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Decidability results for ATL* with imperfect information and perfect recall

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Abstract

Alternating-time Temporal Logic (ATL*) is a central logic for multiagent systems. Its extension to the imperfect information setting (ATL*i ) is well known to have an undecidable model-checking problem when agents have perfect recall. Studies have thus mostly focused either on agents without memory, or on alternative semantics to retrieve decidability. In this work we establish new decidability results for agents with perfect recall: We first prove a meta-theorem that allows the transfer of decidability results for classes of multiplayer games with imperfect information, such as games with hierarchical observation, to the model-checking problem for ATL*i . We then establish that model checking ATL* with strategy context and imperfect information is decidable when restricted to hierarchical instances.
Decidability results for ATL with imperfect
information and perfect recall
Raphaël Berthon1, Bastien Maubert2, and Aniello Murano3
1 École Normale Supérieure de Rennes, France
raphael.berthon@ens-rennes.fr
2 Università degli Studi di Napoli Federico II, Italy
bastien.maubert@gmail.com
3 Università degli Studi di Napoli Federico II, Italy
murano@na.infn.it
Abstract
Alternating-time Temporal Logic (ATL) is a central logic for multiagent systems. It has been
extended in various ways, notably with imperfect information (ATL
i). Since the model-checking
problem against ATL
ifor agents with perfect recall is undecidable, studies have mostly focused
either on agents without memory, or on alternative semantics to retrieve decidability. In this
work, we establish new, strong decidability results for agents with perfect recall. We first prove a
meta-theorem that allows the transfer of decidability results for classes of multiplayer games with
imperfect information, such as games with hierarchical observation, to the model-checking prob-
lem for ATL
i. We also establish that model checking ATLwith strategy context and imperfect
information for hierarchical instances is decidable.
1 Introduction
In formal system verification, model checking is a well-established method to automatically
check the correctness of a system [
8
,
31
,
9
]. It consists in modelling the system as a
mathematical structure, expressing its desired behaviour as a formula from some suitable
logic, and checking whether the model satisfies the formula. In the nineties, interest has
arisen in the verification of multiagent systems (MAS), in which various entities (the agents)
interact and can form coalitions to attain certain objectives. This led to the development of
logics that allow reasoning about strategic abilities in MAS [2, 27, 19, 35, 1, 7].
Alternating-time Temporal Logic (
ATL
), introduced by Alur, Henzinger, and Kupfer-
man [
2
], plays a central role in this line of work. This logic, interpreted on concurrent game
structures, extends
CTL
with strategic modalities. These modalities allow one to reason
about the existence of strategies for coalitions of agents to force the system’s behaviour to
satisfy certain temporal properties.
ATL
has been extended in many ways, and among these
extensions an important one is
ATL
with strategy context [
6
,
25
]. In
ATL
, strategies of all
agents are forgotten at each new strategic modality. In
ATL
with strategy context (
ATL
sc
)
instead they are stored in a strategy context, and are forgotten only when replaced by a new
strategy or when the formula explicitly unbinds the agent from her strategy. Thanks to this
additional expressive power,
ATL
sc
can express important game theoretic concepts such as
the existence of Nash Equilibria [25].
This project has received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No 709188.
arXiv:1805.12582v2 [cs.LO] 3 Sep 2018
2 Decidability results for ATL with imperfect information and perfect recall
In many real-life scenarios, such as poker, agents do not always know precisely what is
the current state of the system. Instead, they have a partial view, or observation, of the state.
This fundamental feature of MAS is called imperfect information, and it is known to quickly
bring about undecidability when involved in strategic problems, especially when agents have
perfect recall of the past, which is a usual and important assumption in games with imperfect
information and epistemic temporal logics [
13
]. For instance solving multiplayer games
with imperfect information and perfect recall, i.e., deciding the existence of a distributed
winning strategy in such games, is already undecidable for reachability objective, as proven by
Peterson, Reif and Azhar [
28
]. Since such games are easily captured by
ATL
with imperfect
information (ATL
i), model checking ATL
iwith perfect recall is also undecidable [2].
However it is known that restricting attention to cases where some sort of hierarchy exists
on the different agents’ information yields decidability for several problems related to the
existence of strategies. Synthesis of distributed systems, which implicitly uses perfect recall
and is undecidable in general [
30
], is decidable for hierarchical architectures [
23
]. Actually, for
branching-time specifications, distributed synthesis is decidable exactly on architectures free
from information forks, for which the problem can be reduced to the hierarchical case [
14
]. For
richer specifications from alternating-time logics, being free of information forks is no longer
sufficient, but distributed synthesis is decidable precisely on hierarchical architectures [
32
].
Similarly, solving multiplayer games with imperfect information and perfect recall, i.e.,
checking for the existence of winning distributed strategies, is decidable for
ω
-regular winning
conditions when there is a hierarchy among players, each one observing more than those
below [
29
,
23
]. Recently, it has been proven that this assumption can be relaxed while
maintaining decidability: the problem remains decidable if the hierarchy can change along a
play, or even if transient phases without such a hierarchy are allowed [5].
Our contribution.
In this work we establish several decidability results for model
checking
ATL
i
with perfect recall, with and without strategy context, all related to notions
of hierarchy. Our first result is a theorem that allows the transfer of decidability results for
classes of multiplayer games with imperfect information, such as those mentioned above,
to the model-checking problem for
ATL
i
. This theorem essentially states that if solving
multiplayer games with imperfect information, perfect recall and omega-regular objectives
is decidable on some class of concurrent game structures, then model checking
ATL
i
with
perfect recall is also decidable on this class of models (a simple bottom-up algorithm that
evaluates innermost strategic modalities in every state of the model suffices). As a direct
consequence we easily obtain new decidability results for the model checking of
ATL
i
on
several classes of concurrent game structures.
Our second contribution considers
ATL
with imperfect information and strategy context
(
ATL
sc,i
). Because there are in general infinitely many possible strategy contexts, the
bottom-up approach used for
ATL
i
cannot be used here. Instead we build upon the proof
presented in [
25
] to establish the decidability of model checking
ATL
sc
, by reduction to
the model-checking problem for Quantified
CTL
(
QCTL
). The latter extends
CTL
with
second-order quantification on atomic propositions, and it has been studied in a number
of works [
34
,
21
,
22
,
15
,
24
].
