Admissibility in Quantitative Graph Games
Guillermo A. Pérez
joint work with R. Brenguier, J.-F. Raskin, & O. Sankur
(slides by O. Sankur)
CFV 22/04/16 Seminar @ ULB
Reactive Synthesis: real-world example
R2D2’s goal: Reach the gate G without collisions
uncontrollable
G
controllable
1
Why is C3P0 uncontrollable? See Star Wars Episode II.
Reactive Synthesis: real-world example
R2D2’s goal: Reach the gate G without collisions
uncontrollable
G
controllable
1
Why is C3P0 uncontrollable? See Star Wars Episode II.
Reactive Synthesis: real-world example
R2D2’s goal: Reach the gate G without collisions
uncontrollable
G
controllable
R2D2 can, regardless of what C3P0 does, reach G while avoiding
collisions.
1
Why is C3P0 uncontrollable? See Star Wars Episode II.
Reactive Synthesis: two-player games on graphs
Player ∃ve controls squares and ∀dam controls circles.1 We are “rooting
for” ∃ve.
C
1
(collision)
Our results actually concern n-players.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
3 / 23
Reactive Synthesis: two-player games on graphs
Player ∃ve controls squares and ∀dam controls circles.1 We are “rooting
for” ∃ve.
C
(collision)
Is ∃ve able to perpetually avoid C?
1
Our results actually concern n-players.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
3 / 23
Reactive Synthesis: two-player games on graphs
Player ∃ve controls squares and ∀dam controls circles.1 We are “rooting
for” ∃ve.
C
(collision)
Is ∃ve able to perpetually avoid C?
1
Our results actually concern n-players.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
3 / 23
Formalizing a bit
The set of square vertices is denoted by V∃ .
Strategies
A strategy σ for ∃ve is a function V ∗ V∃ → V ; a strategy τ for ∀dam, a
function V ∗ (V \ V∃ ) → V .
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
4 / 23
Formalizing a bit
The set of square vertices is denoted by V∃ .
Strategies
A strategy σ for ∃ve is a function V ∗ V∃ → V ; a strategy τ for ∀dam, a
function V ∗ (V \ V∃ ) → V .
Winning strategies
A strategy σ for ∃ve is winning for her, w.r.t. to some objective φ, if for all
strategies τ for ∀dam, the resulting play πστ satisfies the objective φ.
Winning strategies are a very robust solution concept.
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Admissibility in Quantitative Graph Games
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Enter admissible strategies
What if ∃ve has no winning strategies?
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Admissibility in Quantitative Graph Games
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Enter admissible strategies
What if ∃ve has no winning strategies?
We would still like to avoid choosing “bad” strategies.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
5 / 23
Enter admissible strategies
What if ∃ve has no winning strategies?
We would still like to avoid choosing “bad” strategies.
Admissible strategies
For a player with objective φ, strategy σ is dominated by σ 0 iff:
for all strategies τ of the other player, if πστ |= φ =⇒ πσ0 τ |= φ; and
for some strategy τ 0 of the other player, πστ 0 6|= φ ∧ πσ0 τ 0 |= φ.
Non-dominated strategies are said to be admissible.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
5 / 23
Enter admissible strategies
What if ∃ve has no winning strategies?
We would still like to avoid choosing “bad” strategies.
Admissible strategies
For a player with objective φ, strategy σ is dominated by σ 0 iff:
for all strategies τ of the other player, if πστ |= φ =⇒ πσ0 τ |= φ; and
for some strategy τ 0 of the other player, πστ 0 6|= φ ∧ πσ0 τ 0 |= φ.
Non-dominated strategies are said to be admissible.
Essentially, dominated strategies are “bad” since there is a better strategy.
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Admissibility in Quantitative Graph Games
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Admissibility: some examples
Even if not a winning strategy, ∃ve should play something which “allows”
her to win.
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Admissibility in Quantitative Graph Games
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Admissibility: some examples
Even if not a winning strategy, ∃ve should play something which “allows”
her to win.
G
Brenguier, Pérez, Raskin, Sankur (ULB)
dom. by
Admissibility in Quantitative Graph Games
G
CFV @ ULB
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Admissibility: some examples
If ∃ve does have a winning strategy σ, then σ is admissible.
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Admissibility in Quantitative Graph Games
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Admissibility: some examples
If ∃ve does have a winning strategy σ, then σ is admissible.
