Mean-payoff games with incomplete information - Semantic Scholar
Mean-payoff games with incomplete information Paul Hunter, Guillermo P´erez, Jean-Franc¸ois Raskin Universit´ e Libre de Bruxelles YR-CONCUR @ Argentina
August, 2013
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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MPGs imperfect information: example
2 1
4 3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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MPGs imperfect information: example
Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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MPGs imperfect information: example Σ = {a, b} and weights on the edges
Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
Example: ∃ve chooses a, ∀dam chooses (1, a, 2); payoff = -1 Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
Example: ∃ve chooses a, ∀dam chooses (1, a, 2); payoff = -1 Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
∃ve only sees colors, ∀dam sees everything Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
Mean-payoff game
Definition (MPGs) Mean-payoff games are 2-player games of infinite duration played on (directed) weighted graphs. ∃ve chooses an action, and ∀dam resolves non-determinism by choosing the next state. ∃ve wants to maximize the average weight of the edges traversed (the MP value). ∀dam wants to minimize the same value.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
4 / 35
Strategies, Mean-payoff value
Definition (Strategies for ∃ve ) An observable strategy for ∃ve is a function from finite sequences (Obs · Σ)∗ Obs to the next action.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
5 / 35
Strategies, Mean-payoff value
Definition (Strategies for ∃ve ) An observable strategy for ∃ve is a function from finite sequences (Obs · Σ)∗ Obs to the next action.
Definition (MP value) Given the transition relation ∆ and the weight function w : ∆ 7→ Z of a P MPG, the MP value is limn→∞ n1 n−1 i=0 w (qi , σi , qi+1 ).
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
5 / 35
Strategies, Mean-payoff value
Definition (Strategies for ∃ve ) An observable strategy for ∃ve is a function from finite sequences (Obs · Σ)∗ Obs to the next action.
Definition (MP value) Given the transition relation ∆ and the weight function w : ∆ 7→ Z of a P MPG, the MP value is limn→∞ n1 n−1 i=0 w (qi , σi , qi+1 ).
Problem (Winner of an MPG) Given a threshold ν ∈ N, the MPG is won by ∃ve iff MP ≥ ν. W.l.o.g assume ν = 0.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
5 / 35
MPGs Theorem (Ehrenfeucht and Mycielski [1979]) MPGs are determined, i.e. if ∃ve doesn’t have a winning strategy then ∀dam does (and viceversa). Positional strategies suffice for either ∀dam or ∃ve to win a MPG.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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MPGs Theorem (Ehrenfeucht and Mycielski [1979]) MPGs are determined, i.e. if ∃ve doesn’t have a winning strategy then ∀dam does (and viceversa). Positional strategies suffice for either ∀dam or ∃ve to win a MPG. Σ = {a, b} Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
6 / 35
MPGs Theorem (Ehrenfeucht and Mycielski [1979]) MPGs are determined, i.e. if ∃ve doesn’t have a winning strategy then ∀dam does (and viceversa). Positional strategies suffice for either ∀dam or ∃ve to win a MPG. Σ = {a, b} ∃ve has a winning strat: play b in 2 and a in 3 Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
6 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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MPG with imperfect information Definition (MPGs with imperfect info.) A MPG with imperfect information is played on a weighted graph given with a coloring of the state space that defines equivalence classes of indistinguishable states (observations).
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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MPG with imperfect information Definition (MPGs with imperfect info.) A MPG with imperfect information is played on a weighted graph given with a coloring of the state space that defines equivalence classes of indistinguishable states (observations). Σ = {a, b} Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
8 / 35
MPG with imperfect information Definition (MPGs with imperfect info.) A MPG with imperfect information is played on a weighted graph given with a coloring of the state space that defines equivalence classes of indistinguishable states (observations). Σ = {a, b} Neither ∃ve nor ∀dam have a winning strategy anymore Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
8 / 35
Motivation and properties
Why consider such a model? MPGs are natural models for systems where we want to optimize the limit-average usage of a resource. Imperfect information arises from the fact that most systems have a limited amount of sensors and input data.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
9 / 35
Motivation and properties
Why consider such a model? MPGs are natural models for systems where we want to optimize the limit-average usage of a resource. Imperfect information arises from the fact that most systems have a limited amount of sensors and input data.
Theorem (Degorre et al. [2010]) MPGs with imperfect info. are no longer “determined”. May require infinite memory to be won by ∃ve . Determining who wins is undecidable.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
9 / 35
The knowledge of ∃ve Definition (Knowledge-based subset construction)
G b,-1
2
Σ,-1
3
Σ,+1
1 Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
10 / 35
The knowledge of ∃ve Definition (Knowledge-based subset construction) ∆K based on where ∃ve might be
GK
G b,-1
2
1 Σ,-1
2
Σ,-1 b
1 3
a
Σ,+1
3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
Σ
MPGs with incomplete info.
