Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Uniform strategies
Strategic Reasoning 2013
Bastien Maubert
Sophie Pinchinat1
Laura Bozzelli2
March 17, 2013
1
2
IRISA, Rennes, France
UPM, Madrid, Spain
1
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
1
Motivation
2
Uniform strategies
3
Synthesizing fully-uniform strategies
2
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Motivation
Two distinct notions of uniform strategies in the literature:
In games with imperfect information:
Uniform strategies [Van Benthem, 2001]
Observation-based strategies [Chatterjee et al., 2006]
In model-checking games for logics of imperfect information:
Uniform strategies [Väänänen, 2007]
Coherent strategies [Grädel, 2012]
Common point
They are different notions of uniformity, but both represent
constraints on strategies (they limit the set of allowed strategies),
and both constraints involve sets of plays.
3
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Games with imperfect information
Arena
Player 1 has a partial observation of plays
Observational equivalence relation on finite plays: ρ ∼ ρ0
Strategies
Player 1 cannot use information she does not have
Strategies must be observation-based, or uniform:
ρ ∼ ρ0 ⇒ σ(ρ) = σ(ρ0 )
Constraint on sets of equivalent plays
4
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Games with imperfect information
Arena
Player 1 has a partial observation of plays
Observational equivalence relation on finite plays: ρ ∼ ρ0
Strategies
Player 1 cannot use information she does not have
Strategies must be observation-based, or uniform:
ρ ∼ ρ0 ⇒ σ(ρ) = σ(ρ0 )
Constraint on sets of equivalent plays
4
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Picture
Imperfect-information games
Games for imperfect-information logics
5
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Picture
Uniform strategies
Imperfect-information games
Games for imperfect-information logics
5
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Picture
Uniform strategies
Games for non-interference
Games for diagnosis
Imperfect-information games
Games for imperfect-information logics
Games with epistemic winning condition
5
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Two-player turn-based game arenas
P rop = {p, q, r, . . .}
G = (V, E, vI , `) where
V = V1 ] V2 : positions
p
E ⊆ V × V : edges
vI ∈ V : initial position
p, q
r
` : V → 2P rop : valuation
π ∈ P laysω
π[0..i] = π[0]π[1] . . . π[i]
q
p, r
Strategy (for Player 1):
σ : V ∗ V1 → V
Out(σ) ⊆ P laysω
6
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Two-player turn-based game arenas
P rop = {p, q, r, . . .}
G = (V, E, vI , `) where
V = V1 ] V2 : positions
p
E ⊆ V × V : edges
vI ∈ V : initial position
p, q
r
` : V → 2P rop : valuation
π ∈ P laysω
π[0..i] = π[0]π[1] . . . π[i]
q
p, r
Strategy (for Player 1):
σ : V ∗ V1 → V
Out(σ) ⊆ P laysω
6
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Specification language: RLTL
LTL for temporal properties
R for grabing sets of plays
Syntax
RLTL : ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | #ϕ | ϕUϕ | Rϕ
Rϕ means: ϕ holds in all related plays.
7
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Examples of formulas
Imperfect-information strategies
Semantics of R: Player 1’s observation
_
R#m)
G(p1 →
m∈M ove
”Whenever it is Player 1’s turn, indistinguishable plays will be
extended with the same move m”
Games with opacity condition
Some positions denote a “secret” information
Player 1 doesn’t want Player 2 to find it
Semantics of R: Player 2’s observation
G¬Rsecret
8
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Semantics
Π ⊆ P laysω : the universe
;⊆ P lays2∗ : any binary relation on finite plays.
Knowledge
Beliefs
Other relations, not related to any kind of “information”
Take π ∈ Π, i ∈ N.
Π, π, i |= ϕ
p, ¬ϕ, ϕ ∧ ψ ...
Π, π, i |= #ϕ
Π, π, i |= ϕUψ
if
...