C
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Admissibility in Quantitative Graph Games
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Admissibility: more motivation
Motivation
useful to compare strategies
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Admissibility in Quantitative Graph Games
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Admissibility: more motivation
Motivation
useful to compare strategies
(even if the goal of the other player is not known, unlike NE)
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
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Admissibility: more motivation
Motivation
useful to compare strategies
(even if the goal of the other player is not known, unlike NE)
if goals and rationality of the players is common knowledge, we can
even iterate
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
8 / 23
Admissibility: more motivation
Motivation
useful to compare strategies
(even if the goal of the other player is not known, unlike NE)
if goals and rationality of the players is common knowledge, we can
even iterate
Synthesis specific motivation
simplifying the synthesis task: dominated strategies will not be
winning, so we can remove them from the start
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
8 / 23
Admissibility: more motivation
Motivation
useful to compare strategies
(even if the goal of the other player is not known, unlike NE)
if goals and rationality of the players is common knowledge, we can
even iterate
Synthesis specific motivation
simplifying the synthesis task: dominated strategies will not be
winning, so we can remove them from the start
good candidate for “assume-guarantee” synthesis
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
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Admissibility: state-of-the-art
For Boolean objectives. . .
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
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Admissibility: state-of-the-art
For Boolean objectives. . .
Admissible strategies [Berwanger 2007]
We might have infinite dominance chains, yet
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
9 / 23
Admissibility: state-of-the-art
For Boolean objectives. . .
Admissible strategies [Berwanger 2007]
We might have infinite dominance chains, yet
every non-admissible strategy is dominated by some admissible one
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
9 / 23
Admissibility: state-of-the-art
For Boolean objectives. . .
Admissible strategies [Berwanger 2007]
We might have infinite dominance chains, yet
every non-admissible strategy is dominated by some admissible one
and iteration does terminate.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
9 / 23
Admissibility: state-of-the-art
For Boolean objectives. . .
Admissible strategies [Berwanger 2007]
We might have infinite dominance chains, yet
every non-admissible strategy is dominated by some admissible one
and iteration does terminate.
The set of admissible strategies is regular.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
9 / 23
Admissibility: state-of-the-art
For Boolean objectives. . .
Admissible strategies [Berwanger 2007]
We might have infinite dominance chains, yet
every non-admissible strategy is dominated by some admissible one
and iteration does terminate.
The set of admissible strategies is regular.
The complexity of related decision problems has been studied [B,R, &
Sassolas 2014].
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
9 / 23
Admissibility: state-of-the-art
For Boolean objectives. . .
Admissible strategies [Berwanger 2007]
We might have infinite dominance chains, yet
every non-admissible strategy is dominated by some admissible one
and iteration does terminate.
The set of admissible strategies is regular.
The complexity of related decision problems has been studied [B,R, &
Sassolas 2014].
Assume-admissible synthesis has also been considered [B,R, & Sankur
2015].
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
9 / 23
Admissibility: capturing all admissible strats
0
1
3
C
2
4
5
G
∃ve: Reach G & ∀dam: Avoid vertex C
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Admissibility in Quantitative Graph Games
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Admissibility: capturing all admissible strats
0
1
3
C
2
4
5
G
∀dam’s admissible strategies: do not take any dotted edges
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Admissibility in Quantitative Graph Games
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Admissibility: capturing all admissible strats
0
1
3
C
2
4
5
G
∃ve’s admissible strategies:
do not take dotted edges
+ infinitely often go to 1 — so that ∀dam can help
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Admissibility in Quantitative Graph Games
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Admissibility: capturing all admissible strats
0
1
3
C
2
4
5
G
Any pair of admissible strategies conforms to this graph + ∃ve eventually
reaching 1.
Note: a play resulting from any such pair of strategies will satisfy both
objectives.
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Admissibility in Quantitative Graph Games
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Outline
Moving to Quantitative Objectives
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Admissibility in Quantitative Graph Games
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Outline
Moving to Quantitative Objectives
1
Preliminaries
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
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Outline
Moving to Quantitative Objectives
1
2
Preliminaries
Value-based characterization of adm. strategies
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
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Outline
Moving to Quantitative Objectives
1
2
3
Preliminaries
Value-based characterization of adm. strategies
Existence of adm. strategies
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
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Outline
Moving to Quantitative Objectives
1
2
3
4
Preliminaries
Value-based characterization of adm. strategies
Existence of adm. strategies
LTL(-ish) characterization of outcomes
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Admissibility in Quantitative Games
Let us assume now a weighted graph and a payoff function Val(·). We will
focus on admissible strategies for ∃ve.
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Admissibility in Quantitative Graph Games
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Admissibility in Quantitative Games
Let us assume now a weighted graph and a payoff function Val(·). We will
focus on admissible strategies for ∃ve.