QCTL
i
, an imperfect-information extension of
QCTL
, has
recently been introduced, and its model-checking problem was proven decidable for the class
of hierarchical formulas [
4
]. In this paper we define a notion of hierarchical instances for the
ATL
sc,i
model-checking problem: informally, an
ATL
sc,i
formula
ϕ
together with a concurrent
game structure
G
is a hierarchical instance if outermost strategic modalities in
ϕ
concern
agents who observe less in
G
. We adapt the proof from [
25
] and reduce the model-checking
problem for
ATL
sc,i
on hierarchical instances to the model-checking problem for hierarchical
R. Berthon, B. Maubert and A. Murano 3
QCTL
i
formulas. We obtain that model checking hierarchical instances of
ATL
sc,i
with
perfect recall is decidable.
Related work.
The model-checking problem for
ATL
i
is known to be decidable when
agents have no memory [
33
], and the case of agents with bounded memory reduces to that of
no memory. Another way to retrieve decidability is to assume that all agents in a coalition
have the same information, either because their observations of the system are the same, or
because they can communicate and share their observations [
16
,
11
,
17
,
20
]. This idea was
also used recently to establish a decidability result for
ATL
sc,i
[
26
] when all agents have the
same observation of the game.
The results we establish here thus strictly extend previously known results on the
decidability of model checking
ATL
i
and
ATL
sc,i
with perfect recall and standard semantics,
and they hold for vast, natural classes of instances, that all rely on notions of hierarchy,
which seems to be inherent to all decidable cases of strategic problems for multiple entities
with imperfect information and perfect recall.
Outline.
After setting some basic definitions in Section 2, we present our transfer theorem
and its various corollaries concerning the model checking problem for
ATL
i
in Section 3. In
Section 4 we prove that when restricted to hierarchical instances, model checking
ATL
sc,i
is
decidable, and we conclude in Section 5.
2 Preleminaries
Let Σbe an alphabet. A finite (resp. infinite)word over Σis an element of Σ
(resp. Σ
ω
).
The empty word is noted
, and Σ
+
= Σ
\ {}
. The length of a word is
|w|
:= 0 if
w
is the
empty word
, if
w
=
w0w1. . . wn
is a finite nonempty word then
|w|
:=
n
+ 1, and for an
infinite word
w
we let
|w|
:=
ω
. Given a word
w
and 0
i, j ≤ |w| −
1, we let
wi
be the
letter at position
i
in
w
and
w
[
i, j
]be the subword of
w
that starts at position
i
and ends at
position
j
. For
nN
we let [
n
] :=
{
1
, . . . , n}
. Finally, let us fix a countably infinite set of
atomic propositions AP and let AP ⊂ AP be some finite subset of atomic propositions.
2.1 Kripke structures
AKripke structure over
AP
is a tuple
S
= (
S, R, `
)where
S
is a set of states,
RS×S
is a
left-total1transition relation and `:S2AP is a labelling function.
Apointed Kripke structure is a pair (
S, s
)where
s∈ S
. A path in a structure
S
= (
S, R, `
)
is an infinite word
λ
over
S
such that for all
iN
,(
λi, λi+1
)
R
. For
sS
,
Paths
(
s
)is
the set of all paths that start in s.
2.2 Infinite trees
Let Xbe a finite set. An X-tree τis a nonempty set of words τX+such that
there exists rX, called the root of τ, such that each uτstarts with r;
if u·xτand u6=, then uτ, and
if uτthen there exists xXsuch that u·xτ.
The elements of a tree
τ
are called nodes. If
u·xτ
, we say that
u·x
is a child of
u
.
Similarly to Kripke structures, a path is an infinite sequence of nodes
λ
=
u0u1. . .
such that
for all
i
,
ui+1
is a child of
ui
, and
P aths
(
u
)is the set of paths that start in node
u
. An
1i.e., for all sS, there exists s0such that (s, s0)R.
4 Decidability results for ATL with imperfect information and perfect recall
AP
-labelled
X
-tree, or (
AP, X
)-tree for short, is a pair
t
= (
τ, `
), where
τ
is an
X
-tree called
the domain of tand `:τ2AP is a labelling.
IDefinition 1
(Tree unfoldings)
.
Let
S
= (
S, R, `
)be a Kripke structure over
AP
, and let
sS
. The tree-unfolding of
S
from
s
is the (
AP, S
)-tree
tS
(
s
) = (
τ, `0
), where
τ
is the set
of all finite paths that start in s, and for every uτ,`0(u) = `(last(u)).
3ATLwith imperfect information
In this section we recall the syntax and semantics of
ATL
with imperfect information and
synchronous perfect-recall semantics, or
ATL
i
for short, and establish a meta-theorem on
the decidability of its model-checking problem.
3.1 Definitions
We first introduce the models of the logics we study. For the rest of the paper, let us fix a
non-empty finite set of agents Ag and a non-empty finite set of moves M.
IDefinition 2.
Aconcurrent game structure with imperfect information (or
CGSi
for short)
over
AP
is a tuple
G
= (
V, E , `, {∼a}aAg
)where
V
is a non-empty finite set of positions,
E
:
V×MAg V
is a transition function,
`
:
V
2
AP
is a labelling function and for each
agent aAg, aV×Vis an equivalence relation.
In a position
vV
, each agent
a
chooses a move
maM
, and the game proceeds to
position
E
(
v, m
), where
mMAg
stands for the joint move (
ma
)
aAg
(note that we assume
E
(
v, m
)to be defined for all
v
and
m2
). For each position
vV
,
`
(
v
)is the finite set of
atomic propositions that hold in
v
, and for
aAg
, equivalence relation
a
represents the
observation of agent
a
: for two positions
v, v0V
,
vav0
means that agent
a
cannot tell
the difference between
v
and
v0
. We may write
v∈ G
for
vV
. A pointed
CGSi
(
G, v
)is a
CGSiGtogether with a position v∈ G.
In Section 3.2 we also use nondeterministic
CGSi
, which are as in Definition 2 except
that they have a transition relation
EV×MAg ×V
instead of a transition function. In a
position
v
, after every agent has chosen a move, forming a joint move
mMAg
, a special
agent called Nature (not in
Ag
) chooses a next position
v0
such that (
v, m, v0
)
E
(see [
5
]
for detail). In the following, unless explicitly specified,
CGSi
always refers to deterministic
CGSi
. The following definitions also concern deterministic
CGSi
, but they can be adapted
to nondeterministic ones in an obvious way.
Afinite (resp. infinite)play is a finite (resp. infinite) word
ρ
=
v0. . . vn
(resp.
π
=
v0v1. . .
)
such that for all
i
with 0
i < |ρ| −
1(resp.
i
0), there exists a joint move
m
such that
E
(
vi,m
) =
vi+1
. A finite (resp. infinite) play
ρ
(resp.
π
)starts in a position
v
if
ρ0
=
v
(resp.
π0
=
v
). We let
Plays
(
G, v
)be the set of plays, either finite or infinite, that start in
v
.
