 3c q 4c q 5c · · ·
q 0 q 1d q 2d q 3d q 4d q 5d · · ·
q 0 q 1e q 2e q 3e q 4e q 5e · · ·
q 0 q 1f q 2f q 3f q 4f q 5f · · ·
q 0 q 1g q 2g q 3g q 4g q 5g · · ·
q 0 q 1h q 2h q 3h q 4h q 5h · · ·
q 0 q 1i q 2i q 3i q 4i q 5i · · ·
3 / 33

Alternation and Complementation
Special case: A in existential mode
A accepts iff ∃ run ρ : ρ fulfills acceptance condition of A
A accepts iff ∀ run ρ : ¬(ρ fulfills acceptance condition of A)
iff ∀ run ρ: ρ fulfills dual acceptance condition of A
⇒ complementation ̂= dualization of:
transition mode
acceptance condition
Want acceptance condition that is closed under dualization.
4 / 33

Outline
1 Weak Alternating Parity Automata
2 Infinite Parity Games
3 Proof of the Complementation Theorem
4 Büchi Complementation Algorithm
5 / 33

Outline
1 Weak Alternating Parity Automata
Definitions and Examples
Dual Automaton
2 Infinite Parity Games
3 Proof of the Complementation Theorem
4 Büchi Complementation Algorithm
6 / 33

Preview
Example ( (b ∗ a) ω )
Büchi automaton B:
b
a
a
q 0 q 1
b
Equivalent WAPA A:
a
b
a, b
q 0
2
b
•
q 1
1
a
q 2
0
7 / 33

Weak Alternating Parity Automaton
☺
Definition (Weak Alternating Parity Automaton)
A weak alternating parity automaton (WAPA) is a tuple
where
Q
Σ
finite set of states
finite alphabet
δ : Q × Σ → B + (Q)
q in
initial state
π : Q → N
A := 〈Q, Σ, δ, q in , π〉
parity function
transition function
(Thomas and Löding, ∼ 2000)
B + (Q): set of all positive Boolean formulae over Q
(built only from elements in Q ∪ {∧ , ∨, ⊤, ⊥})
8 / 33

Transitions
☺
Example (a ω )
a
q 0
2
a
a a
q
• 1
•
1
•
δ : Q × Σ → B + (Q)
〈q 0 , a〉 ↦→ q 0 ∨ (q 1 ∧ q 2 )
〈q 1 , a〉 ↦→ (q 0 ∧ q 1 ) ∨ (q 1 ∧ q 2 )
q 2
0
a
〈q 2 , a〉 ↦→ q 2
Definition (Minimal Models)
Mod ↓ (θ) ⊆ 2 Q : set of minimal models
of θ ∈ B + (Q), i.e. the set of minimal
subsets M ⊆ Q s.t. θ is satisfied by
{
true if q ∈ M
q ↦→
false otherwise
Example
Mod ↓ (q 0 ∨ (q 1 ∧ q 2 ))
= {{q 0 }, {q 1 , q 2 }}
9 / 33

Run Graph (1)
☺
Example (a ω )
Accepting run:
a
q 0
2
a
a a
q
• 1
•
1
•
q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 · · ·
Rejecting run:
q 2
0
q 0 , 0 q 0 , 1 q 0 , 4
q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 · · ·
q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 · · ·
a
10 / 33

Run Graph (2)
☺
Definition (Run)
A run of a WAPA A = 〈Q, Σ, δ, q in , π〉 on a word a 0 a 1 a 2 . . . ∈ Σ ω
is a directed acyclic graph
where
V ⊆ Q × N
R := 〈V , E〉
with 〈q in , 0〉 ∈ V
V contains only vertices reachable from 〈q in , 0〉.
E contains only edges of the form 〈 〈p, i〉, 〈q, i + 1〉 〉 .
For every vertex 〈p, i〉 ∈ V the set of successors is a minimal
model of δ(p, a i )
{
q ∈ Q |
〈
〈p, i〉, 〈q, i + 1〉
〉
∈ E
}
∈ Mod↓ (δ(p, a i ))
11 / 33

Acceptance
☺
Definition (Acceptance)
Let A be a WAPA, w ∈ Σ ω and R = 〈V , E〉 a run of A on w.
An infinite path ρ in R satisfies the acceptance condition of A
iff the smallest occurring parity is even, i.e.
min{π(q) | ∃ i ∈N:〈q, i〉 occurs in ρ} is even.
R is an accepting run iff every infinite path ρ in R satisfies the
acceptance condition.
A accepts w iff there is some accepting run of A on w.
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Infinitely many a’s
Example ( (b ∗ a) ω )
a
b
a, b
q 0
2
b
•
q 1
1
a
q 2
0
Run on b ω :
q 0 , 0
b
q 0 , 1
b
q 0 , 2
b
q 0 , 3
b
q 0 , 4
b
q 0 , 5
b
q 0 , 6
b
· · ·
q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 · · ·
Run on (ba) ω :
q 0 , 0
b
q 0 , 1
a
q 0 , 2
b
q 0 , 3
a
q 0 , 4
b
q 0 , 5
a
q 0 , 6
b
· · ·
q 1 , 1 q 1 , 3 q 1 , 5 · · ·
q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · ·
13 / 33

