Knowledge Constructions for Artificial
Intelligence
Ahti Pietarinen
Department of Philosophy, University of Helsinki
P.O. Box 9, FIN-00014 University of Helsinki
[email protected]
Abstract. Some new types of knowledge constructions in epistemic
logic are defined, and semantics given by combining game-theoretic notions with modal models. One such notion introduced is focussed knowledge, which arises from imperfect information in quantified epistemic
logics. This notion is useful in knowledge representation schemes in artificial intelligence and multi-agent systems with uncertainty. In general,
in all the logics considered here, the imperfect information is seen to
give rise to partiality, including partial common and partial distributed
knowledge. A game-theoretic method of creating non-monotonicity will
then be suggested, based on the partialised notion of ‘only knowing’ and
inaccessible possible worlds. The overall purpose is to show the extent in
which games combine with a given variety of knowledge constructions.
1
Introduction
The underlying motivation for this work can perhaps be illustrated by noting
that classical logic is a logic of perfect information transmission. This fact is of
course true of propositional and predicate logics, but it is also true of intensional
modal logics and logics of epistemic notions (knowledge and belief). By perfect
information transmission, it is meant that in transmitting semantic information
from one logical component to another, that information is never lost.
The aim is to show that once we adopt semantics that is suitable not only
for the received logics with perfect information, we are able to produce new
logics with new, expressive resources that capture an interesting variety of constructions of knowledge. Some of such constructions are needed in representing
knowledge of multi-agent systems. The distinction between perfect and imperfect information transmission can be made precise within the framework of gametheoretic semantics (gts, see e.g. [3,7,8]), which operationalises a semantic game
between two players, the team of Verifiers (V , ∃loise) and the team of Falsifiers
(F , ∀belard). These semantic games provide an evaluation method that can be
defined for a variety of logics.
Research in epistemic logic and reasoning about knowledge has played an
important role in AI, and uncertainty has been a major topic in reasoning even
longer. The purpose of this paper is to combine the two. It is argued that logics
of knowledge can represent multi-agent uncertainty, and it is suggested how
M.-S. Hacid et al. (Eds.): ISMIS 2002, LNAI 2366, pp. 303–311, 2002.
c Springer-Verlag Berlin Heidelberg 2002
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A. Pietarinen
a unifying semantics based on the notion of games can be defined to a wide
variety of knowledge constructions. These notions include focussed, common
and distributed knowledge, and the nonmonotonic notion of ‘only knowing’.
Epistemic logics with imperfect information are partial, that is, there are
sentences that are neither true nor false. Partiality itself has game-theoretic
roots: if the associated semantic games are non-determined, all attempts of trying
to verify or falsify a formula can be defeated. In contrast to received partial
modal logics [5], games give rise to partiality even if the underlying models are
complete.
One outcome is that some new multi-agent logics for notions of knowledge
that need games for their interpretation will be available. Far from being a purely
technical enterprise, these logics are motivated by those knowledge representation schemes in multi-agent systems that virtually necessitate the introduction of
new knowledge constructions. For example, focussed knowledge involves inherent
uncertainty, and is argued to play an important role in representing knowledge
in multi-agent systems where agents, such as communicating processors, do not
always know the content of a message that has been sent to them.
2
From Perfect to Imperfect Information: Knowledge and
Multi-agent Systems
The inauguration of epistemic logic [2] has led to a proliferation of knowledge
in logic, philosophy, computer science and artificial intelligence. Nowadays we
find notions like common, shared and distributed knowledge. Furthermore, the
distinction between de dicto and de re knowledge is widely spread.
The well-formed formulas of ordinary propositional epistemic logic L are
constructed by φ ::= p | ϕ ∨ ψ | ¬ϕ | Ki ϕ. Ki ϕ is read ‘an agent i knows ϕ’.
Let ϕ, ψ be formulas of classical propositional epistemic logic L. A model
is M = W, R, g, where g is a total valuation function g : W → (Φ →
{True, False}), assigning to each proposition letter a subset of a set of possible worlds W = {w0 . . . wn } for which {wi | g(w)(p) = True, w ∈ W}.
R = {ρ1 . . . ρn } is a set of accessibility relations for each i = 1 . . . n, ρi ⊆ W × W.
Let w1 ∈ [w0 ]ρi denote that a possible world w1 is i-accessible from w0 .