In this work we consider agents with synchronous perfect recall, meaning that the
observational equivalence relation for each agent
a
is extended to finite plays the following
way:
ρaρ0
if
|ρ|
=
|ρ|0
and
ρiaρ0
i
for every
i∈ {
0
,...,|ρ| −
1
}
. A strategy for agent a
is a function
σ
:
V+M
such that
σ
(
ρ
) =
σ
(
ρ0
)whenever
ρaρ0
. The latter constraint
captures the essence of imperfect information, which is that agents can base their strategic
2
This assumption, as well as the choice of a unique set of moves for all agents, is made to ease presentation.
All the results presented here also hold when the set of available moves depends on the agent and the
position.
R. Berthon, B. Maubert and A. Murano 5
choices only on the information available to them, and removing this constraint yields the
semantics of classic ATL with perfect information.
Astrategy profile for a coalition
AAg
is a mapping
σA
that assigns a strategy to each
agent
aA
; for
aA
, we may write
σa
instead of
σA
(
a
). An infinite play
π
follows a
strategy profile
σA
for a coalition
A
if for all
i
0, there exists a joint move
m
such that
E
(
πi,m
) =
πi+1
and for each
aA
,
ma
=
σa
(
π
[0
, i
]). For a strategy profile
σA
and a
position
vV
, we define the outcome
Out
(
v, σA
)of
σA
in
v
as the set of infinite plays that
start in vand follow σA.
The syntax of
ATL
i
is the same as that of
ATL
, and is given by the following grammar:
ϕ::= p| ¬ϕ|ϕϕ| hAiϕ|Xϕ|ϕUϕ,
where p∈ AP and AAg.
X
and
U
are the classic next and until operators, respectively, while the strategic operator
hAiquantifies on strategy profiles for coalition A.
The semantics of
ATL
i
is defined with regards to a
CGSiG
= (
V, E , `, {∼a}aAg
), an
infinite play πand a position i0along this play, by induction on formulas:
G, π, i |=pif p`(πi)
G, π, i |=¬ϕif G, π, i 6|=ϕ
G, π, i |=ϕϕ0if G, π, i |=ϕor G, π, i |=ϕ0
G, π, i |=hAiϕif there exists a strategy profile σAs.t.
for all π0Out(πi, σA),G, π0,0|=ϕ
G, π, i |=Xϕif G, π, i + 1 |=ϕ
G, π, i |=ϕUϕ0if there exists jis.t. G, π, j |=ϕ0and,
for all ks.t. ik < j,G, π, k |=ϕ.
An
ATL
i
formula
ϕ
is closed if every temporal operator (
X
or
U
) in
ϕ
is in the scope of
a strategic operator
hAi
. Since the semantics of a closed formula
ϕ
does not depend on the
future, we may write G, v |=ϕif G, π, 0|=ϕfor any infinite play πthat starts in v.
The model-checking problem for
ATL
i
consists in deciding, given a closed
ATL
i
formula
ϕand a finite pointed CGSi(G, v), whether G, v |=ϕ.
3.2 Model checking ATL
i
It is well known that the model-checking problem for
ATL
i
is undecidable for agents with
perfect recall [
2
], as it can easily express the existence of distributed winning strategies for
multiplayer reachability games with imperfect information and perfect recall, which was
proved undecidable by Peterson, Reif and Azhar [
28
]. A direct proof of this undecidability
result for
ATL
i
is also presented in [
12
]. However, there are classes of multiplayer games with
imperfect information that are decidable. For many years, the only known decidable case
was that of hierarchical games, in which there is a total preorder among players, each player
observing at least as much as those below her in this preorder [
29
,
23
]. Recently, this result
has been extended by relaxing the assumption of hierarchical observation. In particular,
it has been shown that the problem remains decidable if the hierarchy can change along a
play, or if transient phases without such a hierarchy are allowed [
5
]. We establish that these
results transfer to the model-checking problem for ATL
i.
We remind that a concurrent game with imperfect information is a pair ((
G, v
)
, W
)where
(
G, v
)is a pointed nondeterministic
CGSi
and
W
is a property of infinite plays called the
winning condition. The strategy problem is, given such a game, to decide whether there exists
6 Decidability results for ATL with imperfect information and perfect recall
a strategy profile for the grand coalition
Ag
to enforce the winning condition against Nature
(for more details see, e.g., [5]).
Before stating our transfer theorem we need to introduce a couple of additional notions.
First we introduce a notion of abstraction over a group of agents. Informally, abstracting a
CGSiG
over an agent consists in erasing her from the group of agents and letting Nature
play for her in G.
IDefinition 3
(Abstraction)
.
Let
AAg
be a group of agents and let
G
= (
V, E , `, {∼a}aAg
)
be a
CGSi
. The abstraction of
G
from
A
is the nondeterministic
CGSi
over set of agents
Ag \Adefined as G ↑A:= (V, E 0, `, {∼a}aAg\A), where for every vVand mMAg\A,
(v, m, v0)E0if m0MAs.t. E(v , (m,m0)) = v0.
Thanks to this notion we can define the following problem:
IDefinition 4
(
A
-strategy problem)
.
The
A
-strategy problem takes as input a pointed
CGSi
(
G, v
), a set
AAg
of agents and a winning condition
W
, and returns the answer to the
strategy problem for the game ((G ↑Ag\A, v), W ).
The
A
-strategy problem for (
G, v
)with winning condition
W
thus consists in deciding whether
there is a strategy profile for agents in Ato enforce Wagainst everybody else.
Finally we introduce the following notion, which simply captures the change of initial
position in a game from a position vto another position v0reachable from v:
IDefinition 5
(Initial shifting)
.
Let
G
be a
CGSi
and let
v, v0∈ G
. The pointed
CGSi
(
G, v0
)
is an initial shifting of (G, v)if v0is reachable from vin G.
We are now ready to state our first result.
ITheorem 6.
If
C
is a class of pointed
CGSi
closed under initial shifting and such that the
A
-strategy problem with
ω
-regular objective is decidable on
C
, then model checking
ATL
i
is
decidable on C.
Proof.
Let
C
be such a class of pointed
CGSi
, and let (
ϕ,
(
G, v
)) be an instance of the
model-checking problem for
ATL
i
on
C
. A bottom-up algorithm consists in evaluating each
innermost subformula of
ϕ
of the form
hAiϕ0
, where
ϕ0
is thus an LTL formula, on each
position
v0
of
G
reachable from
v
. Evaluating
hAiϕ0
on
v0
amounts to solving an instance of
the
A
-strategy problem
3
with
ω
-regular objective (recall that LTL properties are
ω
-regular).