3 Σ
August, 2013
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The knowledge of ∃ve Definition (Knowledge-based subset construction) ∆K based on where ∃ve might be w K makes sense only in visible games (Degorre et al. [2010]) GK
G b,-1
2
1 Σ,-1
2
Σ,-1 b,-1
1 3
a,-1
Σ,+1
3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
Σ,-1
3 Σ,+1
August, 2013
10 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
11 / 35
Don’t lie to ∃ve Definition A game of imperfect information is of incomplete information if for every (q, σ, q 0 ) ∈ ∆, then for every s 0 in the same observation as q 0 there is a transition (s, σ, s 0 ) ∈ ∆ where s is in the same observation as q.
a
3
1 4 2 5
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Don’t lie to ∃ve Definition A game of imperfect information is of incomplete information if for every (q, σ, q 0 ) ∈ ∆, then for every s 0 in the same observation as q 0 there is a transition (s, σ, s 0 ) ∈ ∆ where s is in the same observation as q.
3
a
1 a
2
4 a
5
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
12 / 35
Don’t lie to ∃ve Lemma (imperfect to incomplete info.) imperfect information can be turned into incomplete information with a possible exponential blow-up (via its knowledge-based subset construction). G0
G b,-1
2
b,-1
Σ,-1
1
2
Σ,-1
3
Σ,+1
1 b,-1
Σ,-1
3
Σ,+1
a,-1
3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
Σ,+1
August, 2013
13 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
14 / 35
Incomplete information peculiarities
Observe that in an MPG of incomplete information: 1
the view ∃ve has of a play in the game is o0 σ0 o1 σ1 . . .,
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Incomplete information peculiarities
Observe that in an MPG of incomplete information: 1
the view ∃ve has of a play in the game is o0 σ0 o1 σ1 . . .,
2
given current oi the game could be in any q ∈ oi (not true in imperfect information),
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
15 / 35
Incomplete information peculiarities
Observe that in an MPG of incomplete information: 1
the view ∃ve has of a play in the game is o0 σ0 o1 σ1 . . .,
2
given current oi the game could be in any q ∈ oi (not true in imperfect information),
3
∀dam can have a two step strategy: choose observations first,
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
15 / 35
Incomplete information peculiarities
Observe that in an MPG of incomplete information: 1
the view ∃ve has of a play in the game is o0 σ0 o1 σ1 . . .,
2
given current oi the game could be in any q ∈ oi (not true in imperfect information),
3
∀dam can have a two step strategy: choose observations first,
4
“delay” the specific choice of states for later!
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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∀dam and determinacy Definition Observable strategies: we let ∀dam reveal to ∃ve only the (Obs × Σ)+ 7→ Obs version of his strategy. Let γ be a function mapping observation-action sequences to concrete state-action ones.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
16 / 35
∀dam and determinacy Definition Observable strategies: we let ∀dam reveal to ∃ve only the (Obs × Σ)+ 7→ Obs version of his strategy. Let γ be a function mapping observation-action sequences to concrete state-action ones.
Definition (New winning condition) Let ψ be a play in the game. ∃ve wins if all paths in γ(ψ) are winning for her. ∀dam wins if there is some path which is winning for him.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
16 / 35
∀dam and determinacy Definition Observable strategies: we let ∀dam reveal to ∃ve only the (Obs × Σ)+ 7→ Obs version of his strategy. Let γ be a function mapping observation-action sequences to concrete state-action ones.
Definition (New winning condition) Let ψ be a play in the game. ∃ve wins if all paths in γ(ψ) are winning for her. ∀dam wins if there is some path which is winning for him.
Theorem (Observable determinacy) The new winning condition is a projection of the perfect information game winning condition (via γ). The new winning condition is coSuslin and hence determined∗ . P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
16 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
17 / 35
Function-Reachability game Definition (Function sequence classification) A function sequence is good (bad) if a function is pointwise bigger or equal (smaller) then a previous one – same observation. Σ,-3 a,-1
2
Σ,-1
1
4 b,-1
3
Σ,+1
Σ,-1
Σ,-1
obs: blue play: fI cur. f: fI (1) = 0 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
18 / 35
Function-Reachability game Definition (Function sequence classification) A function sequence is good (bad) if a function is pointwise bigger or equal (smaller) then a previous one – same observation. Σ,-3
2
a,-1
Σ,-1
1
4 b,-1
3
Σ,+1
Σ,-1
Σ,-1
obs: blue-a-yellow play: fI a f1 cur. f: f1 (2) = −3, f1 (3) = −1 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
18 / 35
Function-Reachability game Definition (Function sequence classification) A function sequence is good (bad) if a function is pointwise bigger or equal (smaller) then a previous one – same observation. Σ,-3 a,-1
2
Σ,-1
1
4 b,-1
3
Σ,+1
Σ,-1
Σ,-1
obs: blue-a-yellow-b-green play: fI a f1 b f2 cur. f: f2 (4) = −4 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Function-Reachability game Definition (Function sequence classification) A function sequence is good (bad) if a function is pointwise bigger or equal (smaller) then a previous one – same observation. Σ,-3 a,-1
2
Σ,-1
1
4 b,-1
3
Σ,+1
Σ,-1
Σ,-1
obs: blue-a-yellow-b-green-a-green play: fI a f1 b f2 a f3 good cur. f: f3 (4) = −3 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Unfolding a MPG with incomplete information fI σ0
σ1
o0
o2
o3
o5
“Unfold” G, stop when a good or bad sequence is reached.
