Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games...
Transcript of Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games...
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Playing Infinite Games in Finite TimeJoint work with John Fearnley,
Daniel Neider, and Roman Rabinovich
Martin Zimmermann
RWTH Aachen University
November 17th, 2011
Algosyn Workshop 2011, Rolduc
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Introduction
How can you play infinite games in finite time? Aren’t they infinitefor a reason?
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Introduction
How can you play infinite games in finite time? Aren’t they infinitefor a reason?
Introductory example
Player 0 wins play iff no red vertex is visited ⇒ safety game.
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Introduction
How can you play infinite games in finite time? Aren’t they infinitefor a reason?
Introductory example
Player 0 wins play iff no red vertex is visited ⇒ safety game.
Observations:
If a red vertex is reached, there is no point in continuing.
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Introduction
How can you play infinite games in finite time? Aren’t they infinitefor a reason?
Introductory example
Player 0 wins play iff no red vertex is visited ⇒ safety game.
Observations:
If a red vertex is reached, there is no point in continuing.
Positional determinacy: stop a play when some vertex isvisited for the second time.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 2/16
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Introduction
How can you play infinite games in finite time? Aren’t they infinitefor a reason?
Introductory example
Player 0 wins play iff no red vertex is visited ⇒ safety game.
Observations:
If a red vertex is reached, there is no point in continuing.
Positional determinacy: stop a play when some vertex isvisited for the second time.
Generalizes to finite-state determinacy: stop a play whensome memory state is repeated.
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Introduction
How can you play infinite games in finite time? Aren’t they infinitefor a reason?
Introductory example
Player 0 wins play iff no red vertex is visited ⇒ safety game.
Observations:
If a red vertex is reached, there is no point in continuing.
Positional determinacy: stop a play when some vertex isvisited for the second time.
Generalizes to finite-state determinacy: stop a play whensome memory state is repeated.
Can we play in finite time without relying on a memory structure?
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Muller Games
Inspired by previous work of McNaughton on playing infinite gamesin finite time, we consider Muller games (A,F0,F1):
arena A and partition (F0,F1) containing the loops of A.
Player i wins ρ iff Inf(ρ) = {v | ∃ωn s.t. ρn = v} ∈ Fi .
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Muller Games
Inspired by previous work of McNaughton on playing infinite gamesin finite time, we consider Muller games (A,F0,F1):
arena A and partition (F0,F1) containing the loops of A.
Player i wins ρ iff Inf(ρ) = {v | ∃ωn s.t. ρn = v} ∈ Fi .
Running example
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Player 0 has a winning strategy from every vertex: alternatebetween 0 and 2.
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Muller Games
Inspired by previous work of McNaughton on playing infinite gamesin finite time, we consider Muller games (A,F0,F1):
arena A and partition (F0,F1) containing the loops of A.
Player i wins ρ iff Inf(ρ) = {v | ∃ωn s.t. ρn = v} ∈ Fi .
Running example
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Player 0 has a winning strategy from every vertex: alternatebetween 0 and 2.
RemarkMuller games are not reducible to safety games.
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Outline
1. Playing Muller Games in Finite Time
2. Solving Muller Games by Solving Safety Games
3. Conclusion
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Scoring Functions
Let F ⊆ V , F 6= ∅.
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Scoring Functions
Let F ⊆ V , F 6= ∅. For v ∈ V define
ScF (v) =
{
1 if F = {v},
0 otherwise,
and
AccF (v) =
{
∅ if F = {v},
F ∩ {v} otherwise.
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Scoring Functions
Let F ⊆ V , F 6= ∅. For v ∈ V and w ∈ V+ define
ScF (wv) =
0 if v /∈ F ,
ScF (w) if v ∈ F ∧AccF (w) 6= (F \ {v}),
ScF (w) + 1 if v ∈ F ∧AccF (w) = (F \ {v}),
and
AccF (wv) =
∅ if v /∈ F ,
AccF (w) ∪ {v} if v ∈ F ∧AccF (w) 6= (F \ {v}),
∅ if v ∈ F ∧AccF (w) = (F \ {v}).
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1}
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1}
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0}
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2}
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0} {0}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0} {0} {0, 1}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0} {0} {0, 1} {0, 1}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0} {0} {0, 1} {0, 1} {0, 1}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0} {0} {0, 1} {0, 1} {0, 1} {0, 1}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0} {0} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0} {0} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} ∅
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Scoring Functions cont’d
ScF (w): maximal k ∈ N such that F is visited k times sincelast vertex in V \ F (reset).
AccF (w): set A ⊂ F of vertices seen since last increase orreset of ScF .
