Energy parity games:
Bounded vs Unbounded1
Arno Pauly
Université libre du Bruxelles
Highlights 2016
1
Joint work with Stéphane Le Roux.
An example
v0
a
d
Vertex
a
b
c
...
Parity
0
3
2
...
b
e
Energy-δ
+5
−2
−1
...
c
Goal
P
Unbounded case: Player I wants to ensure that ∀t ti=0 δi ≥ 0
and that the least priority visited infinitely many times is even.
Bounded case: Let E0 = 0, Et+1 = min{B, Et + δt }. Player I
wants that ∀tEt ≥ 0, and that the least priority visited infinitely
many times is even.
Goal
P
Unbounded case: Player I wants to ensure that ∀t ti=0 δi ≥ 0
and that the least priority visited infinitely many times is even.
Bounded case: Let E0 = 0, Et+1 = min{B, Et + δt }. Player I
wants that ∀tEt ≥ 0, and that the least priority visited infinitely
many times is even.
Finite memory determinacy
K.Chatterjee & L.Doyen.
Energy Parity Games.
Theoretical Computer Science 2012.
n: number of vertices, W := max |δi |, d number of parities
Theorem
Unbounded energy parity games are finite memory determined,
and 4dWn states suffice.
Finite memory determinacy
K.Chatterjee & L.Doyen.
Energy Parity Games.
Theoretical Computer Science 2012.
n: number of vertices, W := max |δi |, d number of parities
Theorem
Unbounded energy parity games are finite memory determined,
and 4dWn states suffice.
Finite memory determinacy
K.Chatterjee & L.Doyen.
Energy Parity Games.
Theoretical Computer Science 2012.
n: number of vertices, W := max |δi |, d number of parities
Theorem
Unbounded energy parity games are finite memory determined,
and 4dWn states suffice.
Lower bound
start
a
b
c
d
δa = +1, δi = −W , priority(d) = 0, other priorities are 1
Proposition (Chatterjee and Doyen)
2W (n − 1) bits of memory are required.
Lower bound
start
a
b
c
d
δa = +1, δi = −W , priority(d) = 0, other priorities are 1
Proposition (Chatterjee and Doyen)
2W (n − 1) bits of memory are required.
Why bounded?
I
Bounded energy can be tracked by a finite automaton.
I
Easy to prove: B states suffice for determinacy.
I
Extension to multiplayer multioutcome games possible.
I
Objectives expressible by finite automata can be combined
nicely, cf Stéphane Le Roux’s talk.
Why bounded?
I
Bounded energy can be tracked by a finite automaton.
I
Easy to prove: B states suffice for determinacy.
I
Extension to multiplayer multioutcome games possible.
I
Objectives expressible by finite automata can be combined
nicely, cf Stéphane Le Roux’s talk.
Why bounded?
I
Bounded energy can be tracked by a finite automaton.
I
Easy to prove: B states suffice for determinacy.
I
Extension to multiplayer multioutcome games possible.
I
Objectives expressible by finite automata can be combined
nicely, cf Stéphane Le Roux’s talk.
Why bounded?
I
Bounded energy can be tracked by a finite automaton.
I
Easy to prove: B states suffice for determinacy.
I
Extension to multiplayer multioutcome games possible.
I
Objectives expressible by finite automata can be combined
nicely, cf Stéphane Le Roux’s talk.
Equivalence
Theorem
Player 1 can win an unbounded energy parity game iff she can
win the bounded energy parity game with B ≥ 2nW .
Corollary
Unbounded energy parity games are finite memory determined,
and 2Wn states suffice. (This is optimal!)
Equivalence
Theorem
Player 1 can win an unbounded energy parity game iff she can
win the bounded energy parity game with B ≥ 2nW .
Corollary
Unbounded energy parity games are finite memory determined,
and 2Wn states suffice. (This is optimal!)
Reference
S. Le Roux & A. Pauly .
Extending finite memory determinacy: General techniques
and an application to energy parity games.
arXiv 1602.08912.