Agents with Perfect and Truly Perfect
Recall
Nils Bulling
Computational Intelligence Group
Department of Informatics
TU Clausthal
March 3, 2014
(based on joint work with Wojtek Jamroga and Matei Popovici)
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
1
Outline
1 Introduction
2 Reasoning about Strategic Ability
Alternating-Time Temporal Logic
Imperfect Information
3 Comparing Standard Semantics
Overview of the Results
Tree Unfoldings: Perfect vs. Imperfect Recall
4 Perfect Recall and No Forgetting
ATL with Truly Prefect Recall
Expressivity and Validities
5 Conclusions
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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1 Introduction
1. Introduction
1
Introduction
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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1 Introduction
Dependencies in Strategic Reasoning
Logic: Alternating-Time Temporal Logics [Alur et al., 2002]
What does an agents decision affect?
memory: perfect / imperfect
knowledge: perfect / imperfect
behavior in subgames ; forgetting (nested quantification in
FOL)
ability: objective / subjective
Further aspects:
limited resources
strategy commitment
social and normative constraints
...
knowledge
memory
forgetting
Nils Bulling · Agents with Perfect and Truly Perfect Recall
subjective ability
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1 Introduction
Memory Matters
q0
n
lea
c
de
liv
er
q1
q2
clean
delivered
Strategies without memory: Q → Act
Strategies with memory: Q ∗ → Act
Strategic ability depends on memory
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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1 Introduction
Knowledge Matters
win
)
)
R
(nop,pickL )
ck
R
ick
,p
op
(n
q30
(nop,pickL )
(nop,pickL )
q3
(n
op
,p
i
q2
q3
q20
g
(n
op
,p
ick
R)
win
)
R
ck
(nop,pickL )
q20
q2
i
,p
(n
op
q30
Shell game: Ball in left shell (q2 ) are right shell (q20 )
Strategic ability depends on knowledge
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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1 Introduction
Remembering Things in Subgames Matters
q0
(putR ,nop)
(close,nop)
q2
(nop,pickL )
q3
q10
(close,nop)
(putL ,nop)
q20
g
(no
p,p
ick
R)
win
(nop,pickL )
q1
)
kR
ic
p,p
(no
q30
Strategic ability depends on further assumptions...
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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1 Introduction
Motivation for This Work
Focus is on the logics; i.e., on the level of valid sentences.
Validities capture general properties of games.
Wich logics characterize the same kind of ability in games?
Propose a new no-forgetting semantics forgetting
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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2 Reasoning about Strategic Ability
2. Reasoning about Strategic Ability
2
Reasoning about Strategic Ability
Alternating-Time Temporal Logic
Imperfect Information
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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2 Reasoning about Strategic Ability
2.1 Alternating-Time Temporal Logic
2.1 Alternating-Time Temporal Logic
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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2 Reasoning about Strategic Ability
2.1 Alternating-Time Temporal Logic
Temporal Logic and Reactive Systems
1
2
pos0
LTL: Modelling linear time
2
1
computation: q0 q1 q2ω
pos1
1
pos2
2
“Some property p holds in some future
state’’
q0
pos0
ATL? admits a fine-grained quantification over computations
Nils Bulling · Agents with Perfect and Truly Perfect Recall
q2
pos2
q1
pos1
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2 Reasoning about Strategic Ability
2.1 Alternating-Time Temporal Logic
Concurrent Game Structures and ATL
wait,wait
push,push
wait,wait
push,push
pu
sh
cooperate
q2
pos2
q0
pos0
it
wa
sh,
h
pu
us
it,p
wa
execute actions
,w
ait
wa
it,p
us
h
Agents:
wait,push
push,wait
q1
wait,wait
push,push
pos1
Strategic logic ATL (Alur et al. 1997-2002):
hhAiiγ
“Group A has a strategy to guarantee γ”
ATL: ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | hhAii hϕ | hhAii2ϕ | hhAiiϕUϕ
ATL∗ : Allows arbitrary combinations of cooperation and temporal
modalities (e.g. hhAii23ϕ).
