A Note On Syntactic Characterization of Incomplete
Information in ATEL
Thomas Ågotnes
Department of Informatics, University of Bergen
PB. 7800, N-5020 Bergen, Norway
[email protected]
Abstract
It has been argued that the class of semantic structures of Alternating-time Temporal Epistemic Logic (ATEL) are too general and should be restricted in order to properly integrate the
semantics of epistemic logic and Alternating-time Temporal Logic (ATL). One of the semantic
restrictions is that an agent should have the same choices, or actions, available in indiscernible
states. van der Hoek and Wooldridge [10] present a syntactic property claimed to be given by the
semantic restriction. This paper points out that the claim does not hold; the semantic restriction
is in fact neither sufficient nor necessary for the suggested syntactic property. Furthermore, the
difficulty of expressing such local properties of agents seems to be due to a fundamental limit of
the expressiveness of ATEL. A redefinition of the ATEL semantics in order to be able to express
such properties is briefly discussed.
1
Introduction
Alternating-time Temporal Epistemic Logic (ATEL) is an extension of Alternating-time Temporal
Logic (ATL) with epistemic operators. It has been pointed out [5, 10] that in some contexts there is a
problem with the integration of the semantics of the temporal and the epistemic operators: epistemic
logic presupposes that agents have incomplete information while the notions of choices and strategies
in ATL in general do not. Incomplete information in ATL, and ATEL, can be modeled by imposing
semantic restrictions on the structures. One of the needed restrictions is that a player must have the
same choices, or actions, available in indiscernible states. The topic of this paper is expressing this restriction syntactically. Syntactic expression of semantic properties are interesting i.e. for constructing
formal axiomatizations. van der Hoek and Wooldridge have suggested a syntactic representation of
the mentioned property [10]. The main contribution of this paper is to point out that the semantic condition is neither necessary nor sufficient for the syntactic property. Furthermore, I argue that it seems
to be impossible to represent such semantic conditions syntactically in ATEL due to a lack of expressiveness regarding “local” properties of individual agents in ATL. In Section 5 I briefly introduce a
redefinition of the semantic structures in order to be able to express these kinds of properties.
First, the language and semantics of ATEL are presented, along with a discussion about incomplete
information.
1
2
Background
Alternating Epistemic Transition Systems (AETSs) [9] is a generalization of Alternating Transition
Systems (ATSs) [1]. An AETS is a tuple
(Σ, Q, Π, π, δ, ∼1 , . . . , ∼n )
where
• Σ = {a1 , . . . , an } is a set of agents
• Q is a finite, non-empty set of states
• Π is a finite, non-empty set of propositions
• π : Q → 2Π
Q
• δ : Q × Σ → 22 ; the transition function. It is required that for every q ∈ Q, a ∈ Σ and
Qa ∈ δ(q, a): | ∩a∈Σ Qa | = 1.
• For each a ∈ Σ, ∼a ⊆ Q × Q; the epistemic accessibility relation. ∼a is required to be an
equivalence relation.
The transition function in ATSs gives a set of choices for each agent in each state, where a choice
is a set of states. The system is completely controlled by the agents, and the result of each agent
making a choice is the single state in the intersection of the choices. A computation is a sequence of
states λ = q0 q1 · · · such that for each k, for each a ∈ Σ there is a Qa ∈ δ(qk , a) such that qk+1 =
∩a∈Σ Qa ; the kth element of λ is denoted λ[k]. A strategy for agent a is a function fa : Q+ → 2Q
such that fa (λq) ∈ δ(q, a). Given a state q and a set of strategies for a group of agents G ⊆ Σ,
f~G = {fi : i ∈ G}, out(q, f~G ) denotes the set of possible computations starting in state q where the
agents in G use the strategies f~G and each agent a ∈ Σ \ G is only restricted, in state q 0 , to choices
from δ(q 0 , a). The set of all strategies for a group G is denoted Str (G).
The syntax of the ATEL language is defined over a set of propositions Π and a set of agents Σ.