By assumption (
G, v
)
∈ C
, and because
C
is closed by initial shifting and
v0
is reachable
from
v
, we have that (
G, v0
)
∈ C
. Also by assumption, the
A
-strategy problem for
ω
-regular
winning conditions is decidable on
C
. We thus have an algorithm to evaluate each
hAiϕ0
on
each
v0
. One can then mark positions of the game with fresh atomic propositions indicating
where these formulas hold, and repeat the procedure until all strategic operators have been
eliminated. It then remains to evaluate a boolean formula in the initial position v.J
Let us recall for which classes of nondeterministic
CGSi
the strategy problem is known
to be decidable. A (nondeterministic or deterministic)
CGSiG
has hierarchical observation
if there exists a total preorder
4
over
Ag
such that if
a4b
and
vav0
, then
vbv0
.
3
Observe that if
A
=
Ag
then
G ↑Ag\A
=
G
, and Nature thus does not do anything. This is coherent with
the fact that for agents with perfect recall
hAgiϕEϕ
, where
E
is the CTL path quantifier, even for
imperfect information.
R. Berthon, B. Maubert and A. Murano 7
This notion was refined in [
5
] to take into account the agents’ memory, using the notion
of information set: for a finite play
ρPlays
(
G, v
)and an agent
a
, the information set of
agent
a
after
ρ
is
Ia
(
ρ
) :=
{ρ0Plays
(
G, v
)
|ρaρ0}
. A finite play
ρ
yields hierarchical
information if there is a total preorder
4
over
Ag
such that if
a4b
, then
Ia
(
ρ
)
Ib
(
ρ
). If
all finite plays in
Plays
(
G, v
)yield hierarchical information for the same preorder over agents,
(
G, v
)yields static hierarchical information. If this preorder can vary depending on the play,
(
G, v
)yields dynamic hierarchical information. The last generalisation consists in allowing for
transient phases without hierarchy: if every infinite play in
Plays
(
G, v
)has infinitely many
prefixes that yield hierarchical information, (G, v)yields recurring hierarchical information.
IProposition 7.
Hierarchical observation as well as static, dynamic and recurring hierarch-
ical information are preserved by abstraction.
Proof.
Abstraction removes agents without affecting observations of remaining ones. The
result thus follows from the respective definitions of hierarchical observation and of static,
dynamic and recurring hierarchical information. J
IProposition 8.
Hierarchical observation as well as static, dynamic and recurring hierarch-
ical information are preserved by initial shifting.
This is obvious for hierarchical observation. For the other cases we establish Lemma 9
below. It is then easy to check that Proposition 8 holds.
ILemma 9.
If a finite play
v·ρ·v0·ρ0
yields hierarchical information in (
G, v
), so does
v0·ρ0in (G, v0), with the same preorder among agents.
Proof.
Assume that
v·ρ·v0·ρ0
yields hierarchical information in (
G, v
)with preorder
4
over
Ag
. Suppose towards a contradiction that there are agents
a, b Ag
such that
a4b
but
Ia
(
v0·ρ0
)
6⊆ Ib
(
v0·ρ0
). This means that there is
v0·ρ00 Plays
(
G, v0
)such that
v0·ρ0av0·ρ00
but
v0·ρ06∼bv0·ρ00
. By definition of synchronous perfect recall relations we
then have that
v·ρ·v0·ρ0av·ρ·v0·ρ00
and
v·ρ·v0·ρ06∼bv·ρ·v0·ρ00
. This implies that
Ia
(
v·ρ·v0·ρ0
)
6⊆ Ib
(
v·ρ·v0·ρ0
), which contradicts the fact that
a4b
. Therefore for all
agents
a, b
such that
a4b
we have
Ia
(
v0·ρ0
)
Ib
(
v0·ρ0
), and thus
v0·ρ0
yields hierarchical
information with preorder 4.J
Let
Cobs
(resp.
Cstat
,
Cdyn
,
Crec
) be the class of pointed
CGSi
with hierarchical observation
(resp. static, dynamic, recurring hierarchical information). We instantiate Theorem 6 to
obtain three decidability results for ATL
i.
ITheorem 10.
Model checking
ATL
i
is decidable on the class of
CGSi
with hierarchical
observation.
Proof.
By Proposition 8,
Cobs
is closed under initial shifting. It is proven in [
23
] that the
strategy problem is decidable for games with hierarchical observation and
ω
-regular objectives.
Since, by Proposition 7, all pointed nondeterministic
CGSi
obtained by abstracting agents
from
CGSi
in
Cobs
also yield hierarchical observation, we get that the
A
-strategy problem
with
ω
-regular objectives is decidable on
Cobs
. We can therefore apply Theorem 6 on
Cobs
.
J
It is proven in [
5
] that the strategy problem with
ω
-regular objectives is also decidable
for games with static hierarchical information and for games with dynamic hierarchical
information. Since Proposition 7 and Proposition 8 also hold for
Cstat
and
Cdyn
, with the
same argument as in the proof of Theorem 10, we obtain the following results as consequences
of Theorem 6:
8 Decidability results for ATL with imperfect information and perfect recall
ITheorem 11.
Model checking
ATL
i
is decidable on the class of
CGSi
with static hierarchical
information.
ITheorem 12.
Model checking
ATL
i
is decidable on the class of
CGSi
with dynamic
hierarchical information.
Note that in fact, since
Cobs ⊂ Cstat ⊂ Cdyn
, Theorem 10 and Theorem 11 are also obtained
as corollaries of Theorem 12, but we wanted to illustrate how Theorem 6 can be applied to
obtain decidability results for different classes of CGSi.
I
Remark. The last result in [
5
] establishes that the strategy problem is decidable for games
with recurring hierarchical information, but only for observable
ω
-regular winning conditions,
i.e., when all agents can tell whether a play is winning or not. Now considering
ATL
i
on
Cdyn
we could require atomic propositions to be observable for all agents; in that case we could
evaluate the inner-most strategy quantifiers using the above-mentioned result. But then the
fresh atomic propositions that mark positions where these subformulas hold (see the proof of
Theorem 6) would not, in general, be observable by all agents. So on
Crec
we could obtain a
decision procedure for the fragment of
ATL
i
without nested non-trivial strategy quantifiers,
where non-trivial means for coalitions other than the empty coalition or the one made of all
agents (which, we recall, are simply the CTL path quantifiers). We do not state it explicitly
because it does not seem of much interest.
Concerning complexity, the strategy problem for games with imperfect information and
hierarchical observation is already nonelementary [30, 28], hence the following result:
ICorollary 13.
Model checking
ATL
i
is nonelementary on games with hierarchical observa-
tion, hence also for games with static or dynamic hierarchical information.