Dual Automaton (1)
☺
Definition (Dual Automaton)
The dual of a WAPA A = 〈Q, Σ, δ, q in , π〉 is
A := 〈Q, Σ, δ, q in , π〉
where
δ(q, a) is obtained from δ(q, a) by exchanging ∧ , ∨ and ⊤, ⊥
π(q) := π(q) + 1
for all q ∈ Q and a ∈ Σ
14 / 33

Dual Automaton (2)
Example ( (b ∗ a) ω )
WAPA A:
a
b
a, b
δ(q 0 , a) = q 0
δ(q 0 , b) = q 0 ∧q 1
q 0
2
b
•
q 1
1
a
q 2
0
δ(q 1 , a) = q 2
δ(q 1 , b) = q 1
δ(q 2 , a) = q 2
δ(q 2 , b) = q 2
Dual A:
a, b
b
a, b
δ(q 0 , a) = q 0
δ(q 0 , b) = q 0 ∨q 1
q 0
3
b
q 1
2
a
q 2
1
δ(q 1 , a) = q 2
δ(q 1 , b) = q 1
δ(q 2 , a) = q 2
δ(q 2 , b) = q 2
15 / 33

Complementation Theorem
Main statement of this talk:
Theorem (Complementation)
The dual A of a WAPA A accepts its complement, i.e.
L(A) = Σ ω \ L(A)
(Thomas and Löding, ∼ 2000)
16 / 33

Outline
1 Weak Alternating Parity Automata
2 Infinite Parity Games
3 Proof of the Complementation Theorem
4 Büchi Complementation Algorithm
17 / 33

Automaton vs. Pathfinder
c
b
a
player A
find accepting run R
player P
find rejecting path in R
18 / 33

Infinite Parity Game (1)
☺
Example (a ω )
a
q 0
a a
q
• 1
•
1
A: •
2 a
w = a ω
q 2
a
0
Game G A,w :
q 0 , 0 {q 0 }, 0 q 0 , 1 {q 0 }, 1 q 0 , 2 · · ·
{q 1 , q 2 }, 0
{q 1 , q 2 }, 1 · · ·
q 1 , 1 {q 0 , q 1 }, 1 q 1 , 2 · · ·
q 2 , 1 {q 2 }, 1 q 2 , 2 · · ·
19 / 33

Infinite Parity Game (2)
☺
Definition (Game)
A game for a WAPA A = 〈Q, Σ, δ, q in , π〉 and w = a 0 a 1 a 2 . . . ∈ Σ ω
is a directed graph
where
G A,w := 〈V A ˙∪ V P , E〉
V A := Q × N (decision nodes of player A)
V P := 2 Q × N (decision nodes of player P)
E ⊆ (V A × V P ) ∪ (V P × V A )
s.t. the only contained edges are
• 〈 〈q, i〉, 〈M, i〉 〉 iff M ∈ Mod ↓ (δ(q, a i ))
• 〈 〈M, i〉, 〈q, i + 1〉 〉 iff q ∈ M
for q ∈ Q, M ⊆ Q, i ∈ N
(Thomas and Löding, ∼ 2000)
20 / 33

Playing a Game
☺
Definition (Play)
A play γ in a game G A,w is an infinite path starting with 〈q in , 0〉.
Definition (Winner)
The winner of a play γ is
player A iff the smallest parity of occurring V A -nodes is even
player P · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · odd
X ∈ {A, P}: a player,
Definition (Strategy)
X : its opponent
A strategy f X : V X → V X
for player X selects for every
decision node of player X one of its successor nodes in G A,w .
f X is a winning strategy iff player X wins every play γ that is
played according to f X .
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Strategies
Example
Winning strategy for player A (so far):
parities
q 0 , 0 {q 0 }, 0 q 0 , 1 {q 0 }, 1 q 0 , 2 · · · q 0 ↦→ 2
{q 1 , q 2 }, 1 · · ·
q 1 , 1 {q 0 , q 1 }, 1 q 1 , 2 · · · q 1 ↦→ 1
{q 1 , q 2 }, 0
q 2 , 1 {q 2 }, 1 q 2 , 2 · · · q 2 ↦→ 0
Not a winning strategy for player A:
q 0 , 0 {q 0 }, 0 q 0 , 1 {q 0 }, 1 q 0 , 2 · · ·
{q 1 , q 2 }, 1 · · ·
q 1 , 1 {q 0 , q 1 }, 1 q 1 , 2 · · ·
{q 1 , q 2 }, 0
q 2 , 1 {q 2 }, 1 q 2 , 2 · · ·
22 / 33