M, w |= p
M, w |= ¬ϕ
M, w |= ϕ ∨ ψ
M, w |= Ki ϕ
iff
iff
iff
iff
{wi | g(w)(p) = True, w ∈ W}, p ∈ Φ.
M, w |= ϕ.
M, w |= ϕ or M, w |= ψ.
M, w |= ϕ, for all w ∈ [w]ρi .
Let Kj ψ be an L-formula, and let A = {K1 . . . Kn }, Ki ∈ A, i ∈ {1 . . . n} such
that Kj is in the syntactic scope of Ki . Now if B ⊆ A, then (Kj /B) ψ ∈ L∗ , Kj ∈
/
B. For example, K1 (K2 /K1 ) ϕ and K1 (ϕ ∧ (K2 /K1 ) ψ) are wffs of L∗ .
Every L∗ -formula ϕ defines a game G(ϕ, w, g) on a model M between two
teams of players, the team of falsifiers F = {F1 . . . Fn } and the team of verifiers
V = {V1 . . . Vk }, where w is a world and g is an assignment to the propositional
letters. The game G(ϕ, w, g) is defined by the following rules.
Knowledge Constructions for Artificial Intelligence
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(G.¬) If ϕ = ¬ψ, V and F change roles, and the next choice is in G(ψ, w, g).
(G.∨) If ϕ = (ψ∨θ), V chooses Left or Right, and the next choice is in G(ψ, w, g)
if Left and G(θ, w, g) if Right.
(G.Ki ) If ϕ = Ki ψ, and the game has reached w, Fj ∈ F chooses w1 ∈ [w]ρi ,
and the next choice is in G(ψ, w1 , g).
(G.(Ki /B)) If ϕ = (Ki /B) ψ, Ki ∈ B, and the game has reached w, then Fl ∈ F
chooses w1 ∈ W ‘independently’ of the choices made for the elements in B.
(G.at) If ϕ is atomic, the game ends, and V wins if ϕ true, and F wins if ϕ
false.
The formulas (Ki /B) ψ signal imperfect information: player choosing for Ki on
the left-hand side of the slash is not informed of the choices made for the elements
in B earlier in the game. Nothing is said about the accessibility relation, since
we want to leave the interpretation of these modalities open.
The purpose of V is to show that ϕ is true in M (M, w |=+ ϕ), and the
purpose of F is to show that ϕ is false in M (M, w |=− ϕ). If M, w |=+ ϕ,
V wins, and if M, w |=− ϕ, F wins. A strategy for a player in G(ϕ, w, g)
is a function assigning to each non-atomic subformula a member of the team,
outputting a possible world, a value in {Left, Right} (the connective information),
or an instruction to change roles (negation). A winning strategy is a strategy by
which a player can make operational choices such that every play results in a
win for him or her, no matter how the opponent chooses.
Let ϕ be an L∗ -formula. For any M, w ∈ W, M, w |=+ ϕ iff a strategy exists
which is winning for V in G(ϕ, w, g), and M, w |=− ϕ iff a strategy exists which
is winning for F in G(ϕ, w, g). A game is determined, iff for every play on ϕ,
either V has a winning strategy in G or F has a winning strategy in G. It is easy
to see that games for L∗ are not determined. From non-determinacy it follows
that the law of excluded middle ϕ ∨ ¬ϕ fails in L∗ . This is a common thing to
happen in logics with imperfect information.
Non-determinacy is related to partiality. A partial model is a triple M =
W, R, g, where g is a partial valuation function g : W → (Φ → {True, False}),
assigning to each proposition letter in Φ a subset g(Φ) of a set of possible worlds
W = {w0 . . . wn }. Partiality means that
M, w |=+ Ki ϕ iff M, w |=+ ϕ for all w ∈ W, w ∈ [w]ρi .
M, w |=− Ki ϕ iff M, w |=− ϕ for some w ∈ W, w ∈ [w]ρi .
An alternative way to approach partiality is by gts of imperfect information,
where partiality arises at the level of complex formulas, dispensing with partial models. One consequence is that semantic games generalise received partial
modal logics [5] to complete models.
3
Focussed Knowledge and Multi-agent Systems
There are important non-technical motivations as to why one should be interested in combining games with various modalities. For one thing, in quantified
extensions the combination gives rise to focussed knowledge.