We now turn to ATL with imperfect information and strategy context, and study its
model-checking problem.
4ATLiwith strategy context
While in ATL strategies for all agents are forgotten each time a new strategy quantifier is
met, in ATL with strategy context (
ATLsc
) [
6
,
10
,
25
] agents keep using the same strategy as
long as the formula does not say otherwise. In this section we consider
ATLsc
with imperfect
information (
ATLsc,i
). As far as we know, the only existing work on this logic is [
26
], which
proved its model-checking problem to be decidable in the case where all agents have the
same observation of the game. We extend significantly this result by establishing that the
model-checking problem is decidable as long as strategy quantification is hierarchical, in the
sense that if there is a strategy quantification for agent
a
nested in a strategy quantification
for agent
b
, then
b
should observe no more than
a
. In other terms, innermost strategic
quantifications should concern agents who observe more.
4.1 Syntax and semantics
The models are still
CGSi
. To remember which agents are currently bound to a strategy, and
what these strategies are, the semantics uses strategy contexts. Formally, a strategy context for
a set of agents
BAg
is a strategy profile
σB
. We define the composition of strategy contexts
as follows. If
σB
is a strategy context for
B
and
σA
is a new strategy profile for coalition
A
,
we let
σAσB
be the strategy context for
AB
defined as
σAB
:
a7→ (σA(a)if aA,
σB(a)otherwise
.
R. Berthon, B. Maubert and A. Murano 9
So if
a
is assigned a strategy by
σA
, her strategy in
σAσB
is
σA
(
a
). If she is not assigned
a strategy by σAher strategy remains the one given by σB, if any.
Also, given a strategy context
σB
and a set of agents
AAg
, we let (
σB
)
\A
be the
strategy context obtained by restricting σBto the domain B\A.
Finally, because agents who do not change their strategy keep playing the one they were
assigned, if any, we cannot forget the past at each strategy quantifier, as in the semantics of
ATL
i
(see Section 3.1). We thus define the outcome of a strategy profile
σA
after a finite
play
ρ
, written
Out
(
ρ, σA
), as the set of infinite plays
π
that start with
ρ
and then follow
σA
:
πOut
(
ρ, σA
)if
π
=
ρ·π0
for some
π0
, and for all
i≥ |ρ| −
1, there exists a joint move
mMAg such that E(πi,m) = πi+1 and for each aA,ma=σa(π[0, i]).
To differentiate from
ATL
, in
ATL
sc
the strategy quantifier for a coalition
A
is written
h·A·i
instead of
hAi
.
ATL
sc
also has an additional operator, (
|A|
), that releases agents in
A
from their current strategy, if they have one. The syntax of
ATL
sc,i
is the same as that of
ATL
sc and is thus given by the following grammar:
ϕ::= p| ¬ϕ|ϕϕ| h·A·iϕ|(|A|)ϕ|Xϕ|ϕUϕ,
where p∈ AP and AAg.
I
Remark. In [
25
] the syntax of
ATL
sc
contains in addition operators
h·A·i
and (
|A|
)for
complement coalitions. While they add expressivity when the set of agents is not fixed, and
are thus of interest when considering expressivity or satisfiability, they are redundant if we
consider model checking, which is our case in this work. To simplify presentation we thus
choose not to consider them here.
The semantics of
ATL
sc,i
is defined with regards to a
CGSiG
= (
V, E , `, {∼a}aAg
), an
infinite play
π
, a position
iN
along this play, and a strategy context
σB
. The semantics is
defined by induction on formulas:
G, π, i |=σBpif p`(πi)
G, π, i |=σB¬ϕif G, π , i 6|=σBϕ
G, π, i |=σBϕϕ0if G, π , i |=σBϕor G, π, i |=σBϕ0
G, π, i |=σBh·A·iϕif there exists a strategy profile σAs.t.
for all π0Out(π[0, i], σAσB),G, π0, i |=σAσBϕ
G, π, i |=σB(|A|)ϕif G, π , i |=(σB)\Aϕ
G, π, i |=σBXϕif G, π , i + 1 |=σBϕ
G, π, i |=σBϕUϕ0if there exists jis.t. G, π , j |=σBϕ0
and, for all ksuch that ik < j,G, π , k |=σBϕ.
The notion of closed formula is as defined in Section 3.1 and once more, the semantics
of a closed formula
ϕ
being independent from the future, we may write
G, v |
=
σBϕ
instead
of
G, π,
0
|
=
σBϕ
for any infinite play
π
that starts in position
v
. We also write
G, v |
=
ϕ
if
G, v |=σϕ, that is if ϕholds in vwith the empty strategy context.
The model-checking problem for
ATL
sc,i
consists in deciding, given a closed
ATL
sc,i
formula
ϕand a finite pointed CGSi(G, v), whether G, v |=ϕ.
We now present
QCTL
with imperfect information, or
QCTL
i
for short, before proving our
main result on the model-checking problem for
ATL
sc,i
by reducing it to the model-checking
problem for a decidable fragment of QCTL
i.
4.2 QCTLwith imperfect information
Quantified
CTL
, or
QCTL
for short, is an extension of
CTL
with second-order quantifiers
on atomic propositions that has been well studied [
34
,
21
,
22
,
24
]. It has recently been
10 Decidability results for ATL with imperfect information and perfect recall
further extended to take into account imperfect information, resulting in the logic called
QCTL
with imperfect information, or
QCTL
i
[
3
,
4
]. We briefly present this logic, as well as
a decidability result on its model-checking problem proved in [
3
,
4
] and that we rely on to
establish our result on the model checking of ATL
sc,i.
Imperfect information is incorporated into
QCTL
by considering Kripke models with
internal structure in the form of local states, like in distributed systems (see for instance [
18
]),
and then parameterising quantifiers on atomic propositions with observations that define
what portions of the states a quantifier can “observe”. The semantics is then adapted to
capture the idea of quantification on atomic propositions being made with partial observation.
Let us fix a collection
{Li}i[n]
of
n
disjoint finite sets of local states. We also let
Xn=L1×. . . ×Ln.
IDefinition 14.
Acompound Kripke structure (CKS) over
AP
is a Kripke structure
S
=
(S, R, `)such that SXn.
The syntax of
QCTL
i
is that of
QCTL
, except that quantifiers over atomic propositions
are parameterised by a set of indices that defines what local states the quantifier can “observe”.
It is thus defined by the following grammar:
ϕ:= p| ¬ϕ|ϕϕ|Eϕ| ∃op. ϕ |Xϕ|ϕUϕ
where
p∈ AP
and
oN
is a finite set of indices. We use standard abbreviations:
>
:=
p¬p
,
:= ¬>,Fϕ:= >Uϕ,Gϕ:= ¬F¬ϕand Aϕ:= ¬E¬ϕ.