Outline
1 Weak Alternating Parity Automata
2 Infinite Parity Games
3 Proof of the Complementation Theorem
Lemma 1
Lemma 2
Lemma 3
Sublemma
Putting it All Together
4 Büchi Complementation Algorithm
23 / 33

Lemma 1
Let A be a WAPA and w ∈ Σ ω .
Lemma 1
Player A has a winning strategy in G A,w iff A accepts w.
Explanation (oral):
Player A wins every play γ
played according to f A .
G A,w :
· · · q, i + 1
There is a run graph R in which
every path ρ is accepting.
q, i + 1
p, i {q, q ′ , q ′′ }, i q ′ , i + 1
· · · q ′′ , i + 1
R:
p, i q ′ , i + 1
q ′′ , i + 1
24 / 33

Lemma 2
Let A be a WAPA and w ∈ Σ ω .
Lemma 2
Player P has a winning strategy in G A,w iff A does not accept w.
Explanation (oral):
Player P wins every play γ
played according to f P .
G
· · ·
A,w :
{.., q, ..}, i q, i + 1
· · ·
· · ·
p, i {.., q ′ , ..}, i q ′ , i + 1
· · ·
· · ·
{.., q ′′ , ..}, i q ′′ , i + 1
· · ·
(pointed out by Jan Leike)
Every run graph R contains a
rejecting path ρ.
R:
· · ·
p, i q, i + 1
· · ·
· · ·
R ′ : p, i q ′ , i + 1
· · ·
· · ·
R ′′ : p, i q ′′ , i + 1
· · ·
25 / 33

Sublemma
☺
Let θ ∈ B + (Q) be a formula over Q.
Sublemma
S ⊆ Q is a model of θ iff for all M ∈ Mod ↓ (θ): S ∩ M ≠ ∅.
Proof:
W.l.o.g. θ is in DNF, i.e.
∨
θ =
Then θ is in CNF, i.e.
θ =
M∈Mod ↓ (θ)
∧
M∈Mod ↓ (θ)
∧
q
q∈M
∨
q
q∈M
Thus S ⊆ Q is a model of θ iff it contains at least one
element from each disjunct of θ.
26 / 33

Lemma 3 (1)
Let A be a WAPA, A its dual and w = a 0 a 1 a 2 . . . ∈ Σ ω .
Lemma 3
Player A has a winning strategy in G A,w
iff player P has a winning strategy in G A,w
.
Proof:
⇒ Construct a winning strategy f P for player P in G A,w
.
· · ·
⇐ Construct a winning strategy f A for player A in G A,w .
· · ·
27 / 33

Lemma 3 (2)
⇒ Construct a winning strategy f P for player P in G A,w
.
At position 〈S, i〉∈V P
in G A,w
:
· · · · · ·
p, · · · i S, i q, · · · i + 1
· · · · · ·
f A : winning strategy for player A in G A,w
Assume there is 〈p, i〉∈V A occurring in a
play γ in G A,w played according to f A s.t.
S ∈ Mod ↓ (δ(p, a i )) (otherwise don’t care).
f A
(
〈p, i〉
)
=〈M, i〉 ⇒ M ∈ Mod↓ (δ(p, a i ))
(sublemma)
=⇒ There exists a q ∈S ∩ M.
Define f P
(
〈S, i〉
)
:= 〈q, i + 1〉
in G A,w :· · · · · ·
p, i M, · · · i q, · · · i + 1
· · · · · ·
∀ γ: play in G A,w
played according to f P
∃ γ: play in G A,w played according to f A
s.t. γ and γ contain the same V A -nodes.
• Player A wins γ in G A,w .
• ∀ q ∈Q : π(q) = π(q) + 1
⇒ Player P wins γ in G A,w
.
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Lemma 3 (3)
⇐ Construct a winning strategy f A for player A in G A,w .
At position 〈p, i〉∈V A
in G A,w :· · · q, i + 1
p, i M, · · · i q ′ , i + 1
· · · q ′′ , i + 1
in G A,w
:
· · ·
S, i q, i + 1
· · ·
· · ·
p, i S ′ , i q ′ , i + 1
· · ·
· · ·
S ′′ , i q ′′ , i + 1
· · ·M∗
f P : winning strategy for player P in G A,w
M ∗ := { q ∈ Q | ∃ S ∈Mod ↓ (δ(p, a i )) :
f P
(
〈S, i〉
)
= 〈q, i + 1〉
}
(sublemma)
=⇒ M ∗ is a model of δ(p, a i ).
M: subset of M ∗ that is a minimal model
M ⊆ M ∗ , M ∈ Mod ↓ (δ(p, a i ))
Define f A
(
〈p, i〉
)
:= 〈M, i〉
∀ γ: play in G A,w played according to f A
∃ γ: play in G A,w
played according to f P
s.t. γ and γ contain the same V A -nodes.
• Player P wins γ in G A,w
.
• ∀ q ∈Q : π(q) = π(q) − 1
⇒ Player A wins γ in G A,w .
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All Three Lemmas
☺
Let A be a WAPA, A its dual and w ∈ Σ ω .
Lemma 1
Player A has a winning strategy in G A,w iff A accepts w.
Lemma 2
Player P has a winning strategy in G A,w iff A does not accept w.
Lemma 3
Player A has a winning strategy in G A,w
iff player P has a winning strategy in G A,w
.
30 / 33