Example: M, q0 |= hh1ii2¬pos1
Nils Bulling · Agents with Perfect and Truly Perfect Recall
M, q0 6|= hh1ii3pos1
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2 Reasoning about Strategic Ability
2.1 Alternating-Time Temporal Logic
Strategies and Their Outcome
Perfect information perfect recall strategy for agent a
(IR-strategy):
sa : Q + → Act .
Perfect information memoryless strategy for agent a
(Ir -strategy):
sa : Q → Act
Outcome:
out(q, sA )= set of all paths/executions possible if A follows sA .
Nils Bulling · Agents with Perfect and Truly Perfect Recall
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2 Reasoning about Strategic Ability
2.1 Alternating-Time Temporal Logic
Perfect Information Semantics
Let x ∈ {r, R}.
M, q |=Ix hhAiiΦ
M, λ |=Ix
hϕ
M, λ |=Ix 3ϕ
iff there is a collective Ix-strategy sA such that,
for each path λ ∈ out(q, sA ), we have M, λ |=Ix
Φ.
iff M, λ[1, ∞] |=Ix ϕ;
iff M, λ[i, ∞] |=Ix ϕ for some i ≥ 0;
Logics: ATLIr , ATLIR , ATL∗Ir , ATL∗IR
Theorem 1 ( [Schobbens, 2004, Alur et al., 2002])
For every ϕ ∈ ATL: M, q |=IR ϕ iff M, q |=Ir ϕ
ATL = ATLIr = ATLIR
ATL∗Ir 6= ATL∗IR
Nils Bulling · Agents with Perfect and Truly Perfect Recall
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2 Reasoning about Strategic Ability
2.2 Imperfect Information
2.2 Imperfect Information
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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2 Reasoning about Strategic Ability
2.2 Imperfect Information
Modeling Imperfect Information
We combine ATL∗ and epistemic logic.
indistinguishability relations ∼a ⊆ Q × Q:
q ∼a q 0 : q and q 0 give the same observation to a.
hhAiiγ: Agents A (mutually) know that they can enforce γ.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
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2 Reasoning about Strategic Ability
2.2 Imperfect Information
Example 2 (Robots and Carriage)
2
2
1
it
h
us
wait,push
push,wait
pos2
wa
sh,
pu
q2
it,p
2
wait,wait
push,push
2
pos1
pos0
wa
1
pos2
q0
,w
ait
wa 1
it,p
us
h
wait,wait
push,push
pos0
pu
sh
1
q1
wait,wait
push,push
pos1
Can Agt enforce to be in q1 in the next step?
M, q0 |= Ir hhAgtii hpos1
M, q0 6|= ir hhAgtii hpos1
Strategies should be executable ; uniform strategies
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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2 Reasoning about Strategic Ability
2.2 Imperfect Information
Uniform Strategies and Subjective Ability
Memoryless strategies(ir ):
if q ∼a q 0 then sa (q) = sa (q 0 )
Perfect recall strategies (iR):
if h ≈a h0 then sa (h) = sa (h0 )
Subjective ability: All paths from all indistinguishable states are
taken into account:
S
outi (q, sA ) = q∼A q0 out(q0 , sA )
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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2 Reasoning about Strategic Ability
2.2 Imperfect Information
Imperfect Information Semantics
The ix-semantics, x ∈ {r, R}, is defined as follows:
M, q |=ix hhAiiϕ iff
there is a collective ix-strategy sA
such that, for each path λ ∈ outi (q 0 , sA ),
we have M, λ |=ix ϕ
Logics: ATLir , ATLiR , ATL∗ir , ATL∗iR
Notes:
This definition models that “everybody in A knows that ϕ”.
hhAii implicitly includes a knowledge operator!
We can define Ka ψ as hhaiiψUψ.
We can define EA ψ as hhAiiψUψ.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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3 Comparing Standard Semantics
3. Comparing Standard Semantics
3
Comparing Standard Semantics
Overview of the Results
Tree Unfoldings: Perfect vs. Imperfect Recall
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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3 Comparing Standard Semantics
This is joint work with
Wojtek Jamroga (University of Luxembourg)
Bulling, N. and Jamroga, W. (2014).
Comparing variants of strategic ability: how uncertainty and
memory influence general properties of games.