Example:
w 0 0 1 1 0 0 1 2
Sc{0,1} 0 0 1 1 2 2 3 0
Acc{0,1} {0} {0} ∅ {1} ∅ {0} ∅ ∅
Sc{0,1,2} 0 0 0 0 0 0 0 1
Acc{0,1,2} {0} {0} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} ∅
RemarkF = Inf(ρ) ⇔ lim infn→∞ ScF (ρ0 · · · ρn) = ∞
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 6/16
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Finite-time Muller Games
Two properties of scoring functions (informal versions):
1. If you play long enough (i.e., k |V | steps), some score value willbe high (i.e., k).
2. At most one score value can increase at a time.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 7/16
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Finite-time Muller Games
Two properties of scoring functions (informal versions):
1. If you play long enough (i.e., k |V | steps), some score value willbe high (i.e., k).
2. At most one score value can increase at a time.
DefinitionFinite-time Muller game: (A,F0,F1, k) with threshold k ≥ 3.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 7/16
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Finite-time Muller Games
Two properties of scoring functions (informal versions):
1. If you play long enough (i.e., k |V | steps), some score value willbe high (i.e., k).
2. At most one score value can increase at a time.
DefinitionFinite-time Muller game: (A,F0,F1, k) with threshold k ≥ 3.
Rules:
Players move a token through the arena.
Stop play w as soon as score of k is reached for the first time.
There is a unique F such that ScF (w) = k (see above).
Player i wins w iff F ∈ Fi .
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 7/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 (w.l.o.g.)
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
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Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 70: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/70.jpg)
Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 71: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/71.jpg)
Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1 0 1
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 72: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/72.jpg)
Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1 0 1 2
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 73: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/73.jpg)
Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1 0 1 2 Sc{0,1,2} = 2
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 74: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/74.jpg)
Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1 0 1 2 Sc{0,1,2} = 2
3
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 75: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/75.jpg)
Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1 0 1 2 Sc{0,1,2} = 2
3 0
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 76: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/76.jpg)
Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1 0 1 2 Sc{0,1,2} = 2
3 0 2
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 77: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/77.jpg)
Two Examples
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Losing player (Player 1) can enforce score of two:
1 2 2 1 Sc{1,2} = 2
0 1 2
3
F0 = {{0, 1}, {1, 2},{0, 1, 2, 3}}
F1 = {{0, 1, 2}, {0, 2, 3}}
Losing player (Player 1) is the first to reach a score of two:
3 0 2
1 0 1 2 Sc{0,1,2} = 2
3 0 2 Sc{0,2,3} = 2
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 8/16
![Page 78: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/78.jpg)
Results
Theorem (FZ10)
The winning regions in a Muller game (A,F0,F1) and in the
finite-time Muller game (A,F0,F1, 3) coincide.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 9/16
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Results
Theorem (FZ10)
The winning regions in a Muller game (A,F0,F1) and in the
finite-time Muller game (A,F0,F1, 3) coincide.
Stronger statement, which implies the theorem:
Lemma (FZ10)
On her winning region, Player i can prevent her opponent from
ever reaching a score of 3 for every set F ∈ F1−i .
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 9/16
![Page 80: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/80.jpg)
Results
Theorem (FZ10)
The winning regions in a Muller game (A,F0,F1) and in the
finite-time Muller game (A,F0,F1, 3) coincide.
Stronger statement, which implies the theorem:
Lemma (FZ10)
On her winning region, Player i can prevent her opponent from
ever reaching a score of 3 for every set F ∈ F1−i .
Corollary
Two “reductions”: Muller game to ..
1. .. reachability game on unravelling up to score 3.
2. .. safety game: see next slides.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 9/16
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Outline
1. Playing Muller Games in Finite Time
2. Solving Muller Games by Solving Safety Games
3. Conclusion
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 10/16
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“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 83: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/83.jpg)
“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Idea: track of Player 1’s scores and avoid ScF = 3 for F ∈ F1.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 84: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/84.jpg)
“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
Idea: track of Player 1’s scores and avoid ScF = 3 for F ∈ F1.
Ignore scores of Player 0.Identify plays having the same scores and accumulators forPlayer 1: w =F1 w
′ iff last(w) = last(w ′) and for all F ∈ F1:
ScF (w) = ScF (w′) and AccF (w) = Acc(w ′)
Build =F1-quotient of unravelling up to score 3 for Player 1.Winning condition for Player 0: avoid ScF = 3 for all F ∈ F1.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 85: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/85.jpg)
“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
Sc{0,1} = 0, Acc{0,1} = {0}Sc{1,2} = 0, Acc{1,2} = ∅
Sc{0,1} = 0, Acc{0,1} = {1}Sc{1,2} = 0, Acc{1,2} = {1}
Sc{0,1} = 0, Acc{0,1} = ∅Sc{1,2} = 0, Acc{1,2} = {2}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 86: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/86.jpg)
“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
Sc{0,1} = 1, Acc{0,1} = ∅Sc{1,2} = 0, Acc{1,2} = ∅
Sc{0,1} = 0, Acc{0,1} = ∅Sc{1,2} = 1, Acc{1,2} = ∅
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 87: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/87.jpg)
“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
Sc{0,1} = 1, Acc{0,1} = ∅Sc{1,2} = 0, Acc{1,2} = ∅
Sc{0,1} = 1, Acc{0,1} = ∅Sc{1,2} = 0, Acc{1,2} = {1}
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 88: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/88.jpg)
“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 89: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/89.jpg)
“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 90: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/90.jpg)
“Reducing“ Muller games to Safety Games
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 11/16
![Page 91: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/91.jpg)
Results
Theorem (NRZ11)
1. Player i wins the Muller game from v iff she wins the safety
game from [v ]=F1.