Complementation Theorem
☺
Theorem (Complementation)
The dual A of a WAPA A accepts its complement, i.e.
L(A) = Σ ω \ L(A)
Proof:
A accepts w
(lemma 1)
⇐⇒
(lemma 3)
⇐⇒
(lemma 2)
⇐⇒
(Thomas and Löding, ∼ 2000)
player A has a winning strategy in G A,w
player P has a winning strategy in G A,w
A does not accept w
31 / 33

Outline
1 Weak Alternating Parity Automata
2 Infinite Parity Games
3 Proof of the Complementation Theorem
4 Büchi Complementation Algorithm
32 / 33

Büchi Complementation Algorithm
☺
BA B
2Ω(n log n)
BA B
O(n 2 )
WAPA A
O(1)
2 O(n)
WAPA A
Total complexity: 2 O(n2 )
Can reach 2 O(n log n) (lower bound) by improving A → B.
33 / 33

References
Thomas, W. (1999)
Complementation of Büchi Automata Revisited.
In J. Karhumäki et al., editors, Jewels are Forever, Contributions on Th.
Comp. Science in Honor of Arto Salomaa, pages 109–122, Springer.
Klaedtke, F. (2002)
Complementation of Büchi Automata Using Alternation.
In E. Grädel et al., editors, Automata, Logics, and Infinite Games, LNCS
2500, pages 61-77. Springer.
Löding, C. and Thomas, W. (2000)
Alternating Automata and Logics over Infinite Words.
In J. van Leeuwen et al., editors, IFIP TCS 2000, LNCS 1872, pages
521–535. Springer.
Kupferman, O. and Vardi, M. Y. (2001)
Weak Alternating Automata Are Not that Weak.
In ACM Transactions on Computational Logic, volume 2, No. 3, July
2001, pages 408–429.

Appendix

From BA to WAPA
Given:
B = 〈Q, Σ, δ, q in , F 〉: BA
n = |Q|
☺
Construction (BA → WAPA)
where
A := 〈 Q ×{0, . . . , 2n} , Σ, δ ′ , 〈q
} {{ }
in , 2n〉, π 〉
⎧∨
⎪⎨ ∨
δ ′ (〈p, i〉, a) := ∨
π(〈p, i〉) := i
⎪⎩ ∨
O(n 2 )
q∈δ(p,a)
q∈δ(p,a)
q∈δ(p,a)
q∈δ(p,a)
〈q, 0〉 if i = 0
〈q, i〉 ∧ 〈q, i − 1〉 if i even, i > 0
〈q, i〉
〈q, i − 1〉
if i odd, p /∈ F
if i odd, p ∈ F
for p ∈ Q, a ∈ Σ, i ∈ {0, . . . , 2n}
(Thomas and Löding, ∼ 2000)
Back

From WAPA to BA
Given:
A = 〈Q, Σ, δ, q in , π〉: stratified WAPA, i.e.
∀p ∈Q ∀a∈Σ : δ(p, a) ∈ B +( {q ∈ Q | π(p) ≥ π(q)} )
E ⊆ Q: all states with even parity
Construction (WAPA → BA)
where
B := 〈 2
} Q ×2
{{ Q
}
, Σ, δ ′ , 〈{q in }, ∅〉, 2 Q ×{∅} 〉
2 O(n) {
〈M ′ , M ′ \E〉 ∣ M ′ ( ∧
∈ Mod ↓
δ ′ (〈M, ∅〉, a) :=
δ ′ (〈M, O〉, a) :=
{
〈M ′ , O ′ \E〉
for a ∈ Σ, M, O ⊆ Q, O ≠ ∅
Back
q∈M
δ(q, a))}
∣ M ′ (
∈ Mod ↓
∧q∈M δ(q, a)) ,
O ′ ⊆ M ′ ,
O ′ ( ∧
∈ Mod ↓ q∈O
δ(q, a))}
(Miyano and Hayashi, 1984)
☺