Π, π, i + 1 |= ϕ
Π, π, i |= Rϕ
if
for all π 0 ∈ Π, j ∈ N, such that
π[0..i] ; π 0 [0..j], Π, π 0 , j |= ϕ
9
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Two possible universes
Yellow lines: σ
Rϕ
ϕ
Π = Out(σ) : Strict uniformity
ϕ
Rϕ
ϕ
Π = P laysω : Full uniformity
10
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Two possible universes
Yellow lines: σ
Rϕ
ϕ
Π = Out(σ) : Strict uniformity
ϕ
Rϕ
ϕ
Π = P laysω : Full uniformity
10
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Two possible universes
Yellow lines: σ
Rϕ
ϕ
Π = Out(σ) : Strict uniformity
ϕ
Rϕ
ϕ
Π = P laysω : Full uniformity
10
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Two possible universes
Yellow lines: σ
Rϕ
ϕ
Π = Out(σ) : Strict uniformity
ϕ
Rϕ
ϕ
Π = P laysω : Full uniformity
10
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Two notions of uniform strategies
As strategy σ is ϕ-strictly uniform if:
for all π in Out(σ),
Out(σ), π, 0 |= ϕ
A strategy σ is ϕ-fully uniform if:
for all π in Out(σ),
P laysω , π, 0 |= ϕ
11
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Utilisations of strictly and fully
Observation based strategies: strict uniformity.
Coherent strategies: strict uniformity.
Games with epistemic winning condition (LTLK)
Agent knows the strategy: Strict uniformity.
Agent ignores the strategy: Full uniformity.
Example: study strategies of Player 1, while the objective
concerns the knowledge of Player 2.
Question:
How to decide the existence of a uniform strategy?
12
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Synthesizing fully-uniform strategies
The fully-uniform strategy problem (FUS)
Input: Finite arena G, relation ;, ϕ ∈ RLTL.
Output: Yes if there is a ϕ-fully uniform strategy in G.
How to finitely represent a binary relation?
13
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Synthesizing fully-uniform strategies
The fully-uniform strategy problem (FUS)
Input: Finite arena G, relation ;, ϕ ∈ RLTL.
Output: Yes if there is a ϕ-fully uniform strategy in G.
How to finitely represent a binary relation?
⇒ Finite state transducers (FST)
They recognize rational relations.
13
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Finite state transducers and rational relations
a/a, b/b
q0
a/b
b/a
/a
a/
q1
q2
/b
b/
Relation recognized by this transducer:
w ∼ w0 if |w|a = |w0 |a
14
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
FUS with rational relations
FUSrat
Input:
Finite arena G
FST T representing a rational relation ;
ϕ ∈ RLTL
Output: Yes if there is a ϕ-fully uniform strategy in G.
Theorem
FUSrat is decidable
15
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Reduction to LTL games
Let G, T, ϕ be an instance of FUSrat (T recognizes ;).
R depth
dR (ϕ) is the maximum nesting of R modalities in ϕ.
Reduction to LTL games
Iterate dR (ϕ) times:
Powerset construction from G and T
If ψ ∈ LTL, Rψ can be evaluated positionally
⇒ Elimination of innermost Rψ subformulas of ϕ
Solve the LTL game, synthesize a finite-memory strategy.
16
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Complexity
Solving LTL games:
2Exptime-complete [Pnuelli & Rosner, 85]
A non-elementary decision procedure, essentially optimal.
FUSrat is non-elementary-complete
More precisely, if we assume that dR (ϕ) ≤ d, then it is
2Exptime-complete if d ≤ 2
dExptime-complete if d > 2
Lower bounds: encoding of exp[d]-space bounded alternating
Turing Machines.
17
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Conclusion
Two general notions of uniform strategies
Subsumes both previous notions of uniform strategies
A language (RLTL) to specify uniformity constraints
Enables to represent constraints on strategies, or winning
conditions with epistemic or belief features
The fully-uniform strategy problem is decidable, for a very
general class of relations.
18
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Current and future work
Additional assumptions on ; that make FUS elementary
Decidability status of the strictly-uniform strategy problem,
for various classes of relations
n agents
Language allowing both strict and full semantics: Rs and Rf
Add ATL-like modalities?