2. Player 0’s winning region in the safety game can be turned
into finite-state winning strategy for her in the Muller game.
3. Size of the safety game (n!)3.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 12/16
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Results
Theorem (NRZ11)
1. Player i wins the Muller game from v iff she wins the safety
game from [v ]=F1.
2. Player 0’s winning region in the safety game can be turned
into finite-state winning strategy for her in the Muller game.
3. Size of the safety game (n!)3.
Remarks:
Size of parity game in LAR-reduction n!. But: safety gamesallow much simpler algorithms.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 12/16
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Results
Theorem (NRZ11)
1. Player i wins the Muller game from v iff she wins the safety
game from [v ]=F1.
2. Player 0’s winning region in the safety game can be turned
into finite-state winning strategy for her in the Muller game.
3. Size of the safety game (n!)3.
Remarks:
Size of parity game in LAR-reduction n!. But: safety gamesallow much simpler algorithms.
2. does not hold for Player 1.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 12/16
![Page 94: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/94.jpg)
Results
Theorem (NRZ11)
1. Player i wins the Muller game from v iff she wins the safety
game from [v ]=F1.
2. Player 0’s winning region in the safety game can be turned
into finite-state winning strategy for her in the Muller game.
3. Size of the safety game (n!)3.
Remarks:
Size of parity game in LAR-reduction n!. But: safety gamesallow much simpler algorithms.
2. does not hold for Player 1.
Not a reduction in the classical sense: not every play of theMuller game can be mapped to a play in the safety game.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 12/16
![Page 95: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/95.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 96: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/96.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game..
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 97: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/97.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 98: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/98.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 99: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/99.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 100: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/100.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 101: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/101.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 102: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/102.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 103: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/103.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 104: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/104.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 105: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/105.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 106: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/106.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 107: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/107.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 108: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/108.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 109: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/109.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 110: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/110.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 111: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/111.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 112: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/112.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 113: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/113.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 114: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/114.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 115: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/115.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the safety game.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 116: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/116.jpg)
Proof Idea: Safety to Muller
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
ScF for F ∈ F1 in Muller game bounded by 2 ⇒ winning strategy
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 13/16
![Page 117: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/117.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 118: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/118.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two..
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 119: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/119.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 120: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/120.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 121: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/121.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 122: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/122.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 123: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/123.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 124: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/124.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 125: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/125.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 126: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/126.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 127: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/127.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 128: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/128.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 129: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/129.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 130: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/130.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 131: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/131.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 132: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/132.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 133: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/133.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Pick a winning strategy for the Muller game that bounds Player 1’sscores by two.. and let’s play.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 134: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/134.jpg)
Proof Idea: Muller to Safety
10 2F0 = {{0, 1, 2}, {0}, {2}}
F1 = {{0, 1}, {1, 2}}
[0]
[1]
[2]
[10]
[12]
[01]
[21]
[101]
[100]
[122]
[121]
[1010]
[1001]
[1221]
[1212]
[10101]
[10010]
[12212]
[12121]
[101010]
[100101]
[122121]
[121212]
Score of three is avoidable from every prefix ⇒ red vertices neverreached ⇒ winning strategy.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 14/16
![Page 135: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/135.jpg)
Outline
1. Playing Muller Games in Finite Time
2. Solving Muller Games by Solving Safety Games
3. Conclusion
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 15/16
![Page 136: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/136.jpg)
Conclusion
You can play Muller games in finite time!
New algorithm for Muller games: just solve the safety game.
New memory structure for Muller games: maximal elementsof winning region suffice (antichain).
New concept: permissive strategies for Muller games.
Same constructions applicable for many other types of games.
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 16/16
![Page 137: Playing Infinite Games in Finite Timezimmermann/slides/AlgoSyn_2011.pdf · Playing Infinite Games in Finite Time JointworkwithJohnFearnley, DanielNeider,andRomanRabinovich Martin](https://reader035.fdocuments.us/reader035/viewer/2022062415/604198fc2991771ba669b7fd/html5/thumbnails/137.jpg)
Conclusion
You can play Muller games in finite time!
New algorithm for Muller games: just solve the safety game.
New memory structure for Muller games: maximal elementsof winning region suffice (antichain).
New concept: permissive strategies for Muller games.
Same constructions applicable for many other types of games.
Ongoing and future work:
Progress measure algorithm for Muller games?
Is there a tradeoff between size and quality of a strategy?
Can you play infinite games in infinite arenas in finite time?
Martin Zimmermann RWTH Aachen University Playing Infinite Games in Finite Time 16/16