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3.1
A. Pietarinen
Language and Semantics
Let the syntax for first-order epistemic logic Lωω consist of a signature, a logical
vocabulary, and rules for building up formulas: φ ::= P | Ki ϕ | ∀xϕ | ∃xϕ |
ϕ ∨ ψ | ¬ϕ | x y.
Let Qψ, Q ∈ {∀xj , ∃yj , Ki } be an Lωω -formula in the syntactic scope of
the elements in A = {K1 . . . Kn , ∀xk , ∃yk }. Then L∗ωω consists of wffs of Lωω
together with: if B ⊆ A, then (Q/B) ψ is an L∗ωω -formula, Q ∈ B. For example,
K1 ∃y(K2 /K1 y) Sxy ∈ L∗ωω . This hides the information about the choice for K1
and y at K2 .
Models and valuations have to take the world-boundedness of individuals into
account. A non-empty world-relative domain consisting of aspects of individuals
is Dwj . We skip the formal definitions here.
The semantics needs to be enriched by a finite number of identifying functions
(world lines), which extend the valuation g to a (partial) mapping from worlds to
W
, such that if w ∈ W and g is an identifying
individuals, that is, to g : X → Dw
i
function, then g(w) ∈ Dw . These functions imply that individuals have local
manifestations in any possible world.
The interpretation of the equality sign (identifying functional) is:
M, w0 , g |= x y iff for some wi , wj ∈ W, ∃f ∃h such that f (wi ) = h(wj ).
That is, two individuals are identical iff there are world lines f and h that pick the
same individuals in wi and in wj . World lines can meet at some world but then
pick different individuals in other worlds: the two-place identifying functional
operation spells out when they meet. Individuals within a domain of a possible
world are local and need to be cross-identified in order to be global and specific.
The informal game rules for L∗ωω are:
(G.∃x . . . Ki ) If Ki is in the syntactic scope of ∃x and the game has reached
w, the individual picked for x by a verifying player V has to be defined and
exist in all worlds accessible from the current one.
This rule is motivated by the fact that the course of the play reached at a certain
point is unbeknownst to F choosing for Ki . This approach leads to the notion
of specific focus.
(G.Ki . . . ∃x) If ∃x is in the scope of Ki , the individual picked for x has to be
defined and exist in the world chosen for Ki .
This leads to the notion of non-specific focus. Finally, the rule for the hidden
information says that
(G.Q/B) If ϕ = (Q/B) ψ, Q ∈ {∀x, Ki }, and the game has reached w, then if
Q = ∀x, F1 ∈ F chooses an individual from Dw1 , where w1 is the world from
which the first world in B has departed. The next choice is in G(ψ, w, g).
If Q = K1 then F1 ∈ F chooses a world w1 ‘independently’ of the choices
made for the elements in B, and the next choice is in G(ψ, w1 , g). Likewise
for V1 ∈ V .
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The notion of choosing independently is explained below. Other game rules are
ordinary.
Independent modalities mean that the player choosing for Ki is not informed
of the choices made for Kj , and hence Ki ’s are exempted from the syntactic scope
of Kj . This can be captured by taking hints from the theory of games. We will
apply a partitional information structure (Ii )i∈N in the corresponding extensiveform games, which partitions sequences of actions (histories) h ∈ H into equivalence classes (information sets) {Sji | Sji ∈ (Ii )i∈N , h ∼i h ∈ Sji , h, h ∈ H}. The
purpose of equivalence classes is to denote which histories are indistinguishable
to players.
Payoff functions ui (h) associate a pair of truth-values in {1, −1} to terminal
h ∈ H. Strategies are functions fi : P −1 ({i}) → A from histories where players
move to sequences of actions in A. If i is planning his decisions within the
equivalence class Sji annotated for him, his strategies are further required to be
uniform on indistinguishable histories h, h ∈ Sji , that is, fi (h) = fi (h ), i ∈ N .
This leads to an informal observation: tracing uniform strategies along the
game histories reveals in which worlds the specific focus is located. To see this,
it suffices to correlate information sets of an extensive game with world lines.
The clause ‘choosing independently’ that appeared in the above game rules
now means that players’ strategies have to be uniform on indistinguishable histories, that is, on worlds that players cannot distinguish.