f2
.. .
.. .
f1
We are left with a new reachability game
σ0
σ1 .. .
o5
Not all branches will be labelled. . .
f3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Strategy transfer
Let H be the reachability game played on the unfolding of G,
Theorem (Strategy transfer for ∃ve ) ∃ve has a finite memory winning strategy in G if and only if she has a winning strategy in H.
Theorem (Strat. transfer for ∀dam ) If ∀dam has a winning observable strategy in H then he also has a winning strategy in G.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Finite memory, Adeq. Pure, Pure games All based on function sequences (branches) of the associated reachability game H.
Definition 1
Finite memory games: ∃ve can force good leaves or ∀dam can force bad leaves.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Finite memory, Adeq. Pure, Pure games All based on function sequences (branches) of the associated reachability game H.
Definition 1
Finite memory games: ∃ve can force good leaves or ∀dam can force bad leaves.
2
Adequately pure games: ∃ve ( ∀dam ) can force good (bad) branches where all but 2 functions have different support.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Finite memory, Adeq. Pure, Pure games All based on function sequences (branches) of the associated reachability game H.
Definition 1
Finite memory games: ∃ve can force good leaves or ∀dam can force bad leaves.
2
Adequately pure games: ∃ve ( ∀dam ) can force good (bad) branches where all but 2 functions have different support.
3
Pure games [structural]: the unfolding of G is finite and in all branches, all but 2 functions have different support.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
21 / 35
Relevant problems
Let A be a class of MPGs with incomplete (or imperfect) information. Given MPG with incomplete (imperfect) information G,
Problem (Class membership) Is G a member of A?
Problem (Winner determination) Does ∃ve have a winning strategy in G?
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Summary
Finite memory Information ClassUndec1 membership Winnerdet.
1
R-c
Adequately pure
Pure
incomplete imperfect PSPACE- NEXPcomplete hard, in EXPSPACE PSPACE- EXPcomplete complete
incomplete imperfect coNPcoNEXPcomplete complete NP coNP
∩
EXPcomplete
gray=Degorre et al. [2010],white & yellow are our results
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
23 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
24 / 35
Does ∃ve win pure G?
Theorem Deciding if ∃ve has a winning strategy in a given pure MPG with incomplete information is in NP ∩ coNP.
Based on Bj¨orklund et al. [2004]. Observe∗ that positional strategies suffice for ∃ve to win pure games with incomplete information.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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Is G pure?
Theorem The class membership problem for pure games with incomplete information is coNP-complete.
Proof. One can “guess” a branch in H (of size at most |Obs| + 1) and in polynomial time check that it is neither good nor bad. For hardness we reduce from the HAMILTONIAN-CYCLE problem.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
26 / 35
HAM-CYCLE as an MPG
−1
S 0
−N/ + N
v0
+1/ − 1
vn
+1/ − 1
+1/ − 1
vn−1
v1
+1/ − 1 +1/ − 1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
v2
+1/ − 1
···
MPGs with incomplete info.
vn−2
August, 2013
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Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
28 / 35
Solving a pure G
Let G 0 be the equivalent game with incomplete information, then
Remark In pure games, the values of functions in F can be assumed to range over −W · |Obs 0 | and W · |Obs 0 | where W is the biggest transition weight. New reachability game R = hF, fI , Σ, ∆succ , goodi.
Theorem Determining if ∃ve wins a pure MPG with incomplete information is EXP-complete.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
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Is G pure?
Theorem Deciding if an MPG with imperfect information is pure is coNEXPTIME-complete.
Proof. Membership: non-deterministically guess a branch in H, check that it not good nor bad. For hardness we reduce from the SUCCINCT HAMILTONIAN-CYCLE problem.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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SUCCINCT HAM-CYCLE
Definition (Galperin and Wigderson [1983]) G = hV , E i with m ≥ 2n vertices, each labelled with a distinct n-bit string. A circuit CG receives two n-bit inputs and outputs 1 if there is an edge. CG has r = O(nk ) gates. 2n
.. .
f = 1 iff (u, v ) ∈ E O(nk ) gates
0
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
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SUCCINCT HAM-CYCLE
Theorem (Exponential blow-up) Most problems (reducible as a “projection”) have an exponential blow-up when the graph is represented succinctly. SUCCINCT HAM-CYCLE is NEXPTIME-complete. 2n
.. .
f = 1 iff (u, v ) ∈ E O(nk )
gates
0
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
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Is G pure?