What if we take µ-calculus instead of LTL?
19
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Thank you
Questions?
20
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Games with Imperfect Information
p1
vI
vI
a
a
pa
vI , a
b
p1
b
v1
v2
a
pa
a
a
p1
v4
a
a
v2
pb
v1 , a
b
v3
p1
v1
pa
v1 , b
pb
v2 , a
v2 , b
p1
v3
pa
v4
pa
v3 , a
v4 , a
21
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Games with Imperfect Information
p1
vI
vI
a
a
pa
vI , a
b
p1
b
v1
v2
a
pa
a
a
p1
v4
a
a
v2
pb
v1 , a
b
v3
p1
v1
pa
v1 , b
pb
v2 , a
v2 , b
p1
v3
pa
v4
pa
v3 , a
v4 , a
21
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Games with Imperfect Information
p1
vI
vI
a
a
pa
vI , a
b
p1
b
v1
v2
a
pa
a
a
p1
v4
a
a
v2
pb
v1 , a
b
v3
p1
v1
pa
v1 , b
pb
v2 , a
v2 , b
p1
v3
pa
v4
pa
v3 , a
v4 , a
21
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Games with Imperfect Information
p1
vI
pa
vI , a
p1
p1
v1
pa
v2
pb
v1 , a
p1
pa
v1 , b
pb
v2 , a
v2 , b
a∈Act
p1
v3
pa
π[0, i] ; π 0 [0, j] if π[0, i]
and π 0 [0, j] are
observationally equivalent
W
ϕ = G(p1 → R # pa )
v4
pa
v3 , a
v4 , a
Theorem
A strategy σ for Player 1 is “uniform” iff
σ is ϕ-strictly uniform.
22
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Dependence Logic
∀x0 ∀x1 ϕ0
∅
∀x1 ϕ0
{x0 7→ 0}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 0}
x1 = x0
{x0 7→ 0,
x1 7→ 0}
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 0}
d, p0
∀x1 ϕ0
{x0 7→ 1}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 1}
x1 = x0
{x0 7→ 0,
x1 7→ 1}
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 1}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 2}
x1 = x0
{x0 7→ 0,
x1 7→ 2}
∀x1 ϕ0
{x0 7→ 2}
...
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 2}
d, p1
d, p2
π[0, i] ; π 0 [0, j] if π[i] = (dep(t1 , . . . , tn ), s),
π 0 [j] = (dep(t1 , . . . , tn ), s0 ) and s and s0 agree on t1 , . . . , tn−1 .
W
ϕ = G(d →
R(tn = a))
a∈Dom
Theorem
A strategy σ is “uniform” iff it is ϕ-strictly uniform
23
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Dependence Logic
∀x0 ∀x1 ϕ0
∅
∀x1 ϕ0
{x0 7→ 0}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 0}
x1 = x0
{x0 7→ 0,
x1 7→ 0}
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 0}
d, p0
∀x1 ϕ0
{x0 7→ 1}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 1}
x1 = x0
{x0 7→ 0,
x1 7→ 1}
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 1}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 2}
x1 = x0
{x0 7→ 0,
x1 7→ 2}
∀x1 ϕ0
{x0 7→ 2}
...
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 2}
d, p1
d, p2
π[0, i] ; π 0 [0, j] if π[i] = (dep(t1 , . . . , tn ), s),
π 0 [j] = (dep(t1 , . . . , tn ), s0 ) and s and s0 agree on t1 , . . . , tn−1 .
W
ϕ = G(d →
R(tn = a))
a∈Dom
Theorem
A strategy σ is “uniform” iff it is ϕ-strictly uniform
23
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Dependence Logic
∀x0 ∀x1 ϕ0
∅
∀x1 ϕ0
{x0 7→ 0}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 0}
x1 = x0
{x0 7→ 0,
x1 7→ 0}
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 0}
d, p0
∀x1 ϕ0
{x0 7→ 1}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 1}
x1 = x0
{x0 7→ 0,
x1 7→ 1}
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 1}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 2}
x1 = x0
{x0 7→ 0,
x1 7→ 2}
∀x1 ϕ0
{x0 7→ 2}
...