The notion of uniformity puts some constraints on allowable models. At any
modal depth (defined in a standard way) there has to be the same number of
departing worlds. If we assume that players can observe the set of available
choices, the uniformity of strategies also requires that the departing worlds have
to coincide for all indistinguishable worlds. Independent Ki ’s can either refer to
simultaneous worlds accessible from the current one, or to detached submodels of
M . In the latter case we evaluate formulas in M, (w01 , w02 . . . w0n ), g. The models
would break into concurrent submodels, whence the designated worlds in each
submodel become independent.
3.2
A Case for Imperfect Information in Multi-agent Systems
Understanding knowledge of communicating multi-agent system benefits from
the kind of concurrency outlined above. Suppose a process U2 sends a message
x to U1 . We ought to report this by saying that ‘U2 knows what x is’, and
‘U1 knows that it has been sent’ (U1 might knows this because a communication
channel is open). This is already a rich situation involving all kinds of knowledge.
However, the knowledge involved in this two-agent system cannot be captured
in ordinary (first-order) epistemic logic.
In this system, what is involved is ‘U2 knows what has been sent’, as well as
‘U1 knows that something has been sent’. However, what is not involved is ‘U1
knows that U2 knows’, nor ‘U2 knows that U1 knows’. How do we combine these
clauses? It is easy to see that the three formulas ∃xKU2 Mess(x) ∧ KU1 ∃yMess(y),
KU1 ∃x(Mess(x) ∧ KU2 Mess(x)), and ∃x(KU2 Mess(x) ∧ KU1 ∃yMess(y) ∧ x 308
A. Pietarinen
y) all fail. So does an attempt that distinguishes between a message whose
content is known (‘Cont(x)’), and a message that has been sent (‘Sent(y)’):
∃x∃y((KU1 Cont(x) = y) ∧ KU2 Sent(x)). For now U2 comes to know what has
been sent, which is too strong.
Hence, what we need is information hiding:
∃xKU2 (KU1 /KU2 x)(∃y/KU2 x) (Mess(x) ∧ x y).
(1)
In concurrent processing for quantified multi-modal epistemic logic, the notion
of focussed knowledge is thus needed.
4
Further Knowledge Constructions and Semantic Games
Here we move back to propositional logics and observe how games can be applied to various other constructions of knowledge. What we get is a range of
partial logics for different modalities. The purpose is to show that (i) gts is
useful for epistemic logics in artificial intelligence as it unifies the semantic outlook to different notions of knowledge; (ii) if games are non-determined, one
gets partialised versions of these logics; (iii) if the possible-worlds semantics is
augmented with inaccessible worlds, non-monotonic epistemic logic can be built
upon game-theoretic principles.
4.1
Partial Only Knowing
Let us begin with the logic of only knowing, partialise it, and then define semantics game rules for it. ‘Only knowing’ (Oi ϕ, or ‘exactly knowing’) means that we
do not have worlds in a model where ϕ could be true other than the accessible
ones [4,6]. Informally, such a description picks a model where the set of possible
worlds is as large as possible. This is because the larger set of possible worlds,
the less knowledge an agent has.
To partialise the logic of only knowing, we define
M, w |=+ Oi ϕ iff M, w |=+ ϕ ⇔ w ∈ [w]ρi , for all w ∈ W.
M, w |=− Oi ϕ iff M, w |=− ϕ ⇔ w ∈ [w]ρi , for some w ∈ W.
Let a logic based on these be LO . The operator Oi can also be understood in
terms of another operator Ni : Oi ϕ ::= Ki ϕ ∧ Ni ¬ϕ.
M, w |=+ Ni ϕ iff for all w ∈ W, M, w |=+ ϕ,
M, w |=− Ni ϕ iff for some w ∈ W, M, w |=− ϕ,
where W is the set of inaccessible worlds, W = W ∗ ∪ W, W ∗ is the set of
accessible worlds.
A game G(ϕ, w, g) for LO -formulas ϕ, with a world w ∈ W and an assignment
g to the propositional letters is defined as a set of classical rules plus:
(G.Oi ) If ϕ = Oi ψ, and the game has reached w ∈ W, F chooses between Ki ψ
and Ni ¬ϕ, and the game continues with respect to that choice.
Knowledge Constructions for Artificial Intelligence
309
(G.Ni ) If ϕ = Ni ¬ψ, and the game has reached w ∈ W, F chooses w ∈ W,
and the next choice is in G(¬ψ, w , g).