Theorem Deciding if an MPG with imperfect information is pure is coNEXPTIME-complete.
Proof. Membership: non-deterministically guess a branch in H, check that it not good nor bad. For hardness we reduce from the SUCCINCT HAMILTONIAN-CYCLE problem. “mixed” path ⇒ simulated 2N transitions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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coNEXP-hardness proof
−1
S 0
−2N
O1
0
O2
+1
−N
Chk (+1)
G0 (+N)
−1 −1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G1 (+1)
−1
···
MPGs with incomplete info.
−1
Gk (+1)
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Summary 1
Done: incomplete info., observable determinacy, subclasses
2
Cooking: other asymmetric information types, other quantitative games, mixed strategies
Finite memory Information ClassUndec1 membership Winnerdet. 1
R-c
Adequately pure
Pure
incomplete imperfect PSPACE- NEXPcomplete hard, in EXPSPACE PSPACE- EXPcomplete complete
incomplete imperfect coNPcoNEXPcomplete complete NP coNP
∩
EXPcomplete
gray=Degorre et al. [2010],white & yellow are our results
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
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References I
Bj¨ orklund, H., Sandberg, S., and Vorobyov, S. (2004). Memoryless determinacy of parity and mean payoff games: a simple proof. Theoretical Computer Science, 310(1):365–378. Degorre, A., Doyen, L., Gentilini, R., Raskin, J.-F., and Toru´ nczyk, S. (2010). Energy and mean-payoff games with imperfect information. In Computer Science Logic, pages 260–274. Springer. Ehrenfeucht, A. and Mycielski, J. (1979). Positional strategies for mean payoff games. International Journal of Game Theory, 8:109–113. Galperin, H. and Wigderson, A. (1983). Succinct representations of graphs. Information and Control, 56(3):183–198.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
35 / 35
coNEXP-hardness proof
xN
x1 x2 ··· x1 x2
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
O1 xN
MPGs with incomplete info.
August, 2013
36 / 35
coNEXP-hardness proof
−1
S 0
−2N
O1
0
O2
+1
−N
Chk (+1)
G0 (+N)
−1 −1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G1 (+1)
−1
···
MPGs with incomplete info.
−1
Gk (+1)
August, 2013
37 / 35
coNEXP-hardness proof xN
x1 x2 ··· σ
x1 x2 −N
xN
+ τN+1
σ −N
O1
xN+1
yN+1
x N+1 − τN+1
.. .
σ −N
+ τ2N
x2N
y2N − τ2N
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G0
MPGs with incomplete info.
x 2N
August, 2013
38 / 35
coNEXP-hardness proof
−1
S 0
−2N
O1
0
O2
+1
−N
Chk (+1)
G0 (+N)
−1 −1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G1 (+1)
−1
···
MPGs with incomplete info.
−1
Gk (+1)
August, 2013
39 / 35
coNEXP-hardness proof
xl xl
···
···
−1
Gj−1
x2N+j−1
x1 x2
σ
x1 x2 xr xr
−1
v1 τ +
σ σ
v2 τ
−1 −1
σ
x 2N+j−1
v1 τ−
Gj v3 χ
v2 τ+
v0
v0 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
v3
τ+
−1 σ
χ
−
τ
x2N+j −
τ +, τ −
MPGs with incomplete info.
x 2N+j August, 2013
40 / 35
coNEXP-hardness proof
xl
···
···
−1
Gj−1 xl
x2N+j−1
x1 x2
σ
x1 x2 xr xr
−1
v2
v1 τ +
σ σ
τ −1 −1
σ
x 2N+j−1
Gj v3
v1 τ−
χ
v2
τ +, τ −
v0
v0 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
v3
τ+
−1 σ
χ
−
τ
x2N+j
+
τ−
MPGs with incomplete info.
x 2N+j August, 2013
40 / 35
coNEXP-hardness proof
−1
S 0
−2N
O1
0
O2
+1
−N
Chk (+1)
G0 (+N)
−1 −1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G1 (+1)
−1
···
MPGs with incomplete info.