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 2}
d, p1
d, p2
π[0, i] ; π 0 [0, j] if π[i] = (dep(t1 , . . . , tn ), s),
π 0 [j] = (dep(t1 , . . . , tn ), s0 ) and s and s0 agree on t1 , . . . , tn−1 .
W
ϕ = G(d →
R(tn = a))
a∈Dom
Theorem
A strategy σ is “uniform” iff it is ϕ-strictly uniform
23
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Dependence Logic
∀x0 ∀x1 ϕ0
∅
∀x1 ϕ0
{x0 7→ 0}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 0}
x1 = x0
{x0 7→ 0,
x1 7→ 0}
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 0}
d, p0
∀x1 ϕ0
{x0 7→ 1}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 1}
x1 = x0
{x0 7→ 0,
x1 7→ 1}
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 1}
x1 = x0 ∨ dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 2}
x1 = x0
{x0 7→ 0,
x1 7→ 2}
∀x1 ϕ0
{x0 7→ 2}
...
dep(x0 , x1 )
{x0 7→ 0,
x1 7→ 2}
d, p1
d, p2
π[0, i] ; π 0 [0, j] if π[i] = (dep(t1 , . . . , tn ), s),
π 0 [j] = (dep(t1 , . . . , tn ), s0 ) and s and s0 agree on t1 , . . . , tn−1 .
W
ϕ = G(d →
R(tn = a))
a∈Dom
Theorem
A strategy σ is “uniform” iff it is ϕ-strictly uniform
23
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
When is full uniformity needed?
R: Player 2’s knowledge.
Player 1: perfect information.
S ⊆ V : a secret property.
Player 1 wants to prevent Player 2 from knowing S
Let ϕ = G¬RS.
Opacity guarantee
Player 1 can protect the secret if
she has a ϕ-fully uniform strategy
Player 2 can consider possible plays that are not induced by
Player 1’s strategy.
24
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Powerset construction
T : transducer that recognizes ;
“Product” of the arena G with T
We simulate the nondeterministic execution of T along the
game
T reads the current play and nondeterministicaly outputs
related plays
25
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Powerset construction
What we want: information sets, to evaluate Rϕ positionally.
Information set = set of last positions of equivalent plays
What we need to remember:
For the execution of T : The set S of states in which the
transducer may be (nondeterminism)
For computing I: For each q ∈ S, for each possible execution
of T that ends in q, remember the last
letter/position on the output. It is the set
Last(q).
A position: (v, S, Last)
Computing I:
I(v, S, Last) =
S
Last(q)
q∈S∩QF
26
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Idea of the Rϕ elimination
ϕ ∈ LTL
Positionality
Whether π, i |= Rϕ or not does not depend on π[i + 1, . . .]
(quantification over plays).
π[0 . . . i] determines the set of equivalent plays.
In the powerset construction, the relevant information on the
past is in the position
⇒ For full uniformity the semantics of Rϕ can be defined
positionally:
Positional semantics
(v, S, Last) |= Rϕ if v 0 |= Aϕ for all v 0 ∈ I(v, S, Last)
27
Motivation
Uniform strategies
Synthesizing fully-uniform strategies
Idea of the Rϕ elimination
Take G, ;, ϕ an instance. Let Gb be the powerset contruction.
Marking
Let Rϕ1 , . . . , Rϕn be all the subformula of ϕ such that
ϕi ∈ LTL
b v)
For each vb ∈ Vb , for all i, if vb |= Rϕi , let pRϕi ∈ `(b
ϕ
b := ϕ[pRϕ1 /Rϕ1 , . . . , pRϕn /Rϕn ]
We have d(ϕ)
b = d(ϕ) − 1
b Tb
Lift T to G:
Player 1 has a ϕ-fully uniform strategy in G with [T ]
iff
she has a ϕ-fully
b
uniform strategy in Gb with [Tb]
28