Autonomous Agents and Multi-Agent Systems, 28(3):474–518.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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3 Comparing Standard Semantics
3.1 Overview of the Results
3.1 Overview of the Results
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Dependence Logic@KNAW@NL, March 3, 2014
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3 Comparing Standard Semantics
3.1 Overview of the Results
Overview of Variants of ATL
?
objective
perfect recall
ATL⇤io R
ATLio R
ATL⇤io r
memoryless
subjective
ATL⇤is R
ATL⇤Ir 6= ATL⇤IR
ATLis R
ATLIR = ATLIr
ATLio r
Nils Bulling · Agents with Perfect and Truly Perfect Recall
language
ATL⇤is r
ATLis r
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3 Comparing Standard Semantics
3.1 Overview of the Results
Overview ATL∗
perfect inf.
and recall
imperfect inf.
perfect recall
ATL⇤IR
perfect inf.
ATL⇤Ir memoryless
ATL⇤is R
imperfect inf.
memoryless
Nils Bulling · Agents with Perfect and Truly Perfect Recall
ATL⇤is r
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3 Comparing Standard Semantics
3.1 Overview of the Results
Overview ATL
perfect inf.
imperfect inf.
perfect recall
ATLis R
imperfect inf.
memoryless
ATLis r
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3 Comparing Standard Semantics
3.1 Overview of the Results
Examples of Important (In)Validities
Validities are not only important conceptually but are also helpful to
construct efficient decision procedures (e.g. model checking
algorithms). For example:
hhaii3p ↔ p ∨ hhaii hhhaii3p
Invalid in all variants with imperfect information.
Valid for perfect information.
They are useful to prove strict inclusion.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
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3 Comparing Standard Semantics
3.1 Overview of the Results
Example: Perfect vs. Imperfect Information
Subjective incomplete information vs. perfect information.
Proposition 3
Val (ATLis r ) ( Val (ATLIr )
⊆: Every CGS can be seen as a special iCGS.
6⊇: M, q0 6|=is r (win ∨ hhaii hhhaii3win) → hhaii3win (Ir-valid)
q4
q5
q0
pi
ck
R
q2
kR
c
pi
pickL
pickL
look
q1
a
look
q3
win
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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3 Comparing Standard Semantics
3.2 Tree Unfoldings: Perfect vs. Imperfect Recall
3.2 Tree Unfoldings: Perfect vs.
Imperfect Recall
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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3 Comparing Standard Semantics
3.2 Tree Unfoldings: Perfect vs. Imperfect Recall
Perfect Information
Now we compare memoryless and perfect recall strategies.
Is there a class of models in which both types are equivalent?
Yes, for tree-like model ; tree unfolding.
Tree unravelling
q1
(α, α)
(β, α)
q1
q1
(α, α)
q2
q2
(α, β)
q1
Nils Bulling · Agents with Perfect and Truly Perfect Recall
q2
q1
q2
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3 Comparing Standard Semantics
3.2 Tree Unfoldings: Perfect vs. Imperfect Recall
Tree-like models:
M, q |= Ir ϕ iff M, q |= IR ϕ
Proposition 4
Val (ATL∗Ir ) ( Val (ATL∗IR )
+
(Even: Val (ATL+
Ir ) ( Val (ATLIR ))
If ϕ is not ATL∗IR -valid consider the tree unfolding of a witnessing
¬ϕ-model...
This technique is well known from modal/temporal logic.
More complicated for strategic ability and imperfect information.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
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3 Comparing Standard Semantics
3.2 Tree Unfoldings: Perfect vs. Imperfect Recall
Imperfect Information Naive Tree Unfoldings
q4
q0
pi
ck
R
kR
c
pi
q2
pickL
pickL
look
q5
q1
a
M, q0 6|=is r hhaii hhhaii hhhaii hwin
look
but it is true in this naive tree
unfolding
q3
win
0
To (M1 , q0 )
1 To (M1 , q1 )
⇠a
02 03 04
..
.
15 12 13
..
. 040
.
151 ..
0402 0403 0404
..
.
..
.
..
.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
..
.
1512 1513 1515
..
.
..
.
..
.
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3 Comparing Standard Semantics
3.2 Tree Unfoldings: Perfect vs. Imperfect Recall
Pando Unfolding
0
To (M1 , q0 )
1 To (M1 , q1 )
⇠a
02 03 04
..