A finite set
oN
is called an observation, and two states
s
= (
l1, . . . , ln
)and
s0
=
(l0
1, . . . , l0
n)are o-indistinguishable, written sos0, if for all i[n]o, it holds that li=l0
i.
The intuition is that a quantifier with observation
o
must choose the valuation of atomic
propositions uniformly with respect to
o
. Note that in [
3
], two semantics are considered for
QCTL
i
, just like in [
24
] for
QCTL
: the structure semantics and the tree semantics. In the
former, formulas are evaluated directly on the structure, while in the latter the structure is
first unfolded into an infinite tree. Here we only present the tree semantics, as it is this one
that allows us to capture agents with perfect recall. But we first need a few more definitions.
For
p∈ AP
, two labelled trees
t
= (
τ, `
)and
t0
= (
τ0, `0
)are equivalent modulo
p
, written
tpt0
, if
τ
=
τ0
and for each node
uτ
,
`
(
u
)
\ {p}
=
`0
(
u
)
\ {p}
. So
tpt0
if they are the
same trees, except for the labelling of proposition p.
This notion of equivalence modulo
p
is the one used to define quantification on atomic
propositions in
QCTL
: intuitively, an existential quantification over
p
chooses a new labelling
for valuation
p
, all else remaining the same, and the evaluation of the formula continues from
the current node with the new labelling. For imperfect information we need to express the
fact that this new labelling for a proposition is done uniformly with regards to the quantifier’s
observation.
First, we define the notion of indistinguishability between two nodes in the unfolding of a
CKS. Let
o
be an observation, let
τ
be an
Xn
-tree (which may be obtained by unfolding
some pointed CKS), and let
u
=
s0. . . si
and
u0
=
s0
0. . . s0
j
be two nodes in
τ
. The nodes
u
and
u0
are
o
-indistinguishable, written
uou0
, if
i
=
j
and for all
k∈ {
0
, . . . , i}
, we have
skos0
k
. Observe that this definition corresponds to the notion of synchronous perfect
recall in
CGSi
(see Section 3.1). We now define what it means for the labelling of an atomic
proposition to be uniform with regards to an observation.
IDefinition 15.
Let
t
= (
τ, `
)be a labelled
Xn
-tree, let
p∈ AP
be an atomic proposition
and
oN
an observation. Tree
t
is
o
-uniform in
p
if for every pair of nodes
u, u0τ
such
that uou0, we have p`(u)iff p`(u0).
R. Berthon, B. Maubert and A. Murano 11
The satisfaction relation
|
=
t
(
t
is for tree semantics) is now defined as follows, where
t= (τ, `)is a labelled Xn-tree, λis a path in τand iNa position along that branch:
t, λ, i |=tpif p`(λi)
t, λ, i |=t¬ϕif t, λ, i 6|=tϕ
t, λ, i |=tϕϕ0if t, λ, i |=tϕor t, λ, i |=tϕ0
t, λ, i |=tEϕif there exists λ0P aths(λi)such that t, λ0,0|=tϕ
t, λ, i |=top. ϕ if there exists t0ptsuch that t0is o-uniform in pand t0, λ, i |=tϕ
t, λ, i |=tXϕif t, λ, i + 1 |=tϕ
t, λ, i |=tϕUϕ0if there exists jisuch that t, λ, j |=tϕ0and for ik < j, t, λ, j |=tϕ
Similarly to
ATL
sc,i
, we say that a
QCTL
i
formula is closed if all temporal operators are
in the scope of a path quantifier. The semantics of such formulas depending only on the
current node, for a closed formula
ϕ
we may write
t|
=
tϕ
for
t, r |
=
tϕ
, where
r
is the root of
t
, and given a pointed CKS (
S, s
)and a
QCTL
i
formula
ϕ
, we write
S, s |
=
tϕ
if
tS
(
s
)
|
=
tϕ
.
I
Remark. In [
3
] the syntax is presented with path formulas distinguished from state formulas,
and the semantics is defined accordingly. To make the presentation more uniform with that
of ATLsc,i we chose here a different, but equivalent, presentation.
IRemark. Note that when nis fixed, the propositional quantifier with perfect information
from
QCTL
is equivalent to the
QCTL
i
quantifier that observes all the components, i.e.,
the quantifier parameterised with observation [n].
The model-checking problem for QCTL
iis the following: given a closed QCTL
iformula
ϕand a finite pointed CKS (S, s), decide whether S, s |=tϕ.
We now define the class of
QCTL
i
formulas for which the model-checking problem is
known to be decidable with the tree semantics.
IDefinition 16.
A
QCTL
i
formula
ϕ
is hierarchical if for all subformulas
ϕ1, ϕ2
of the form
ϕ1=o1p1. ϕ0
1and ϕ2=o2p2. ϕ0
2where ϕ2is a subformula of ϕ0
1, we have o1o2.
The following result is proved in [
3
], where
QCTL
i,
is the set of hierarchical
QCTL
i
formulas:
ITheorem 17 ([3]).Model checking QCTL
i,with tree semantics is decidable.
4.3 Model checking ATL
sc,i
We establish that model checking
ATL
sc,i
is decidable on a class of instances whose definition
relies on the notion of hierarchical observation.
IDefinition 18.
Let
G
= (
V, E , `, {∼a}aAg
)be a
CGSi
, and let
a, b Ag
be two agents.
Agent
a
observes no more than agent
b
in
G
, written
a4Gb
, if for every pair of positions
v, v0V
,
vbv0
implies
vav0
. We say that
AAg
is hierarchical in
G
if
4G
is a total
preorder on A.
If a set of agents
A
is hierarchical in a
CGSiG
, we thus may talk about maximal and
minimal agents in A, referring to maximal and minimal elements of Afor the relation 4G.
The essence of the requirement that makes the problem decidable is the same as for the
decidability result on
QCTL
i
(Theorem 17): nesting of quantifiers (here, strategy quantifiers)
should be hierarchical, with those observing more inside those observing less. However, unlike
in
QCTL
i
, in
ATL
sc,i
observations are not part of formulas, but rather they are given by the
models. We thus define the notion of hierarchical ATL
sc,i formula with respect to a CGSi:
12 Decidability results for ATL with imperfect information and perfect recall
IDefinition 19.
