Linear Algebraic Semantics for Multi-agent Communication

Ryo Hatano, Katsuhiko Sano and Satoshi Tojo

School of Information Science, Japan Advanced Institute of Science and Technology,

1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan

Keywords:

Logic of Belief, Belief Revision, Dynamic Epistemic Logic, Propositional Dynamic Logic, Linear Algebra.

Abstract:

When we study multi-agent communication system, it forces us to manage an existence of communication

channels between agents, such as phone numbers or e-mail addresses, while ordinary modal logic for multi-

agent system does not consider the notion of channel. This paper proposes a decidable and semantically

complete logic of belief with communication channels, and then expands the logic with informing action

operators to change agents’ beliefs via communication channels. Moreover, for a better formalism for handling

these semantics efﬁciently, we propose a linear algebraic representation of these. That is, with the help of

Fitting (2003) and van Benthem and Liu (2007), we reformulate our proposed semantics of the doxastic static

logic and its dynamic extensions in terms of boolean matrices. We also implement and publicize a calculation

system of our matrix reformulations as an open system on the web.

1 INTRODUCTION

One of the most important aspects of multi-agent

communication is changes of an agent’s knowledge or

belief (G¨ardenfors, 2003). Nowadays, such changes

are well-discussed in terms of modal logic, as dy-

namic epistemic logic (van Ditmarsch et al., 2007).

For example, public announcement logic, proposed

by Plaza (Plaza, 1989), can capture how an agent’s

knowledge change after a piece of information is pub-

licly announced to all the agents, while we do not as-

sume any structure among agents. On the other hand,

in communication of multiple agents, we can natu-

rally consider the existence of channels between them

(Barwise and Seligman, 1997), e.g., phone numbers

or e-mail addresses. Then, communicability in those

agents can be represented in a directed graph, where

a vertex is an agent and an edge a channel.

There are several studies integrating the notion of

structure among agents into dynamic epistemic logic.

(Seligman et al., 2011) proposes a two-dimensional

modal logic which can handle both agents’ knowl-

edge and a friendship relation between agents. Based

on the two-dimensional framework, (Sano and Tojo,

2013) implemented the idea of communication chan-

nel in terms of a modal operator and studies belief

changes of agents, where they raised the following re-

quirements:

(R1) An effect of an informing action is restricted to

some speciﬁed agents determined by communi-

cation channels.

(R2) An existence of communication channel be-

tween agents depends on a given situation, i.e.,

it is not constant or rigid for all situations.

One of the deﬁciencies of the two dimensional

framework is that it is still unknown whether the re-

sulting logics in (Seligman et al., 2011; Sano and

Tojo, 2013) are decidable, i.e., we can effectively test

if a given formula is a theorem of a given logic. One

of the purposes of this paper is to propose a decidable

multi-agent doxastic logic which satisﬁes the two re-

quirements above and can talk about communication

channels among agents. Instead of communication

channel as a modal operator,we implement the notion

of channel as a constant symbol c

whose reading is

‘there is a channel from agent a to agent b’. More-

over, instead of public announcement operators, this

paper proposes two dynamic operators satisfying the

requirement (R1), called semi-private announcement

and introspective announcement operators.

When we study logic of multi-agent system, it

forces us to manage many indices, such as agent IDs

and names of the worlds in our syntax and its seman-

tics. What seems to be lacking is an introduction

of a better formalism or notation for handling such

many indices. Thus far, (Fitting, 2003) proposed a

linear algebraic reformulation of Kripke semantics of

174

Hatano R., Sano K. and Tojo S..

Linear Algebraic Semantics for Multi-agent Communication.

DOI: 10.5220/0005219001740181

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence (ICAART-2015), pages 174-181

ISBN: 978-989-758-073-4

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

modal logic. (Tojo, 2013) has employed the notion

of boolean matrix and tried to integrate the notion of

communication channel with dynamic logic of mul-

tiple agents’ beliefs in term of linear algebra. In this

research, we give a more rigorous logical formalisms

to (Tojo, 2013). That is, we reformulate our proposed

doxastic logic and its dynamic extensions in terms of

boolean matrices.

To sum up, this paper ﬁrst proposes a decidable

multi-agent doxastic logic and its dynamic extensions

with two informing action operators, and then refor-

mulate our Kripke semantics in terms of boolean ma-

trices.