.
..
. 040
.
151 ..
0402 0403 0404
..
.
To (M1 , q0 )
..
.
1512 1513 1515
..
.
..
.
..
.
..
.
040â1
To (M1 , q1 )
040â02 040â03 040â04
..
.
..
.
040â040
..
.
..
.
151â0
..
.
..
.
..
.
..
.
151â15 151â12 151â13
151â040
..
.
..
.
151â1
To (M1 , q1 )
...
To (M1 , q1 )
151â1512 151â1513 151â1515
151â151
151â0402 151â0403 151â0404
..
.
..
.
040â1512 040â1513 040â1515
151â02 151â03 151â04
..
.
040â15 040â12 040â13
040â151
040â0402 040â0403 040â0404
To (M1 , q0 )
..
.
040â0
..
.
⇠a
15 12 13
..
.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
..
.
..
.
..
.
..
.
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3 Comparing Standard Semantics
3.2 Tree Unfoldings: Perfect vs. Imperfect Recall
The pando unfolding (=subjective information tree unfolding) is
truth preserving.
Val (ATLis r ) ( Val (ATLis R )
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4 Perfect Recall and No Forgetting
4. Perfect Recall and No Forgetting
4
Perfect Recall and No Forgetting
ATL with Truly Prefect Recall
Expressivity and Validities
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4 Perfect Recall and No Forgetting
This is joint work with
Wojtek Jamroga (University of Luxembourg)
Matei Popovici (Clausthal University of Technology)
and work in progress.
We also acknowledge Thomas Ågotnes and Valentin Goranko for
discussions and comments.
Bulling, N., Jamroga, W., and Popovici, M. (2014).
Agents with truly perfect recall in alternating-time temporal logic
(extended abstract, to appear).
In Proceedings of the 13th International Conference on
Autonomous Agents and Multi-Agent Systems (AAMAS 2014),
Paris, France. ACM Press.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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4 Perfect Recall and No Forgetting
4.1 ATL with Truly Prefect Recall
4.1 ATL with Truly Prefect Recall
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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4 Perfect Recall and No Forgetting
4.1 ATL with Truly Prefect Recall
Example 5 (Shell Game)
q0
(putR ,nop)
(close,nop)
q2
(nop,pickL )
q3
q10
(close,nop)
(putL ,nop)
q20
g
(no
p,p
ick
R)
win
(nop,pickL )
q1
)
kR
ic
p,p
(no
q30
M, q0 |=i hhgii3win
g can win against any choice of s
M, q2 6|=i hhgii3win
g cannot ensure winning from q2
M, q0 |=i hhsii3¬hhgii3win s deprives g of the ability of winning
g forgets past events.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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4 Perfect Recall and No Forgetting
4.1 ATL with Truly Prefect Recall
No Forgetting Semantics
Outcome takes the history into account:
outM (h, sA ) = {h ◦ λ | all paths that extend h if A follow sA }.
Plays takes all paths starting from indistinguishable histories into
account:
S
0
for x = i
x
h≈A h0 outM (h , sA )
playsM (h, sA ) =
outM (h, sA )
for x = I
Nils Bulling · Agents with Perfect and Truly Perfect Recall
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4 Perfect Recall and No Forgetting
4.1 ATL with Truly Prefect Recall
ATL with Truly Perfect Recall
ATL (standard) perfect recall semantics with new notation:
M, λ |=x hhAiiϕ iff there is a collective x-strategy sA such that, for
each λ0 ∈ playsx (λ[0], sA ), M, λ0 |=x ϕ
No-forgetting semantics:
M, λ, k |=nf
x hhAiiϕ iff there exists an x-strategy sA such that, for
all paths λ0 ∈ playsx (λ[0, k], sA ), M, λ0 , k |=nf
x ϕ;
M, λ, k |=nf
x
hϕ iff M, λ, k + 1 |=nf ϕ
x
λ[0, k − 1]: history
λ[k]: current state
Logics: ATL?nf,I , ATL?nf,i ,ATLnf,I , ATLnf,i
λ[k + 1, ∞]: future
Nils Bulling · Agents with Perfect and Truly Perfect Recall
Dependence Logic@KNAW@NL, March 3, 2014
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4 Perfect Recall and No Forgetting
4.1 ATL with Truly Prefect Recall
Example 6 (Shell Game)
q0
(putR ,nop)
(close,nop)
q2
(nop,pickL )
q3
q10
(close,nop)
(putL ,nop)
q20
g
(no
p,p
ick
R)
win
(nop,pickL )
q1
)
kR
ic
p,p
(no
q30
M, q0 |=i hhsii3¬hhgii3win
M1 , q0 |=nf
i ¬hhsii3¬hhgii3win
Nils Bulling · Agents with Perfect and Truly Perfect Recall
problem solved!