Let Φbe an
ATL
sc,i
formula and let
G
be a
CGSi
. We say that Φis
hierarchical in Gif:
for every subformula ϕof the form ϕ=h·A·iϕ0,Ais hierarchical in G, and
for all subformulas
ϕ1, ϕ2
of the form
ϕ1
=
h·A1·iϕ0
1
and
ϕ2
=
h·A2·iϕ0
2
where
ϕ2
is a
subformula of ϕ0
1, maximal agents of A1observe no more than minimal agents of A2.
An instance
,
(
G, v
)) of the model-checking problem for
ATL
sc,i
is hierarchical if Φis
hierarchical in G.
In the rest of the section we establish the following:
ITheorem 20. Model checking ATL
sc,i is decidable on the class of hierarchical instances.
We build upon the proof in [
25
] that establishes the decidability of the model-checking
problem for
ATL
sc
by reduction to the model-checking problem for
QCTL
. The main differ-
ence is that we reduce to the model-checking problem for
QCTL
i
instead, using quantifiers
parameterised with observations corresponding to agents’ observations. We also need to
make a couple of adjustments to obtain formulas in the decidable fragment QCTL
i,.
Let
,
(
G, vι
)) be a hierarchical instance of the
ATL
sc,i
model-checking problem, where
G
= (
V, E , `, {∼a}aAg
)is a
CGSi
over
AP
. In the reduction we will transform Φinto an
equivalent
QCTL
i
formula Φ
0
in which we need to refer to the current position in the model
G
, and also to talk about moves taken by agents. To do so, we consider the additional sets of
atomic propositions
APv
:=
{pv|vV}
and
APm
:=
{pa
m|aAg and mM}
, that we
take disjoint from AP.
First we define the CKS
SG
on which Φ
0
will be evaluated. Since the models of the
two logics use different ways to represent imperfect information (equivalence relations on
positions for
CGSi
and local states for CKS) this requires a bit of work. First, for each
vV
and
aAg
, let us define [
v
]
a
as the equivalence class of
v
for relation
a
. Now, noting
Ag
=
{a1, . . . , an}
, we define for each
i
[
n
]the set
Li
:=
{
[
v
]
ai|vV}
of local states for
agent
ai
. Since we need to know the actual position of the
CGSi
to define the dynamics, we
also let
Ln+1
:=
V
. States of
SG
will thus be tuples in
L1×. . . ×Ln×Ln+1
. For each
v∈ G
,
let sv:= ([v]a1,...,[v]an, v)be its corresponding state in SG.
We can now define SG:= (S, R, `0), where
S:= {sv|vV},
R:= {(sv, sv0)| ∃mMAg s.t. E(v, m) = v0}, and
`0(sv) := `(v)∪ {pv}.
To make the connection between finite plays in
G
and nodes in tree unfoldings of
SG
,
let us define, for every finite play
ρ
=
v0. . . vk
, the node
uρ
:=
sv0. . . svk
in
tSG
(
sv0
)(which
exists, by definition of
SG
and of tree unfoldings). Observe that the mapping
ρ7→ uρ
is in
fact a bijection between the set of finite plays starting in a given position
v
and the set of
nodes in tSG(sv).
Now it should be clear that giving to a propositional quantifier in
QCTL
i
observation
oi
:=
{i}
, for
i
[
n
], amounts to giving him the same observation as agent
ai
. Formally,
one can prove the following lemma, simply by applying the definitions of observational
equivalence in the two frameworks:
ILemma 21. For all finite plays ρ, ρ0starting in v,ρaiρ0iff uρoiuρ0in tSG(sv).
R. Berthon, B. Maubert and A. Murano 13
We now describe the translation
4
from
ATLsc,i
formulas to
QCTL
i
formulas. First we
recall the translation from [25] for the perfect-information case.
The translation from
ATLsc
to
QCTL
is parameterised by a coalition
BAg
, that
conveys the set of agents who are currently bound to a strategy. It is defined by induction
on Φas follows:
pB:= p¬ϕB:= ¬ϕB
ϕϕ0B:= ϕBϕ0B(|A|)ϕB:= ϕB\A
XϕB:= XϕBϕUϕ0B:= ϕBUϕ0B
The only non-trivial case is for formulas of the form
h·A·iϕ
. For the rest of the section, we let
M={m1, . . . , ml}. Now, if A={ai1, . . . , aik}, we define
h·A·iϕB:= mai1
1. . . mai1
l. . . maik
1. . . maik
lpout.
Φstrat(A)Φout (AB)A(Gpout ϕAB),
where
Φstrat(A) := ^
aA
AG _
mM
(ma^
m06=m
¬m0a)
and
Φout(A) := pout AG [¬pout AX¬pout]
AG
pout _
vV_
mMA
pvpmAX
_
v0E(v,m)
pv0pout
.
In Φ
out
(
A
), for
m
= (
ma
)
aAMA
, notation
pm
stands for the propositional formula
VaAma
a
which characterizes the joint move
m
that agents in
A
play in
v
. Also,
E
(
v, m
)is
the set of possible next positions when the current one is vand agents in Aplay m, and it
is defined as E(v, m) := {E(v, (m,m0)) |m0MAg\A}.
The idea of this translation is the following: first, for each agent
aA
and each possible
move
mM
, an existential quantification on the atomic proposition
ma
“chooses” for each
finite play
ρ
of (
G, vι
)(or, equivalently, for each node
uρ
of the
tSG
(
svι
)) whether agent
a
plays move
m
in
ρ
or not. Formula Φ
strat
(
A
)ensures that each agent
a
chooses exactly one
move in each finite play, and thus that atomic propositions
ma
characterise a strategy for her.
An atomic proposition
pout
is then used to mark the paths that follow the currently fixed
strategies: formula Φ
out
(
AB
)states that
pout
marks exactly the outcome of strategies just
chosen for agents in
A
, as well as those of agents in
B
, that were chosen previously by a
strategy quantifier “higher” in Φ.
Note that we simplified slightly Φ
strat
(
A
)and Φ
out
(
A
), using the fact that unlike in [
25
],
we have assumed in our definition of
CGSi
that the set of available moves is the same for all
agents in all positions (see Footnote 2).
It is proven in [
25
] that this translation is correct, in the sense that for every
ATLsc
closed
formula
ϕ
and pointed perfect-information concurrent game structure (
G, v
), letting
SG
be
4
Here we abuse language: the construction depends on the model
G
and is therefore not a translation in
the usual sense.
14 Decidability results for ATL with imperfect information and perfect recall
as described above but removing the local states for all agents and keeping only the
Ln+1
component, we have:
G, v |=ϕiff tSG(sv)|=tϕ.