−1
Gk (+1)
August, 2013
41 / 35
August, 2013
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
2 / 35
MPGs imperfect information: example
2 1
4 3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example
Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges
Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
Example: ∃ve chooses a, ∀dam chooses (1, a, 2); payoff = -1 Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
Example: ∃ve chooses a, ∀dam chooses (1, a, 2); payoff = -1 Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
MPGs imperfect information: example Σ = {a, b} and weights on the edges Game to move token: ∃ve chooses σ and ∀dam chooses edge to win ( ∃ve ): maximize average weight of edges traversed
∃ve only sees colors, ∀dam sees everything Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
3 / 35
Mean-payoff game
Definition (MPGs) Mean-payoff games are 2-player games of infinite duration played on (directed) weighted graphs. ∃ve chooses an action, and ∀dam resolves non-determinism by choosing the next state. ∃ve wants to maximize the average weight of the edges traversed (the MP value). ∀dam wants to minimize the same value.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
4 / 35
Strategies, Mean-payoff value
Definition (Strategies for ∃ve ) An observable strategy for ∃ve is a function from finite sequences (Obs · Σ)∗ Obs to the next action.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
5 / 35
Strategies, Mean-payoff value
Definition (Strategies for ∃ve ) An observable strategy for ∃ve is a function from finite sequences (Obs · Σ)∗ Obs to the next action.
Definition (MP value) Given the transition relation ∆ and the weight function w : ∆ 7→ Z of a P MPG, the MP value is limn→∞ n1 n−1 i=0 w (qi , σi , qi+1 ).
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
5 / 35
Strategies, Mean-payoff value
Definition (Strategies for ∃ve ) An observable strategy for ∃ve is a function from finite sequences (Obs · Σ)∗ Obs to the next action.
Definition (MP value) Given the transition relation ∆ and the weight function w : ∆ 7→ Z of a P MPG, the MP value is limn→∞ n1 n−1 i=0 w (qi , σi , qi+1 ).
Problem (Winner of an MPG) Given a threshold ν ∈ N, the MPG is won by ∃ve iff MP ≥ ν. W.l.o.g assume ν = 0.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
5 / 35
MPGs Theorem (Ehrenfeucht and Mycielski [1979]) MPGs are determined, i.e. if ∃ve doesn’t have a winning strategy then ∀dam does (and viceversa). Positional strategies suffice for either ∀dam or ∃ve to win a MPG.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
6 / 35
MPGs Theorem (Ehrenfeucht and Mycielski [1979]) MPGs are determined, i.e. if ∃ve doesn’t have a winning strategy then ∀dam does (and viceversa). Positional strategies suffice for either ∀dam or ∃ve to win a MPG. Σ = {a, b} Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
6 / 35
MPGs Theorem (Ehrenfeucht and Mycielski [1979]) MPGs are determined, i.e. if ∃ve doesn’t have a winning strategy then ∀dam does (and viceversa). Positional strategies suffice for either ∀dam or ∃ve to win a MPG. Σ = {a, b} ∃ve has a winning strat: play b in 2 and a in 3 Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
6 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
7 / 35
MPG with imperfect information Definition (MPGs with imperfect info.) A MPG with imperfect information is played on a weighted graph given with a coloring of the state space that defines equivalence classes of indistinguishable states (observations).
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
8 / 35
MPG with imperfect information Definition (MPGs with imperfect info.) A MPG with imperfect information is played on a weighted graph given with a coloring of the state space that defines equivalence classes of indistinguishable states (observations). Σ = {a, b} Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
8 / 35
MPG with imperfect information Definition (MPGs with imperfect info.) A MPG with imperfect information is played on a weighted graph given with a coloring of the state space that defines equivalence classes of indistinguishable states (observations). Σ = {a, b} Neither ∃ve nor ∀dam have a winning strategy anymore Σ,-1 a,-1
2
b,-1
1
4 b,-1
3
Σ,+1
a,-1
Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
8 / 35
Motivation and properties
Why consider such a model? MPGs are natural models for systems where we want to optimize the limit-average usage of a resource. Imperfect information arises from the fact that most systems have a limited amount of sensors and input data.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
9 / 35
Motivation and properties
Why consider such a model? MPGs are natural models for systems where we want to optimize the limit-average usage of a resource. Imperfect information arises from the fact that most systems have a limited amount of sensors and input data.
Theorem (Degorre et al. [2010]) MPGs with imperfect info. are no longer “determined”. May require infinite memory to be won by ∃ve . Determining who wins is undecidable.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
9 / 35
The knowledge of ∃ve Definition (Knowledge-based subset construction)
G b,-1
2
Σ,-1
3
Σ,+1
1 Σ,-1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
10 / 35
The knowledge of ∃ve Definition (Knowledge-based subset construction) ∆K based on where ∃ve might be
GK
G b,-1
2
1 Σ,-1
2
Σ,-1 b
1 3
a
Σ,+1
3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
Σ
MPGs with incomplete info.