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4 Perfect Recall and No Forgetting
4.2 Expressivity and Validities
4.2 Expressivity and Validities
Nils Bulling · Agents with Perfect and Truly Perfect Recall
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4 Perfect Recall and No Forgetting
4.2 Expressivity and Validities
Perfect Information
M, λ, 0|=nf
I ϕ if, and only if, M, λ |=I ϕ
Expressivity: There is an “equivalent formula” (over all models).
Distinguishing power: Power to distinguish specific pointed models.
Theorem 7 (Expressivity and distinguishing power)
ATL?I and ATL?nf,I are
equally expressive
and have the same distinguishing power.
Theorem 8 (Validities)
ATL?I = ATL?nf,I .
Intuitive: agents cannot forget in the perfect information setting!
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4 Perfect Recall and No Forgetting
4.2 Expressivity and Validities
Imperfect Information (1)
)
(,α)
,β
(
a2
b2
b1
2
(
,β
(,α)
(,α)
)
(α,)
(α,)
a1
b1
(
,β
b0
(α,)
(α,)
a1
a2
a0
b0
)
,β
(
)
(,α)
a0
b2
win
win
M3 , a0 |=i hh1ii hhh2ii hwin
M03 , a0 6|=i hh1ii hhh2ii hwin
Both models cannot be distinguished in ATL?nf,i in a0 .
ATL?nf,i is no more distinguishable as ATL?i
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4 Perfect Recall and No Forgetting
4.2 Expressivity and Validities
Imperfect Information (2)
We also have (model is more complex):
ATL?i is no more distinguishable as ATL?nf,i
Theorem 9
The logics ATL?i and ATL?nf,i have incomparable distinguishing and
expressive powers.
It is different when it comes to general properties of games:
Theorem 10
ATL?i ( ATL?nf,i
ATL?nf,i characterizes games were agents don’t forget the past!
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4 Perfect Recall and No Forgetting
4.2 Expressivity and Validities
Summary
perfect inf.
and recall
imperfect inf.
no forgetting
imperfect inf.
forgetting
Nils Bulling · Agents with Perfect and Truly Perfect Recall
ATL⇤is R
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5 Conclusions
5. Conclusions
5
Conclusions
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5 Conclusions
Overview of the Results
“All” semantic variants are different on the level of general
properties; before our study, it was by no means obvious.
Strong pattern of subsumption (memory, information, forgetting)
In particular: non-validities are interesting.
No forgetting semantics
Future Work:
Strategy commitment
Complexity: model and satisfiability checking wrt. no forgetting
semantics
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5 Conclusions
Thank you for your attention!
Questions?
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5 Conclusions
Some References
Alur, R., Henzinger, T. A., and Kupferman, O. (2002).
Alternating-time Temporal Logic.
Journal of the ACM, 49:672–713.
Bulling, N. and Jamroga, W. (2014).
Comparing variants of strategic ability: how uncertainty and memory influence general properties of
games.
Autonomous Agents and Multi-Agent Systems, 28(3):474–518.
Bulling, N., Jamroga, W., and Popovici, M. (2014).
Agents with truly perfect recall in alternating-time temporal logic (extended abstract, to appear).
In Proceedings of the 13th International Conference on Autonomous Agents and Multi-Agent Systems
(AAMAS 2014), Paris, France. ACM Press.
Schobbens, P. Y. (2004).
Alternating-time logic with imperfect recall.
Electronic Notes in Theoretical Computer Science, 85(2):82–93.
Nils Bulling · Agents with Perfect and Truly Perfect Recall
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