Π are formulae, and if φ1 , φ2 are formulae, G ⊆ Σ and a ∈ Σ then ¬φ1 ,φ1 ∨ φ2 , hhGii φ1 ,
hhGii2φ1 , hhGiiφ1 Uφ2 and Ka φ1 are formulae1 . Satisfiability of a formula ψ in a state q of an
AETS S is defined as follows, where p ∈ Π:
S, q |= p
⇔
p ∈ π(q)
S, q |= ¬φ
⇔
S, q 6|= φ
S, q |= φ1 ∨ φ2
⇔
S, q |= φ1 or S, q |= φ2
S, q |= hhGii φ
⇔
∃f~G ∈Str (G) ∀λ∈out(q,f~G ) S, λ[1] |= φ
S, q |= hhGii2φ
⇔
∃f~G ∈Str (G) ∀λ∈out(q,f~G ) ∀j≥0 S, λ[j] |= φ
S, q |= hhGiiφ1 Uφ2
⇔
∃f~G ∈Str (G) ∀λ∈out(q,f~G ) ∃j≥0 (S, λ[j] |= φ2 and ∀0≤k<j S, λ[k] |= φ1 )
S, q |= Ka φ
⇔
∀q∼a q0 S, q 0 |= φ
1
In addition, ATEL has operators for common knowledge and “everyone knows”, which are not used in the following
and therefore not presented here.
2
2.1
ATEL and Incomplete Information
If we want to model players who have incomplete information about the world, two natural restrictions
on structures and strategies are:
1. An agent must have the same actions available in two states which are indiscernible for that
agent.
2. A strategy must map indiscernible histories of states to the same action.
Alur et. al. [2] propose a restriction on the ATL structures for players with incomplete information,
which includes the two mentioned restrictions, but only in the special case of turn-based synchronous
structures.
Jamroga [5] points out that ATEL does not seem to integrate the semantics of knowledge with the
ATL semantics properly, because neither of these conditions hold in general in AETSs. For example,
if q, q 0 ∈ Q, q 6= q 0 and q ∼a q 0 , agent a can have a strategy fa with fa (λq) 6= fa (λq 0 ) for some λ.
The fact that agents can base their choices on the state of the whole system, seems to contradict the
premise of epistemic logic: that agents have incomplete information. Jamroga argues that a natural
solution to the problem is the two semantic restrictions mentioned above, but that they are difficult to
implement in AETSs since actions are only represented as outcomes, making it difficult to identify
“the same action” in different states. He thus proposes to slightly modify the definition of an AETS
to a tuple2
(Σ, Q, Π, π, ∼1 , . . . , ∼n , ACT , d, δ)
where ACT is a finite set of actions (or action names), da (q) ⊆ ACT for each agent a and state
q, and δ(q, v) ∈ Q when q ∈ Q and v ∈ d1 (q) × · · · × dn (q). Now, the two semantic restrictions
mentioned above can be expressed as
q ∼a q 0 ⇒ da (q) = da (q 0 )
and
∀k (λ[k] ∼i λ0 [k]) ⇒ fa (λ) = fa (λ0 )
respectively.
van der Hoek and Wooldridge [10, p. 144] comment very briefly on the fact that in a specific
context “rational agents . . . make their choices depending on their knowledge”, and propose the same
two semantic restrictions as shown above (with the difference of the representation of actions as
outcomes instead of names). They point out that the second condition on incomplete information
does not seem to be syntactically expressible in ATEL. However the first condition, they argue, gives
a syntactic property. This property is investigated in the next section.
3
A Suggested Syntactic Property
van der Hoek and Wooldridge [10] claim that the first semantic condition on incomplete information
can be expressed syntactically. Quote (p. 144):
2
Jamroga’s modification is based on a version [2] of ATSs, now called concurrent game structures, more recent than
the version AETSs were originally based on. The presentation of ATEL in the current paper is based on the originally published definition using ATSs instead of concurrent game structures because the paper discusses subtle semantic issues. The
difference between Jamroga’s modification and concurrent game structures is, in addition to the two semantic restrictions,
the use of names rather than numbers for actions.
3
q ∼a q 0 ⇒ δ(q, a) = δ(q 0 , a)
(1)
This gives us the following syntactic property.
hhaiiT φ ↔ Ka hhaiiT φ
(2)
where T is a temporal operator. This claim, however, is not true. It is not completely clear what is
meant by “this gives us”, but it seems to be (at least) that (1) implies (2). In any case, (1) is neither a
sufficient nor a necessary condition for (2), as is shown in the rest of this section.