We now explain how to adapt this translation to the case of imperfect information. Observe
that the only difference between
ATL
sc
and
ATL
sc,i
is that in the latter, strategies must be
defined uniformly over indistinguishable finite plays, i.e., a strategy
σ
for an agent
a
must
be such that if
ρaρ0
, then
σ
(
ρ
) =
σ
(
ρ0
). To enforce that the strategies coded by atomic
propositions
ma
in
h·A·iϕB
are uniform, we use the propositional quantifiers with partial
observation of
QCTL
i
. Formally, we define a translation
fB
from
ATL
sc,i
to
QCTL
i
. It is
defined exactly as the one from ATL
sc to QCTL, except for the following inductive case.
If A={ai1, . . . , aik}we let
^
h·A·iϕ
B:= oi1mai1
1. . . mai1
l. . . oikmaik
1. . . maik
lpout.
Φstrat(A)Φout (AB)A(Gpout eϕAB),
where Φ
strat
(
A
)and Φ
out
(
A
)are defined as before, and
pout
is a macro for
{1,...,n+1}pout
(see Remark 4.2).
So the only difference from the previous translation is that now, the labelling of each
atomic proposition
mai
must be
oi
-uniform. This means that if two nodes
u
and
u0
in
tSG
(
svι
)
are
oi
-indistinguishable, then
u
is labelled with
mai
if and only if
u0
also is. In other words,
in the strategy coded by atomic propositions
mai
, agent
ai
plays
m
in
u
if and only if she
also plays it in
u0
, and thus this strategy is uniform (recall that, by Lemma 21, observation
oi
correctly reflects agent
ai
’s observation in
tSG
(
svι
)). It is then clear that this translation
is correct:
G, vι|= Φ iff tSG(svι)|=te
Φ.(1)
However, even if we have taken
,
(
G, vι
)) to be a hierarchical instance,
e
Φ
is not in the
decidable fragment
QCTL
i,
. Indeed, with the current definition of observations
{oi}i[n]
,
hierarchical observation in
G
does not imply hierarchical observation in
SG
: since
oi
=
{i}
,
for
i6
=
j
it is never the case that
oioj
. Still, we note that if agent
aj
observes no more
than agent
ai
, then letting
ai
see also what agent
aj
sees does not increase her knowledge of
the situation:
ILemma 22.
If
aj4Gai
, then for all finite plays
ρ, ρ0
that start in the same position,
uρoiuρ0iff uρoiojuρ0.
Proof. Assume that aj4Gai. It is enough to see that for every pair of states sv, sv0in SG,
we have
svoisv0
iff
svoiojsv0
. The right-to-left implication is obvious: if two states
have the same
i
-th and
j
-th components, in particular they have the same
i
-th component.
For the other direction, assume that
svoisv0
. This means that [
v
]
ai
= [
v0
]
ai
, and thus
that
vaiv0
. Since
aj4Gai
, we also have that
vajv0
, and thus that [
v
]
aj
= [
v0
]
aj
, and it
follows that svoiojsv0.J
In the light of this Lemma 22, we can safely redefine observations as follows: for each
i[n], we let
o0
i:= [
j|aj4Gai
oj.
R. Berthon, B. Maubert and A. Murano 15
Observe that in fact
o0
i
=
{j|aj4Gai}
. Informally, a quantifier with observation
o0
i
sees
what agent
ai
observes (note that
4G
is reflexive), as well as what agents that see no more
than aiobserve.
Let us define a new version of the translation
fB
. First, Φbeing hierarchical in
G
, for
each subformula of Φof the form
h·A·iϕ
we have that
A
is hierarchical in
G
. It is thus possible
to choose for agents in
A
an indexing
A
=
{ai1, . . . , aik}
such that for all 1
c < d k
, we
have aic4Gaid.
Now the translation remains the same as before except for the following inductive case:
If A={ai1, . . . , aik}, where for all 1c < d k, we have aic4Gaid, we let
^
h·A·iϕ
B:= o0
i1mai1
1. . . mai1
l. . . o0
ikmaik
1. . . maik
lpout.
Φstrat(A)Φout (AB)A(Gpout eϕAB),
where Φstrat(A)and Φout (A)are defined as before.
From Lemma 22 we have that this new translation is still correct in the sense of Equa-
tion (1). In addition, for all 1c<dkwe have o0
ico0
id.
Now consider formula
e
Φ
. Because Φis hierarchical in
G
, for every pair of subformulas
ϕ1, ϕ2
of the form
ϕ1
=
h·A1·iϕ0
1
and
ϕ2
=
h·A2·iϕ0
2
where
ϕ2
is a subformula of
ϕ0
1
, maximal
agents of
A1
observe no more than minimal agents of
A2
. It is then easy to see that
e
Φ
would be hierarchical if there were not the perfect-information quantifications on atomic
proposition
pout
that break the monotony of observations along subformulas when there are
nested strategic quantifiers. We explain how to remedy this last problem.
We remove altogether proposition
pout
, and we use instead the formula
ψout
(
A
)defined
below to characterise which paths are in the outcome of the currently-fixed strategies:
ψout(A) := G
^
vV^
mMA
pvpmX_
v0E(v,m)
pv0
.
Clearly, this formula holds in a path
λ
of
tSG
(
svι
)marked with propositions
ma
charac-
terising strategies for agents in
A
, if at each point along
λ
corresponding to some position
v
,
the next point in
λ
corresponds to a position
v0
that can be attained from
v
when agents in
A
each play the move prescribed by their current strategy. The last modification to
fB
is
thus the following:
If A={ai1, . . . , aik}, where for all 1c < d k, we have aic4Gaid, we let
^
h·A·iϕ
B:= o0
i1mai1
1. . . mai1
l. . . o0
ikmaik
1. . . maik
l.Φstrat(A)Aψout (AB)eϕAB,
where Φstrat(A)is defined as before.
It follows from the above considerations that this translation is still correct in the sense
of Equation (1), and one can check that
e
Φ
is a hierarchical
QCTL
i
formula. We conclude
the proof by recalling that by Theorem 17, model checking QCTL
i,is decidable.
Concerning complexity, model checking
ATLsc
being already nonelementary [
25
], so is it
for ATLsc,i.
5 Conclusion
In this work we established new decidability results for the model-checking problem of
ATL
with imperfect information and perfect recall as well as its extension with strategy context.
16 Decidability results for ATL with imperfect information and perfect recall
Should new decidable classes of multiplayer games with imperfect information be discovered,
and assuming the reasonable property of closure under initial shifting, our transfer theorem
(Theorem 6) would entail new decidability results also for
ATL
i
. As for
ATL
sc,i
, it would be
interesting to investigate whether a meaningful notion of hierarchical instances based on,
e.g., dynamic or recurring hierarchical information instead of hierarchical observation as here,
could lead to stronger decidability results.
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