Given strategies σ, τ , let Val(σ, τ ) denote the payoff obtained by ∃ve.
Dominance
Strategy σ for ∃ve is dominated by σ 0 if
for all strategies τ for ∀dam, Val(σ, τ ) ≤ Val(σ 0 , τ ),
there is some strategy τ for ∀dam, Val(σ, τ ) < Val(σ 0 , τ ).
10
s3
s1
Brenguier, Pérez, Raskin, Sankur (ULB)
2
9
3
s5
5
s6
s2
4
s4
Admissibility in Quantitative Graph Games
CFV @ ULB
12 / 23
Admissibility in Quantitative Games
Let us assume now a weighted graph and a payoff function Val(·). We will
focus on admissible strategies for ∃ve.
Given strategies σ, τ , let Val(σ, τ ) denote the payoff obtained by ∃ve.
Dominance
Strategy σ for ∃ve is dominated by σ 0 if
for all strategies τ for ∀dam, Val(σ, τ ) ≤ Val(σ 0 , τ ),
there is some strategy τ for ∀dam, Val(σ, τ ) < Val(σ 0 , τ ).
10
s3
s1
Brenguier, Pérez, Raskin, Sankur (ULB)
2
9
3
s5
5
s6
s2
4
s4
Admissibility in Quantitative Graph Games
CFV @ ULB
12 / 23
Admissibility in Quantitative Games
Let us assume now a weighted graph and a payoff function Val(·). We will
focus on admissible strategies for ∃ve.
Given strategies σ, τ , let Val(σ, τ ) denote the payoff obtained by ∃ve.
Dominance
Strategy σ for ∃ve is dominated by σ 0 if
for all strategies τ for ∀dam, Val(σ, τ ) ≤ Val(σ 0 , τ ),
there is some strategy τ for ∀dam, Val(σ, τ ) < Val(σ 0 , τ ).
10
s3
s1
Brenguier, Pérez, Raskin, Sankur (ULB)
2
9
3
s5
5
s6
s2
4
s4
Admissibility in Quantitative Graph Games
CFV @ ULB
12 / 23
Admissibility in Quantitative Games
Let us assume now a weighted graph and a payoff function Val(·). We will
focus on admissible strategies for ∃ve.
Given strategies σ, τ , let Val(σ, τ ) denote the payoff obtained by ∃ve.
Dominance
Strategy σ for ∃ve is dominated by σ 0 if
for all strategies τ for ∀dam, Val(σ, τ ) ≤ Val(σ 0 , τ ),
there is some strategy τ for ∀dam, Val(σ, τ ) < Val(σ 0 , τ ).
10
s3
s1
Brenguier, Pérez, Raskin, Sankur (ULB)
2
9
3
s5
5
s6
s2
4
s4
Admissibility in Quantitative Graph Games
CFV @ ULB
12 / 23
Admissibility in Quantitative Games
Let us assume now a weighted graph and a payoff function Val(·). We will
focus on admissible strategies for ∃ve.
Given strategies σ, τ , let Val(σ, τ ) denote the payoff obtained by ∃ve.
Dominance
Strategy σ for ∃ve is dominated by σ 0 if
for all strategies τ for ∀dam, Val(σ, τ ) ≤ Val(σ 0 , τ ),
there is some strategy τ for ∀dam, Val(σ, τ ) < Val(σ 0 , τ ).
10
s3
s1
Brenguier, Pérez, Raskin, Sankur (ULB)
2
9
3
s5
5
s6
s2
4
s4
Admissibility in Quantitative Graph Games
CFV @ ULB
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Values of a Game
Values
For play prefix h, strategy σ for ∃ve, let aVal(h) denote the antagonistic
value:
h ),
aVal(h, σ) = inf τ Val(h · πστ
h ).
aVal(h) = supσ inf τ Val(h · πστ
and cooperative value:
h ),
cVal(h, σ) = supτ Val(h · πστ
h ).
cVal(h) = supσ,τ Val(h · πστ
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Admissibility in Quantitative Graph Games
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Values of a Game
Values
For play prefix h, strategy σ for ∃ve, let aVal(h) denote the antagonistic
value:
h ),
aVal(h, σ) = inf τ Val(h · πστ
h ).
aVal(h) = supσ inf τ Val(h · πστ
and cooperative value:
h ),
cVal(h, σ) = supτ Val(h · πστ
h ).
cVal(h) = supσ,τ Val(h · πστ
Tool to study dominance.