I first show the former. That (1) implies (2) can, again, be interpreted in two ways. First, if (1)
holds for a state q in an AETS S then S, q |= hhaiiT φ ↔ Ka hhaiiT φ, or, second, if (1) holds for
every state in an AETS S then S, q |= hhaiiT φ ↔ Ka hhaiiT φ for every state q. I show that the second
interpretation does not hold; this implies that neither the first interpretation holds. Let φ = p ∈ Π. I
show that there is an AETS in which (1) holds for every state, with a state q1 such that
S, q1 6|= hhaii p → Ka hhaii p
(3)
S is illustrated on Fig. 1. S = (Σ, Q, Π, π, δ, ∼a , ∼b ) where
• Σ = {a, b}
• Q = {q1 , q2 , q3 , q4 }
• Π = {p}
{p} q ∈ {q1 , q2 , q3 }
• π(q) =
∅
q = q4
• δ(q1 , a) = δ(q3 , a) = { {q2 , q4 } }
• δ(q1 , b) = { {q2 } }
• δ(q3 , b) = { {q4 } }
• δ(q2 , a) = δ(q2 , b) = { {q2 } }
• δ(q4 , a) = δ(q4 , b) = { {q4 } }
• ∼a = {(q1 , q3 ), (q3 , q1 ), (q1 , q1 ), (q2 , q2 ), (q3 , q3 ), (q4 , q4 )}
• ∼b is the identity relation on Q
Clearly, S is an AETS: for every q ∈ Q if Qa ∈ δ(q, a) and Qb ∈ δ(q, b) then |Qa ∩ Qb | = 1, and ∼a
and ∼b are equivalence relations. (1) holds for any q ∈ Q (and for both a and b). To see that
S, q1 |= hhaii p
(4)
holds, let fa : Q+ → 2Q be any strategy for a. fa (q1 ) = {q2 , q4 } since δ(q1 , a) is singular. Let
λ ∈ out(q1 , {fa }). Clearly, since the only choice available to b in q1 is {q2 }, {λ[1]} = {q2 , q4 } ∩
{q2 } = {q2 }. S, λ[1] |= p, so (4) holds. Assume that
S, q1 |= Ka hhaii p
4
(5)
∼a
p
p
q3
q1
¬p
p
q2
q4
Figure 1: The AETS S.
Since q1 ∼a q3 , S, q3 |= hh{a}ii p. Let ga be any strategy for a. ga (q3 ) = {q2 , q4 } since δ(q3 , a)
is singular. Let λ0 ∈ out(q3 , ga ). Since δ(q3 , b) is singular, {λ0 [1]} = {q2 , q4 } ∩ {q4 } = {q4 }. By 5,
S, λ0 [1] |= p which contradicts the fact that p 6∈ π(q4 ). Thus, (5) does not hold, which shows (3).
Thus, (1) is not a sufficient condition for (2). To show that it is neither a necessary condition, I
construct an AETS S 0 which is a model for (2) for any a, T and φ, for which (1) does not hold for any
q. S 0 is illustrated on Fig. 2. S 0 = (Σ, Q, Π, π, δ 0 , ∼a , ∼b ) where
{ {q2 } } q = q3
0
• δ (q, a) =
δ(q, a)
otherwise
{ {q2 } } q = q3
• δ 0 (q, b) =
δ(q, b)
otherwise
and Σ, Q, Π, π, δ, ∼a and ∼b are as defined above. Clearly, S 0 is still an AETS. Let T ∈ {, 2, F }
∼a
p
p
q3
q1
¬p
p
q2
q4
Figure 2: The AETS S 0 .
and φ be a formula. To show that S 0 , q |= hha0 iiT φ ↔ Ka hha0 iiT φ for any q ∈ Q and a0 ∈ {a, b}, it
suffices to show that
S, q1 |= hhaiiT φ ⇔ S, q3 |= hhaiiT φ
(6)
since it follows trivially for a0 = a and q ∈ {q2 , q4 } and for a0 = b for any q. It is easy to see
that the only computation starting in q1 is λ1 = q1 q2 q2 · · · and the only computation starting in q3
is λ2 = q3 q2 q2 · · · . Satisfaction of a formula ψ in a state q depends only on i) π(q), ii) the states
accessible for each agent from q and iii) the set of possible remaining computations starting in q. For
q1 and q3 , i) π(q1 ) = π(q3 ), ii) q1 ∼a0 q 0 iff q3 ∼a0 q 0 for a0 ∈ {a, b} and iii) the set of possible
remaining computations starting in q1 or q3 is {q2 q2 · · · }. Thus, S, q1 |= ψ ⇔ S, q3 |= ψ for any ψ,
and (6) holds. However, (1) does not hold for S 0 : q1 ∼a q3 but δ 0 (q1 , a) 6= δ 0 (q3 , a).