(G.at) If ϕ is atomic, the game ends. F wins, if M, w |=+ ϕ, w ∈ W, or
if not: M, w |=+ ϕ, w ∈ W ∗ . V wins, if M, w |=− ϕ, w ∈ W, or if
M, w |=+ ϕ, w ∈ W ∗ .
Strategies will operate on all worlds, including inaccessible ones. In general, we
dispense with the accessibility relation and assume that also inaccessible worlds
can be chosen. This is natural, because player knowledge and agent knowledge
mean different things. (Further restrictions can be that any such inaccessible
world can be chosen only once within a play of the game, etc.) Letting ϕ be an
LO -formula, then for any model M , a valuation g, and w ∈ W, ϕ is true iff there
exists a strategy which is winning for V in G(ϕ, w, g), and false iff there exists a
strategy which is winning for F in G(ϕ, w, g).
By imposing the uniformity condition on strategies, the logic of only knowing
becomes partial and the underlying games non-determined, even if the propositions were completely interpreted. Traditionally, the logic of only knowing was
developed in order to create semantic non-monotonicity by using stable sets [6].
An alternative game-theoretic method of creating non-monotonicity can thus be
obtained by assuming that players can choose inaccessible worlds in addition to
the accessible ones.
4.2
Partial Common Knowledge
The modal operator EI ϕ means that ‘everyone in the group of agents I ⊆ Ag
(the set of all agents) knows ϕ’, and CI ϕ means that ‘it is common knowledge among the group of agents I that ϕ’. The partialised version of the logic
augmented with these operators has
M, w |=+ EI ϕ iff M, w |=+ Ki ϕ for all i ∈ I.
M, w |=− EI ϕ iff M, w |=− Ki ϕ for some i ∈ I.
Let EI∗ ϕ be a reflexive and transitive closure on EI0 ϕ∪EI1 ϕ∪· · ·∪EIk+1 ϕ, where
EI0 ϕ = ϕ, EI1 = EI ϕ, and EIk+1 ϕ = EI EIk ϕ. Hence:
M, w |=+ CI ϕ iff M, w |=+ EI∗ ϕ.
M, w |=− CI ϕ iff M, w |=− Ki ϕ for some i ∈ I.
A game-theoretisation of common knowledge is this. A world w is I-reachable
from w if there exists a sequence of worlds w1 . . . wk , w1 = w, wk = w and for all
j, 0 ≤ j ≤ k − 1, there is i ∈ I ⊆ Ag for which wj+1 ∈ [wj ]ρi for some k ≥ 0 [1].
A game G(ϕ, w, g) for formulas of LC , with a world w and an assignment g to
the propositional letters has two additional rules. (The latter is to have duality.
Since there can be infinite plays in the game tree, winning strategies are to be
slightly modified in order to account for this.)
(G.CI ) If ϕ = CI ψ, and the game has reached w, F chooses some w that is
I-reachable from w, and the game continues as G(ψ, w , g).
(G.¬CI ¬) If ϕ = ¬CI ¬ψ, and the game has reached w, V chooses some w
that is I-reachable from w, and the game continues as G(ψ, w , g).
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A. Pietarinen
By letting ϕ be an LC -formula, then for any model M , a valuation g, and w ∈ W,
ϕ is true iff there exists a strategy which is winning for V in G(ϕ, w, g), and false
iff there exists a strategy which is winning for F in G(ϕ, w, g). This is how we
bring gts to bear on common knowledge, certainly an important notion both in
AI and Game Theory.
It is known that common knowledge cannot be attained if communication is
not taken to be reliable. It should be noted that any reference to communication
pertains to agents’ knowledge, not to the information players have, and hence it
is safe for us to consider partialised common knowledge.
4.3
Partial Distributed Knowledge
Let the operator DI ϕ mean ‘it is distributed knowledge among the group of
agents I that ϕ’. Partial distributed knowledge based on this notion can be
defined as follows:
M, w |=+ DI ϕ iff M, w |=+ ϕ for each w ∈ i∈I [w]ρi .
M, w |=− DI ϕ iff M, w |=− ϕ for some w ∈ i∈I [w]ρi .
A game G(ϕ, w, g) for LC+D , with a world w and an assignment g to the propositional letters has two additional rules:
w, F chooses some w ∈
(G.D
I ) If ϕ = DI ψ, and the game has reached
i∈I [w]ρi , and the game continues as G(ψ, w , g).
(G.¬DI ¬)
If ϕ = ¬DI ¬ψ, and the game has reached w, V chooses some
w ∈ i∈I [w]ρi , and the game continues as G(ψ, w , g).