3 Σ
August, 2013
10 / 35
The knowledge of ∃ve Definition (Knowledge-based subset construction) ∆K based on where ∃ve might be w K makes sense only in visible games (Degorre et al. [2010]) GK
G b,-1
2
1 Σ,-1
2
Σ,-1 b,-1
1 3
a,-1
Σ,+1
3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
Σ,-1
3 Σ,+1
August, 2013
10 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
11 / 35
Don’t lie to ∃ve Definition A game of imperfect information is of incomplete information if for every (q, σ, q 0 ) ∈ ∆, then for every s 0 in the same observation as q 0 there is a transition (s, σ, s 0 ) ∈ ∆ where s is in the same observation as q.
a
3
1 4 2 5
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
12 / 35
Don’t lie to ∃ve Definition A game of imperfect information is of incomplete information if for every (q, σ, q 0 ) ∈ ∆, then for every s 0 in the same observation as q 0 there is a transition (s, σ, s 0 ) ∈ ∆ where s is in the same observation as q.
3
a
1 a
2
4 a
5
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
12 / 35
Don’t lie to ∃ve Lemma (imperfect to incomplete info.) imperfect information can be turned into incomplete information with a possible exponential blow-up (via its knowledge-based subset construction). G0
G b,-1
2
b,-1
Σ,-1
1
2
Σ,-1
3
Σ,+1
1 b,-1
Σ,-1
3
Σ,+1
a,-1
3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
Σ,+1
August, 2013
13 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
14 / 35
Incomplete information peculiarities
Observe that in an MPG of incomplete information: 1
the view ∃ve has of a play in the game is o0 σ0 o1 σ1 . . .,
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
15 / 35
Incomplete information peculiarities
Observe that in an MPG of incomplete information: 1
the view ∃ve has of a play in the game is o0 σ0 o1 σ1 . . .,
2
given current oi the game could be in any q ∈ oi (not true in imperfect information),
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
15 / 35
Incomplete information peculiarities
Observe that in an MPG of incomplete information: 1
the view ∃ve has of a play in the game is o0 σ0 o1 σ1 . . .,
2
given current oi the game could be in any q ∈ oi (not true in imperfect information),
3
∀dam can have a two step strategy: choose observations first,
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
15 / 35
Incomplete information peculiarities
Observe that in an MPG of incomplete information: 1
the view ∃ve has of a play in the game is o0 σ0 o1 σ1 . . .,
2
given current oi the game could be in any q ∈ oi (not true in imperfect information),
3
∀dam can have a two step strategy: choose observations first,
4
“delay” the specific choice of states for later!
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
15 / 35
∀dam and determinacy Definition Observable strategies: we let ∀dam reveal to ∃ve only the (Obs × Σ)+ 7→ Obs version of his strategy. Let γ be a function mapping observation-action sequences to concrete state-action ones.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
16 / 35
∀dam and determinacy Definition Observable strategies: we let ∀dam reveal to ∃ve only the (Obs × Σ)+ 7→ Obs version of his strategy. Let γ be a function mapping observation-action sequences to concrete state-action ones.
Definition (New winning condition) Let ψ be a play in the game. ∃ve wins if all paths in γ(ψ) are winning for her. ∀dam wins if there is some path which is winning for him.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
16 / 35
∀dam and determinacy Definition Observable strategies: we let ∀dam reveal to ∃ve only the (Obs × Σ)+ 7→ Obs version of his strategy. Let γ be a function mapping observation-action sequences to concrete state-action ones.
Definition (New winning condition) Let ψ be a play in the game. ∃ve wins if all paths in γ(ψ) are winning for her. ∀dam wins if there is some path which is winning for him.
Theorem (Observable determinacy) The new winning condition is a projection of the perfect information game winning condition (via γ). The new winning condition is coSuslin and hence determined∗ . P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
16 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
17 / 35
Function-Reachability game Definition (Function sequence classification) A function sequence is good (bad) if a function is pointwise bigger or equal (smaller) then a previous one – same observation. Σ,-3 a,-1
2
Σ,-1
1
4 b,-1
3
Σ,+1
Σ,-1
Σ,-1
obs: blue play: fI cur. f: fI (1) = 0 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
18 / 35
Function-Reachability game Definition (Function sequence classification) A function sequence is good (bad) if a function is pointwise bigger or equal (smaller) then a previous one – same observation. Σ,-3
2
a,-1
Σ,-1
1
4 b,-1
3
Σ,+1
Σ,-1
Σ,-1
obs: blue-a-yellow play: fI a f1 cur. f: f1 (2) = −3, f1 (3) = −1 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
18 / 35
Function-Reachability game Definition (Function sequence classification) A function sequence is good (bad) if a function is pointwise bigger or equal (smaller) then a previous one – same observation. Σ,-3 a,-1
2
Σ,-1
1
4 b,-1
3
Σ,+1
Σ,-1
Σ,-1
obs: blue-a-yellow-b-green play: fI a f1 b f2 cur. f: f2 (4) = −4 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
18 / 35
Function-Reachability game Definition (Function sequence classification) A function sequence is good (bad) if a function is pointwise bigger or equal (smaller) then a previous one – same observation. Σ,-3 a,-1
2
Σ,-1
1
4 b,-1
3
Σ,+1
Σ,-1
Σ,-1
obs: blue-a-yellow-b-green-a-green play: fI a f1 b f2 a f3 good cur. f: f3 (4) = −3 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
18 / 35
Unfolding a MPG with incomplete information fI σ0
σ1
o0
o2
o3
o5
“Unfold” G, stop when a good or bad sequence is reached.
f2
.. .