5
In other words; there are models of the schema (2) in which (1) do not hold, and there are structures
in which (1) hold which are not models of the schema (2). Thus, (2) is not a syntactic representation
of the semantic condition (1).
3.1
A Frame Property
The semantic property (1) does not depend on the labeling function π. Instead of comparing the
models of the syntactic property (2) to the model property (1) as in the previous discussion, we can
compare the frames of (2) to the frame property (1). It is not completely clear what a frame for ATEL
is, but in this subsection I consider the situation if we define a frame to be a tuple F = (Σ, Q, Π, δ, ∼1
, . . . , ∼n ) (an AETS without a π). The notions of a structure based on a frame and validity in a frame
are used as usual, and I write (F, π) for the AETS based on F with labeling function π.
It is easy to see that the frame property (1) is not a sufficient condition for the syntactic property
(2): take the frame which S (defined on p. 4) is based on as a counter example.
The other direction does in fact hold: if (2) holds in a frame then (1) holds for that frame. However,
it holds because (2) as a frame condition is really not very interesting: if F = (Σ, Q, Π, δ, ∼1 , . . . , ∼n )
is a frame of (2) then
q ∼a q 0 ⇒ q = q 0
(7)
for each a ∈ Σ. To see this, let F |= (2) and assume that q ∼a q 0 for some q 6= q 0 . Let π be such that
π(q) = {p} and π(q 0 ) = ∅. Then (F, π), q |= (2), particularly3
(F, π), q |= hhaiipUp ↔ Ka hhaiipUp
But (F, π), q |= hhaiipUp and (F, π), q 0 6|= pUp, which is a contradiction showing that the assumption
is impossible. Thus, an agent a in a frame of (2) in each state q either knows everything true in q (if
∼a (q) = {q}) or knows everything (if ∼a (q) = ∅).
Thus, under this definition of a frame, (2) does not hold under the frame property (1), and furthermore it describes a very particular and not very interesting epistemic situation.
4
The Problem
One reason that (2) may intuitively seem to describe (1) is that a statement such as hhaii φ often is
informally interpreted as a statement about agent a’s capabilities. This is an imprecise interpretation.
It may be that hhaii φ holds because the rest of the system will deterministically make φ true in the
next state — no matter what agent a does. In other words, it may be that hh∅ii φ holds — which
trivially implies hhaii φ. For example, (4) above holds because the system S will deterministically
go to state q2 from q1 . a’s capabilities in q1 , described by δ(q1 , a), are to take the system to a state
where either p is true (q2 ) or where p is false (q4 ), but a cannot control which which of these will be
the case. This contrasts with the intuitive interpretation of (4) as a statement about a’s capabilities.
Of course, the fully deterministic system S is not very interesting in practice. As argued above,
the deterministic transition from q1 to q2 in S is a part of the reason that (5) does not hold. This special
case might be ruled out syntactically:
(hhaii p ∧ ¬hh∅ii p) → Ka hhaii p
(8)
3
It is assumed here that this formula fits the intention of the schema 2 since in [10] T is said to be “a temporal operator”
and U is a temporal operator.
6
Indeed (trivially),
S |= (hhaii p ∧ ¬hh∅ii p) → Ka hh{a}ii p
However, there are counter-models of (8) where (1) holds. Such a structure, S 00 , is illustrated on
Figure 3. S 00 = (Σ, Q00 , Π, π 00 , δ 00 , ∼00a , ∼00b ) where Σ and Π is as before and
• Q = {q1 , q2 , q3 , q4 }
{p} q ∈ {q1 , q2 , q3 }
• π(q) =
∅
q ∈ {q4 , q5 }
• δ(q1 , a) = δ(q3 , a) = { {q2 , q4 }, {q5 } }
• δ(q1 , b) = { {q2 , q5 } }
• δ(q3 , b) = { {q4 , q5 } }
• δ(q2 , a) = δ(q2 , b) = { {q2 } }
• δ(q4 , a) = δ(q4 , b) = { {q4 } }
• δ(q5 , a) = δ(q5 , b) = { {q5 } }
• ∼a = {(q1 , q3 ), (q3 , q1 ), (q1 , q1 ), (q2 , q2 ), (q3 , q3 ), (q4 , q4 ), (q5 , q5 )}
• ∼b is the identity relation on Q00
q3
q1
p
q2
∼a
p
p
¬p
¬p
q5
q4
Figure 3: The system S 00 .