This definition amounts to partiality, as games for LC+D are non-determined.
4.4
Some Further Variations
Finally, some further knowledge constructions can be envisaged, by a combination of previous systems. For instance, we can have a logic with ‘only common
knowledge’ (CIO ϕ). To see this, let us start with the notion of ‘everybody only
knows’ (EIO ϕ):
M, w |=+ EIO ϕ iff M, w |=+ Oi ϕ for all i ∈ I.
M, w |=− EIO ϕ iff M, w |=− Oi ϕ for some i ∈ I.
‘Only common knowledge’ is now closure on ‘everybody only knows’, along the
lines already described. For falsification, it suffices that ‘only knowing’ fails.
Thus:
M, w |=+ CIO ϕ iff M, w |=+ EI∗ ϕ, for all i ∈ I.
M, w |=− CIO ϕ iff M, w |=− Oi ϕ, for some i ∈ I.
One can then devise games for these in a straightforward way. Also other combinations are possible. In general, we can have ‘non-standard’ partiality:
M, w |=+ Ki◦ ϕ iff M, w |=+ ϕ for all w ∈ W, w ∈ [w]ρi .
M, w |=− Ki◦ ϕ iff not M, w |=+ ϕ for some w ∈ W, w ∈ [w]ρi .
Knowledge Constructions for Artificial Intelligence
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M, w |=+ Ki# ϕ iff not M, w |=− ϕ for all w ∈ W, w ∈ [w]ρi .
M, w |=− Ki# ϕ iff M, w |=− ϕ for some w ∈ W, w ∈ [w]ρi .
The formula Ki◦ ϕ captures the idea that the sentence is true when ϕ is true
in all accessible worlds, and false when ϕ is not true in some accessible world.
Ki# ϕ, on the other hand, says that the sentence is true when ϕ is not false
in every accessible worlds, and false when ϕ is false in some accessible world.
Clearly the standard interpretation subsumes the truth-conditions for Ki◦ ϕ and
the falsity-conditions for Ki# ϕ. The duals L◦i and L#
i are defined accordingly.
Games for non-standard clauses change the rules for winning conditions to
weaker ones. For ◦-modalities and #-modalities we have, respectively:
(G.at◦ ) If ϕ atomic, game ends. V wins if ϕ true, and F wins if ϕ not true.
(G.at# ) If ϕ atomic, game ends. V wins if ϕ not false, and F wins if ϕ false.
5
Concluding Remarks
Our approach is useful for sentences using non-compositional information hiding.
The general perspective is that games unify the semantics for modalities, and
that their imperfect information versions partialise logics extended with new
operators.
Further uses and computational aspects of the above knowledge constructions have to be left for future occasions. In general, areas of computer science
and AI where these constructions and gts may turn out to be useful include uncertainty in AI and in distributed systems, intensional dimensions of knowledge
representation arising in inter-operation, unification of verification languages for
multi-agent systems [9], reasoning about secure information flow, strategic meanings of programs, modularity, and other information dependencies.
References
1. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge.
Cambridge: MIT Press (1995)
2. Hintikka, J.: Knowledge and Belief. Cornell University Press, Ithaca (1962)
3. Hintikka, J., Sandu, G.: Game-theoretical semantics. In: van Benthem, J., ter
Meulen, A. (eds): Handbook of Logic and Language. Elsevier, Amsterdam (1997)
361–410
4. Humberstone, I.L.: Inaccessible worlds. Notre Dame Journal of Formal Logic 24.
(1981) 346–352
5. Jaspars, J., Thijsse, E.: Fundamentals of partial modal logic. In: Doherty, P. (ed.):
Partiality, Modality, and Nonmonotonicity. Stanford, CSLI (1996) 111–141
6. Levesque, H.J.: All I know: A study in autoepistemic logic. Artificial Intelligence
42. (1990) 263–309
7. Pietarinen, A.: Intentional identity revisited. Nordic Journal of Philosophical Logic
6. (2001) 147–188
8. Sandu, G., Pietarinen, A.: Partiality and games: Propositional logic. Logic Journal
of the IGPL 9. (2001) 107–127.
9. Wooldridge, M.: Semantic issues in the verification of agent communication languages. Journal of Autonomous Agents and Multi-Agent Systems 3. (2000) 9–31