.. .
f1
We are left with a new reachability game
σ0
σ1 .. .
o5
Not all branches will be labelled. . .
f3
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
19 / 35
Strategy transfer
Let H be the reachability game played on the unfolding of G,
Theorem (Strategy transfer for ∃ve ) ∃ve has a finite memory winning strategy in G if and only if she has a winning strategy in H.
Theorem (Strat. transfer for ∀dam ) If ∀dam has a winning observable strategy in H then he also has a winning strategy in G.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
20 / 35
Finite memory, Adeq. Pure, Pure games All based on function sequences (branches) of the associated reachability game H.
Definition 1
Finite memory games: ∃ve can force good leaves or ∀dam can force bad leaves.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
21 / 35
Finite memory, Adeq. Pure, Pure games All based on function sequences (branches) of the associated reachability game H.
Definition 1
Finite memory games: ∃ve can force good leaves or ∀dam can force bad leaves.
2
Adequately pure games: ∃ve ( ∀dam ) can force good (bad) branches where all but 2 functions have different support.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
21 / 35
Finite memory, Adeq. Pure, Pure games All based on function sequences (branches) of the associated reachability game H.
Definition 1
Finite memory games: ∃ve can force good leaves or ∀dam can force bad leaves.
2
Adequately pure games: ∃ve ( ∀dam ) can force good (bad) branches where all but 2 functions have different support.
3
Pure games [structural]: the unfolding of G is finite and in all branches, all but 2 functions have different support.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
21 / 35
Relevant problems
Let A be a class of MPGs with incomplete (or imperfect) information. Given MPG with incomplete (imperfect) information G,
Problem (Class membership) Is G a member of A?
Problem (Winner determination) Does ∃ve have a winning strategy in G?
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
22 / 35
Summary
Finite memory Information ClassUndec1 membership Winnerdet.
1
R-c
Adequately pure
Pure
incomplete imperfect PSPACE- NEXPcomplete hard, in EXPSPACE PSPACE- EXPcomplete complete
incomplete imperfect coNPcoNEXPcomplete complete NP coNP
∩
EXPcomplete
gray=Degorre et al. [2010],white & yellow are our results
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
23 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
24 / 35
Does ∃ve win pure G?
Theorem Deciding if ∃ve has a winning strategy in a given pure MPG with incomplete information is in NP ∩ coNP.
Based on Bj¨orklund et al. [2004]. Observe∗ that positional strategies suffice for ∃ve to win pure games with incomplete information.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
25 / 35
Is G pure?
Theorem The class membership problem for pure games with incomplete information is coNP-complete.
Proof. One can “guess” a branch in H (of size at most |Obs| + 1) and in polynomial time check that it is neither good nor bad. For hardness we reduce from the HAMILTONIAN-CYCLE problem.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
26 / 35
HAM-CYCLE as an MPG
−1
S 0
−N/ + N
v0
+1/ − 1
vn
+1/ − 1
+1/ − 1
vn−1
v1
+1/ − 1 +1/ − 1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
v2
+1/ − 1
···
MPGs with incomplete info.
vn−2
August, 2013
27 / 35
Outline
1
MPG variations Mean-payoff games Imperfect information Incomplete information
2
Observable determinacy
3
Decidable subclasses Pure games with incomplete information Pure games with imperfect information
4
Conclusions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
28 / 35
Solving a pure G
Let G 0 be the equivalent game with incomplete information, then
Remark In pure games, the values of functions in F can be assumed to range over −W · |Obs 0 | and W · |Obs 0 | where W is the biggest transition weight. New reachability game R = hF, fI , Σ, ∆succ , goodi.
Theorem Determining if ∃ve wins a pure MPG with incomplete information is EXP-complete.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
29 / 35
Is G pure?
Theorem Deciding if an MPG with imperfect information is pure is coNEXPTIME-complete.