This paper is organized as follows. Section 2 in-

troduces a static logic of agents’ belief equipped with

the notion of channel between agents and establish

that all the valid formulas on all the ﬁnite Kripke mod-

els for our syntax is completely axiomatizable (The-

orem 1). Moreover, our proposed axiomatization is

decidable (Theorem 2). In order to deal with changes

of agents’ belief via communication channel, Sec-

tion 3 provides two dynamic operators to our syntax

of static logic with sets of reduction axioms. Follow-

ing the idea by (Fitting, 2003), Section 4 reformu-

lates our Kripke semantics in terms of boolean ma-

trix. With the help of (Van Benthem and Liu, 2007),

Section 5 reveals that we can regard our two dynamic

operators as program terms in propositional dynamic

logic and also reformulates the semantics of two op-

erators in terms of boolean matrix. Section 6 use our

boolean matrix reformulation to present an algorithm

for checking agent’s belief at a given world and an al-

gorithm for rewriting a given Kripke model by one of

our dynamic operators. Finally, Section 7 concludes

this paper.

Related Works. Here we comment on linear alge-

braic approach to multi-agent belief revision. (Fitting,

2003) proposed a linear algebraic approach to Kripke

semantics, but he did not consider any dynamic oper-

ators. On the other hand, we reformulate (Van Ben-

them and Liu, 2007)’s idea of relation changer over

propositional dynamic logic in terms of matrices and

provide a linear algebraic treatment with our dynamic

operators. In this sense, this paper can be regarded as

a generalization of (Fitting, 2003) to dynamic exten-

sions. While (Liau, 2004) also used boolean matri-

ces to represent an accessibility relation of an agent

and (Fusaoka et al., 2007) used real-valued matrices

to represent qualitative belief change in multi-agent

setting, both of them did not provide any concrete ax-

iomatization of logics they study.

2 STATIC LOGIC FOR AGENTS’

BELIEF

2.1 Syntax and Semantics

This section introduces a modal epistemic language

which enables us to formalize agents’ beliefs and

communication channels.

Let G be a ﬁxed ﬁnite set of agents. Our syntax

L consists of the following vocabulary: a ﬁnite set

Prop = { p, q, r, ... } of propositional letters; boolean

connectives¬, ∨; belief operators B

(a ∈ G); channel

constants c

(a, b ∈ G). A set of formulas of L is

inductively deﬁned as:

ϕ ::= p | c

| ¬ϕ | ϕ∨ ψ | B

where p ∈ Prop, a, b ∈ G. We deﬁne

ϕ := ¬B

¬ϕ

whose reading is ‘agent a considers it possible that ϕ’.

We also introduce the boolean connectives ∧, →, ↔

as ordinary abbreviations. B

p stands for ‘agent a be-

lieves that p’ and c

is to read ‘there is a communica-

tion channel from a to b’. Then, let us provide Kripke

semantics with our syntax. A model M is a tuple

(W, (R

)

a∈G

, (C

)

a,b∈G

,V) where W is a non-empty

set of worlds, called domain, R

⊆ W × W, C

⊆ W

is a channel relation such that C

= W for all a ∈ G,

and V : Prop → P (W) is a valuation function. Note

that we require C

= W for all a ∈ G in order to cap-

ture our notion of communication channel. A frame

(denoted by F, etc.) is the result of dropping a valua-

tion function from a model.

Given any model M, any world w ∈ W, and any

formula ϕ, we deﬁne the satisfaction relation M, w |=

ϕ inductively as follows:

M, w |= p iff w ∈ V(p)

M, w |= c

iff w ∈ C

M, w |= ¬ϕ iff M, w 6|= ϕ

M, w |= ϕ∨ ψ iff M, w |= ϕ or M, w |= ψ

M, w |= B

ϕ iff M, v |= ϕ for all v with wR

We deﬁne the truth set JϕK

of ϕ in M by JϕK

{w ∈ W |M, w |= ϕ}. ϕ is valid on M if M, w |= ϕ

for all worlds w ∈ W. We say that ϕ is valid in a class

of Kripke models if ϕ is valid on M belongs to the

class. It is clear that c

is always valid in any Kripke

model M. Moreover, given any Kripke model M, it

is easy to see that all the axioms in Table 1 are valid

in M and all the rules of Table 1 preserve validity on

Example 1 (Running Example). Let G = {a, b}. De-

ﬁne M (see Figure 1) by: W = { w

, w

}, R

{(w

, w

), (w

, w

), (w

, w

), (w

, w

), (w

, w

)}, R

= W × W, V(p) = { w

}, C

= { w

, w

}, C

LinearAlgebraicSemanticsforMulti-agentCommunication

175

Table 1: Hilbert-style Axiomatization K

of Static Logic.

(Taut) ϕ, ϕ is a tautology

) B

(ϕ → ψ) → (B

ϕ → B

ψ) (a ∈ G)

(Selfchn) c

(a ∈ G)

(MP) From ϕ and ϕ → ψ, infer ψ

(Nec

) From ϕ, infer B

ϕ (a ∈ G)

Figure 1: Accessibility relations of agents a and b.