For instance, σ is dominated:
(by σ 0 ) if cVal(s1 , σ) < aVal(s1 , σ 0 ),
or, if σ takes an edge s → s 0 with cVal(s 0 ) < aVal(s)
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Characterization of Admissible Strategies
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Examples
Can we represent admissible strategies by removing edges based on aVal
and cVal?
as in the Boolean case
Examples
Can we represent admissible strategies by removing edges based on aVal
and cVal?
as in the Boolean case
Edges to be removed depend on the prefix:
10
s3
s1
2
9
3
s5
5
s6
s2
s4
4
?
s7
Idea: after s1 → s2 , remember that the strategy has committed to get
more than 5.
Examples
Are worst-case optimal strategies always admissible?
σ worst-case optimal if aVal(s, σ) = aVal(s)
Worst-case optimal strategies are admissible if “obtaining more than the
antagonistic value implies a risk”
10
-1
Both strategies are admissible
s2
s1
1
Examples
10
s3
1
s2
s1
aVal(s1 ) = aVal(s2 ) = aVal(s3 ) = 1.
1
Examples
10
s3
1
s2
s1
aVal(s1 ) = aVal(s2 ) = aVal(s3 ) = 1.
1
Examples
10
s3
1
s2
s1
aVal(s1 ) = aVal(s2 ) = aVal(s3 ) = 1.
Not all worst-case optimal strategies are admissible
1
Examples
10
s3
1
s2
s1
1
aVal(s1 ) = aVal(s2 ) = aVal(s3 ) = 1.
Not all worst-case optimal strategies are admissible
Here going left “safely maximizes the cooperative value”:
The antagonistic value is still optimal aVal(s1 , σ) = 1
The cooperative value is good: cVal(s1 , σ) = 10
σ dominated by σ 0 if aVal(h, σ) = cVal(h, σ) = aVal(h, σ 0 ) < cVal(h, σ 0 ).
One last value of the game
Let us define acVal(h) at prefix h:
acVal(h) = sup{cVal(h, σ) | σ strategy s.t. aVal(h, σ) = aVal(h)}
If acVal(h) = aVal(h), then all WCO strategies are admissible.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
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One last value of the game
Let us define acVal(h) at prefix h:
acVal(h) = sup{cVal(h, σ) | σ strategy s.t. aVal(h, σ) = aVal(h)}
If acVal(h) = aVal(h), then all WCO strategies are admissible.
If acVal(h) > aVal(h), then a WCO strategy is admissible iff
cVal(h, σ) > aVal(h).
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Admissibility in Quantitative Graph Games
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Characterization of Admissible Strategies
Assumption: WCO strategies exist: ∀h, ∃σ, aVal(h, σ) = aVal(h).
Characterization of Admissible Strategies
Assumption: WCO strategies exist: ∀h, ∃σ, aVal(h, σ) = aVal(h).
Theorem
A strategy σ for ∃ve is admissible iff at all prefixes h ending in V∃ :
cVal(h, σ) > aVal(h)
or aVal(h, σ) = aVal(h) = acVal(h)
Characterization of Admissible Strategies
Assumption: WCO strategies exist: ∀h, ∃σ, aVal(h, σ) = aVal(h).
Theorem
A strategy σ for ∃ve is admissible iff at all prefixes h ending in V∃ :
cVal(h, σ) > aVal(h)
or aVal(h, σ) = aVal(h) = acVal(h)
10
s3
s1
2
9
3
s5
5
s6
s2
s4
4
9 = cVal(s1 , σ) > aVal(s1 ) = 5,
9 = cVal(s1 s2 s4 , σ) > aVal(s1 s2 s4 ) = 3.
Characterization of Admissible Strategies
Assumption: WCO strategies exist: ∀h, ∃σ, aVal(h, σ) = aVal(h).
Theorem
A strategy σ for ∃ve is admissible iff at all prefixes h ending in V∃ :
cVal(h, σ) > aVal(h)
or aVal(h, σ) = aVal(h) = acVal(h)
10
s3
1
s2
s1
10 = cVal(s1 , σ) > aVal(s1 ) = 1
1
Characterization of Admissible Strategies
Assumption: WCO strategies exist: ∀h, ∃σ, aVal(h, σ) = aVal(h).
Theorem
A strategy σ for ∃ve is admissible iff at all prefixes h ending in V∃ :
cVal(h, σ) > aVal(h)
or aVal(h, σ) = aVal(h) = acVal(h)
10
-1
s2
s1
1
aVal(s1 , σ) = acVal(s1 ) = 1.