Proof. Membership: non-deterministically guess a branch in H, check that it not good nor bad. For hardness we reduce from the SUCCINCT HAMILTONIAN-CYCLE problem.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
30 / 35
SUCCINCT HAM-CYCLE
Definition (Galperin and Wigderson [1983]) G = hV , E i with m ≥ 2n vertices, each labelled with a distinct n-bit string. A circuit CG receives two n-bit inputs and outputs 1 if there is an edge. CG has r = O(nk ) gates. 2n
.. .
f = 1 iff (u, v ) ∈ E O(nk ) gates
0
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
31 / 35
SUCCINCT HAM-CYCLE
Theorem (Exponential blow-up) Most problems (reducible as a “projection”) have an exponential blow-up when the graph is represented succinctly. SUCCINCT HAM-CYCLE is NEXPTIME-complete. 2n
.. .
f = 1 iff (u, v ) ∈ E O(nk )
gates
0
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
31 / 35
Is G pure?
Theorem Deciding if an MPG with imperfect information is pure is coNEXPTIME-complete.
Proof. Membership: non-deterministically guess a branch in H, check that it not good nor bad. For hardness we reduce from the SUCCINCT HAMILTONIAN-CYCLE problem. “mixed” path ⇒ simulated 2N transitions
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
32 / 35
coNEXP-hardness proof
−1
S 0
−2N
O1
0
O2
+1
−N
Chk (+1)
G0 (+N)
−1 −1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G1 (+1)
−1
···
MPGs with incomplete info.
−1
Gk (+1)
August, 2013
33 / 35
Summary 1
Done: incomplete info., observable determinacy, subclasses
2
Cooking: other asymmetric information types, other quantitative games, mixed strategies
Finite memory Information ClassUndec1 membership Winnerdet. 1
R-c
Adequately pure
Pure
incomplete imperfect PSPACE- NEXPcomplete hard, in EXPSPACE PSPACE- EXPcomplete complete
incomplete imperfect coNPcoNEXPcomplete complete NP coNP
∩
EXPcomplete
gray=Degorre et al. [2010],white & yellow are our results
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
34 / 35
References I
Bj¨ orklund, H., Sandberg, S., and Vorobyov, S. (2004). Memoryless determinacy of parity and mean payoff games: a simple proof. Theoretical Computer Science, 310(1):365–378. Degorre, A., Doyen, L., Gentilini, R., Raskin, J.-F., and Toru´ nczyk, S. (2010). Energy and mean-payoff games with imperfect information. In Computer Science Logic, pages 260–274. Springer. Ehrenfeucht, A. and Mycielski, J. (1979). Positional strategies for mean payoff games. International Journal of Game Theory, 8:109–113. Galperin, H. and Wigderson, A. (1983). Succinct representations of graphs. Information and Control, 56(3):183–198.
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
MPGs with incomplete info.
August, 2013
35 / 35
coNEXP-hardness proof
xN
x1 x2 ··· x1 x2
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
O1 xN
MPGs with incomplete info.
August, 2013
36 / 35
coNEXP-hardness proof
−1
S 0
−2N
O1
0
O2
+1
−N
Chk (+1)
G0 (+N)
−1 −1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G1 (+1)
−1
···
MPGs with incomplete info.
−1
Gk (+1)
August, 2013
37 / 35
coNEXP-hardness proof xN
x1 x2 ··· σ
x1 x2 −N
xN
+ τN+1
σ −N
O1
xN+1
yN+1
x N+1 − τN+1
.. .
σ −N
+ τ2N
x2N
y2N − τ2N
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G0
MPGs with incomplete info.
x 2N
August, 2013
38 / 35
coNEXP-hardness proof
−1
S 0
−2N
O1
0
O2
+1
−N
Chk (+1)
G0 (+N)
−1 −1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G1 (+1)
−1
···
MPGs with incomplete info.
−1
Gk (+1)
August, 2013
39 / 35
coNEXP-hardness proof
xl xl
···
···
−1
Gj−1
x2N+j−1
x1 x2
σ
x1 x2 xr xr
−1
v1 τ +
σ σ
v2 τ
−1 −1
σ
x 2N+j−1
v1 τ−
Gj v3 χ
v2 τ+
v0
v0 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
v3
τ+
−1 σ
χ
−
τ
x2N+j −
τ +, τ −
MPGs with incomplete info.
x 2N+j August, 2013
40 / 35
coNEXP-hardness proof
xl
···
···
−1
Gj−1 xl
x2N+j−1
x1 x2
σ
x1 x2 xr xr
−1
v2
v1 τ +
σ σ
τ −1 −1
σ
x 2N+j−1
Gj v3
v1 τ−
χ
v2
τ +, τ −
v0
v0 P. Hunter, G. P´ erez, J.F. Raskin (ULB)
v3
τ+
−1 σ
χ
−
τ
x2N+j
+
τ−
MPGs with incomplete info.
x 2N+j August, 2013
40 / 35
coNEXP-hardness proof
−1
S 0
−2N
O1
0
O2
+1
−N
Chk (+1)
G0 (+N)
−1 −1
P. Hunter, G. P´ erez, J.F. Raskin (ULB)
G1 (+1)
−1
···
MPGs with incomplete info.
−1
Gk (+1)
August, 2013
41 / 35