= C

= W. Agent a believes p in w

and ¬p in

, but he/she is not sure of p or ¬p in w

. On the

other hand, agent b does not believe p nor ¬p at all

the worlds. There are channels from a to b in w

and

, but there is no channel between them in w

2.2 Hilbert-style Axiomatization

The following theorem implies that we can axiom-

atize all the valid formlas on the class of all ﬁnite

Kripke models. The restriction to the ﬁnite models

is important for us, since our matrix representation of

Kripke model is always in terms of ﬁnite matrix.

Theorem 1. For all formulas ϕ in L , ϕ is a theorem

in K

of Table 1 iff ϕ is valid on the class of all ﬁnite

Kripke models.

Proof. (Outline) Since the soundness is easy to es-

tablish, we focus on the completeness with respect to

the class of all ﬁnite Kripke models. We show that

any unprovable formula ϕ in K

is falsiﬁed in a ﬁ-

nite Kripke model. Let ϕ be an unprovable formula

in K

. First, we deﬁne the canonical model M where

ϕ is falsiﬁed at some point of M. Second, since the

domain of the canonical model is inﬁnite, we employ

the technique of ﬁltration to boil the model down to

a ﬁnite model where ϕ is still falsiﬁed at some point.

For both steps, we basically follow the standard tech-

niques, e.g. found in (Blackburn et al., 2002).

We say that a set Γ of formulas is K

-consistent

(for short, consistent) if

′

is unprovable in K

, for

all ﬁnite subsets Γ

′

of Γ, and that Γ is maximally con-

sistent if Γ is consistent and ϕ ∈ Γ or ¬ϕ ∈ Γ for all

formulas ϕ. Note that ψ is unprovable in K

iff ¬ψ

is K

-consistent, for any formula ψ. We deﬁne the

canonical model (W, (R

)

a∈G

, (C

)

a,b∈G

,V) by:

• W is the set of all maximal consistent sets;

• ΓR

∆ iff (B

ψ ∈ Γ implies ψ ∈ ∆) for all ψ;

• C

:= {Γ ∈ W |c

∈ Γ};

• Γ ∈ V(p) iff p ∈ Γ.

Then, we can show the following equivalence (Truth

Lemma (Blackburn et al., 2002, Lemma 4.21)):

M, Γ |= ψ iff ψ ∈ Γ for all formulas ψ and Γ ∈ W,

where we note that we need to use the axiom (K

)

and the rule (Nec

) for the case where ψ is of the

form of B

γ.) Given any unprovable formula ϕ in

, we can ﬁnd a maximal consistent set ∆ such that

¬ϕ ∈ Γ (where we need to use (Taut) and (MP)).

Then, by the equivalence above, ϕ is falsiﬁed at ∆ of

the canonical model M, where we can assure that C

= W for all a ∈ G by the axiom (Selfch). This ﬁnishes

the ﬁrst step of our proof.

Let us move to the second step. Let N =

(W, (R

)

a∈G

, (C

)

a,b∈G

,V) be a Kripke model and Γ

a ﬁnite set of formulas that is closed under taking sub-

formulas. Without loss of generality, we can assume

that Γ contains c

for all agents a occurring in Γ (oth-

erwise, we can just add c

s to Γ for all as occurring

in Γ where note that the number of such as is ﬁnite).

Let us deﬁne an equivalence relation ∼

by w ∼

′

iff (N, w |= ψ iff N, w

′

|= ψ) for all ψ ∈ Γ. Then, we

deﬁne a ﬁnite model N

as follows:

• W

:= { [w]|w ∈ W }, where [w] is the equivalence

class of w with respect to ∼

• [w]R

′

] iff vR

′

for some v ∈ [w] and v

′

∈ [w

′

• C

:= {[w] |w ∈ C

} for c

∈ Γ.

• [w] ∈ V

(p) iff w ∈ V(p) for p ∈ Γ.

Remark that C

always holds, since we assumed that

∈ Γ for all as occurring in Γ. Remark also that

the size of W

is less than or equal to 2

#Γ

, hence

ﬁnite. By induction on ψ ∈ Γ, we can show that

N, w |= ψ iff N, [w] |= ψ for all w ∈ W (the proof can

be found in (Blackburn et al., 2002, Theorem 2.39)).

Recall that any unprovable formula ϕ in K

is falsi-

ﬁed at Γ of the canonical model M. Now we can

apply the ﬁltration technique to obtain a ﬁnite model

where ϕ is falsiﬁed at [∆] and Γ is the union of

| a occurs in ϕ} and the ﬁnite set Sub(ϕ) of all

subformulas of ϕ and this ﬁnishes the second (and

last) step of our proof.

Theorem 2. K

is decidable.

Proof. When ϕ is unprovable in K

, Theorem 2 tells

us that ϕ has a ﬁnite countermodel. Since we can re-

cursively check if a given ