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Energy parity games: Bounded vs Unbounded1
Arno Pauly
Université libre du Bruxelles
Highlights 2016
1Joint work with Stéphane Le Roux.
Energy parity games: Bounded vs Unbounded 1 Arno Pauly Universit - - PowerPoint PPT Presentation
Energy parity games: Bounded vs Unbounded 1 Arno Pauly Universit - - PowerPoint PPT Presentation
Energy parity games: Bounded vs Unbounded 1 Arno Pauly Universit libre du Bruxelles Highlights 2016 1 Joint work with Stphane Le Roux. An example v 0 a c b e d Vertex Parity Energy- a 0 + 5 b 3 2 1 c 2 ... ... ...
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Goal
Unbounded case: Player I wants to ensure that both:
◮ ∀t t i=0 δi ≥ 0 ◮ The least priority visited infinitely often is even.
Bounded case: (with bound B ∈ N) Player I wants to ensure that both:
◮ ∀tEt ≥ 0, where E0 = 0, Et+1 = min{B, Et + δt}. ◮ The least priority visited infinitely often is even.
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Goal
Unbounded case: Player I wants to ensure that both:
◮ ∀t t i=0 δi ≥ 0 ◮ The least priority visited infinitely often is even.
Bounded case: (with bound B ∈ N) Player I wants to ensure that both:
◮ ∀tEt ≥ 0, where E0 = 0, Et+1 = min{B, Et + δt}. ◮ The least priority visited infinitely often is even.
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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 memory states suffice.
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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 memory states suffice.
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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 memory states suffice.
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Lower bound
1/+1 start 1/-W 1/-W 0/-W
Proposition (Chatterjee and Doyen, Energy Parity Games, 2012)
2W(n − 1) memory states are required.
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Lower bound
1/+1 start 1/-W 1/-W 0/-W
Proposition (Chatterjee and Doyen, Energy Parity Games, 2012)
2W(n − 1) memory states are required.
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Why bounded?
◮ Bounded energy can be tracked by a finite automaton. ◮ Easy to prove: B states suffice for finite memory
determinacy.
◮ Extension to multiplayer multioutcome games possible. ◮ Objectives expressible by finite automata can be combined
nicely, cf Stéphane Le Roux’s talk.
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Why bounded?
◮ Bounded energy can be tracked by a finite automaton. ◮ Easy to prove: B states suffice for finite memory
determinacy.
◮ Extension to multiplayer multioutcome games possible. ◮ Objectives expressible by finite automata can be combined
nicely, cf Stéphane Le Roux’s talk.
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Why bounded?
◮ Bounded energy can be tracked by a finite automaton. ◮ Easy to prove: B states suffice for finite memory
determinacy.
◮ Extension to multiplayer multioutcome games possible. ◮ Objectives expressible by finite automata can be combined
nicely, cf Stéphane Le Roux’s talk.
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Why bounded?
◮ Bounded energy can be tracked by a finite automaton. ◮ Easy to prove: B states suffice for finite memory
determinacy.
◮ Extension to multiplayer multioutcome games possible. ◮ Objectives expressible by finite automata can be combined
nicely, cf Stéphane Le Roux’s talk.
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Equivalence
Theorem (Le Roux and Pauly, 2016)
Player 1 can win an unbounded energy parity game iff she can win the bounded energy parity game with B ≥ 2nW.
Corollary (Le Roux and Pauly, 2016)
Unbounded energy parity games are finite memory determined, and 2Wn states suffice. (This is optimal!)
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Equivalence
Theorem (Le Roux and Pauly, 2016)
Player 1 can win an unbounded energy parity game iff she can win the bounded energy parity game with B ≥ 2nW.
Corollary (Le Roux and Pauly, 2016)
Unbounded energy parity games are finite memory determined, and 2Wn states suffice. (This is optimal!)
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Reference
- S. Le Roux & A. Pauly .