Clearly, S 00 is an AETS where (1) holds. S 00 , q1 |= hhaii p and S 00 , q1 |= ¬hh∅ii p, but S 00 , q3 6|=
hhaii p so
S 00 , q1 6|= (hhaii p ∧ ¬hh∅ii p) → Ka hhaii p
Thus, S 00 is a more interesting counter-model of (2) than S is.
It seems that not only is the suggested syntactic property not a proper expression of the first
semantic condition on incomplete information, but also that it is impossible to express this condition
syntactically in ATEL.
Note that the cause of the problem is not the identification of actions with their outcomes; similar
counterexamples to the ones given above can be constructed with Jamroga’s redefinition of AETSs.
The problem seems to be a fundamental property of the expressiveness of ATL: it is difficult
to express local properties of individual agents. Temporal connectives in ATL express something
about future states, without regards to how they come about. Therefore, semantic local properties of
individual agents such as (1) are difficult to express syntactically in ATEL.
7
5
A Possible Solution
The reason a player can have different actions (or choices) available in indiscernible states in ATEL is
that available actions are not considered as something which make states discernible by the players. A
solution to the problem may be to take the possible worlds idea seriously and consider also available
actions as a part of the possible worlds. This can be done by introducing a proposition A for each
potential action for an agent; A is true in a state if and only if the action is available for the agent in
that state. Here, this idea is briefly presented.
Formally, the definition of an AETS can be changed by partitioning Π into disjoint sets of action
propositions (henceforth called just “actions”) for each agent in addition to other propositions, and
defining the da function as a derived rather than primary notion. A structure with action-propositions
is a tuple
(Σ, Q, Π, π, ∼1 , . . . , ∼n , δ)
where (] denotes disjoint union)
Π = Π 0 ] Π1 ] · · · ] Πn
and the rest of the tuple defined as in Jamroga’s version of AETSs (Sec. 2.1) with da defined as
follows (where q ∈ Q and a ∈ Σ):
da (q) = π(q) ∩ Πa
Here, Π0 is the set of “other” primitive propositions, those which constitute Π in a regular AETS.
Of course, introducing actions as propositions brings up a range of questions. Which properties
do we implicitly assign to actions, and what is the relationship between knowledge and action? Here,
I only comment on the properties needed to illustrate the main topic of the paper.
The property
Ka A → A
where A ∈ Πa — if an agent knows that an action is available then it is available — is induced by the
S5 properties of knowledge. The other direction is also interesting (A ∈ Πa ):
A → Ka A
(9)
If an action is available, should the agent know that it is available? In other words, if an agent can
use an action must he know that he can use it? This is precisely the semantic condition that the same
actions must be available in indiscernible states. In fact, it is easy to see that the first condition on
incomplete information
q ∼a q 0 ⇒ da (q) = da (q 0 )
(10)
is a sufficient and necessary condition4 for (9).
Thus, introducing actions as propositions allows precise expression of local semantic properties
such as (10).
6
Comments
The main point in this paper is to point out that the suggested syntactic property is not a proper
expression of the semantic condition, and that just like the second semantic condition (mentioned in
4
To be precise: the schema (10) holds for a structure iff (9) seen as a schema over a ∈ Σ and A ∈ Πa is true in all states
in the structure.
8
the introduction) of incomplete information, it is difficult to express this first condition syntactically
in ATEL due to a lack of expressiveness about “local” properties of agents.
Other discussions of syntactic properties of ATEL include [3], and [10] also discuss many other
properties than the one mentioned.
The paper is a presentation of ongoing research, and two aspects of it in particular are being
addressed further. First, more work is needed in order to determine the expressiveness of ATEL more
precisely in general, and to find a definitive answer to whether 1 is expressible in ATEL in particular.
Second, the properties of structures with action-propositions, from Section 5, is investigated in detail.
Particularly, the interaction between knowledge and action must be analyzed in context of the existing
research [6, 7, 8, 4].
Acknowledgments I thank the anonymous referees for valuable comments.
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