Existence of Admissible Strategies
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Existence of Admissible Strategies
(or why the existence of WCO strategies is a valid
assumption)
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Existence
In general, admissible strategies do not always exist.
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Existence
In general, admissible strategies do not always exist.
Theorem
In a given class of games, if WCO and Co-Op strategies exist, then
admissible strategies always exist.
Otherwise, some games have no admissible strategies.
Cooperatively-optimal (Co-Op) strategy (cVal(s1 , σ) = cVal(s1 ))
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Outcomes Compatible with Admissible Strategies
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Admissible Outcomes: Temporal Logic
Assumption: Prefix-independent objectives (e.g. mean payoff)
Logic
Consider LTL, and in addition consider predicates:
Val > v which holds at infinite play ρ if Val(ρ) > v .
Predicate aValv meaning that aVal at current vertex is v
Predicate acValv
Admissible Outcomes: Temporal Logic
Assumption: Prefix-independent objectives (e.g. mean payoff)
Logic
Consider LTL, and in addition consider predicates:
Val > v which holds at infinite play ρ if Val(ρ) > v .
Predicate aValv meaning that aVal at current vertex is v
Predicate acValv
“Witness” vertices: adversary vertices q where some successor q 0
satisfies cVal(q 0 ) > v
Witnessv (q) ⇔ ∃q 0 , q → q 0 , cVal(q 0 ) > v .
Witnesses that our strategy gave the adversary the opportunity to
help us obtain more than v
Set of Outcomes under Admissible Strategies
Define formulae
φ1 = ∨v ∈aValues (aValv ∧ (Val > v ∨ F(Witnessv ))) ,
φ2 = ∨v ∈aValues (aValv ∧ acValv ∧ Val = v ∧ G (aValv )) .
Recall the characterization of admissible strategies: ∀h, last(h) ∈ V∃ :
cVal(h, σ) > aVal(h)
or aVal(h, σ) = aVal(h) = acVal(h)
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Set of Outcomes under Admissible Strategies
Define formulae
φ1 = ∨v ∈aValues (aValv ∧ (Val > v ∨ F(Witnessv ))) ,
φ2 = ∨v ∈aValues (aValv ∧ acValv ∧ Val = v ∧ G (aValv )) .
And we set Φadm = G (¬V∃ ∨ φ1 ∨ φ2 ) .
Theorem
A play ρ is compatible with some admissible strategy of ∃ve, iff ρ |= Φadm .
Recall the characterization of admissible strategies: ∀h, last(h) ∈ V∃ :
cVal(h, σ) > aVal(h)
or aVal(h, σ) = aVal(h) = acVal(h)
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Conclusion
Value-based characterization of adm. strategies.
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Conclusion
Value-based characterization of adm. strategies.
Admissible actions depend on prefix (unlike for Boolean objectives).
Admissibility is decidable (given a strategy).
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Conclusion
Value-based characterization of adm. strategies.
Admissible actions depend on prefix (unlike for Boolean objectives).
Admissibility is decidable (given a strategy).
Existence of adm. strategies (under hypotheses: WCO and Co-Op
strategies must exist).
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Conclusion
Value-based characterization of adm. strategies.
Admissible actions depend on prefix (unlike for Boolean objectives).
Admissibility is decidable (given a strategy).
Existence of adm. strategies (under hypotheses: WCO and Co-Op
strategies must exist).
LTL+payoff characterization of the outcomes under adm. strats.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
23 / 23
Conclusion
Value-based characterization of adm. strategies.
Admissible actions depend on prefix (unlike for Boolean objectives).
Admissibility is decidable (given a strategy).
Existence of adm. strategies (under hypotheses: WCO and Co-Op
strategies must exist).
LTL+payoff characterization of the outcomes under adm. strats.
allows us to do model checking under admissibility
if we have a prefix-indep. objective, and
if aVal, cVal, and acVal are computable, then
we can model-check the game against an LTL+payoff spec
(decidable even for MP).
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
23 / 23
Conclusion
Value-based characterization of adm. strategies.
Admissible actions depend on prefix (unlike for Boolean objectives).
Admissibility is decidable (given a strategy).
Existence of adm. strategies (under hypotheses: WCO and Co-Op
strategies must exist).
LTL+payoff characterization of the outcomes under adm. strats.
allows us to do model checking under admissibility
if we have a prefix-indep. objective, and
if aVal, cVal, and acVal are computable, then
we can model-check the game against an LTL+payoff spec
(decidable even for MP).
Thank you for your attention.
Brenguier, Pérez, Raskin, Sankur (ULB)
Admissibility in Quantitative Graph Games
CFV @ ULB
23 / 23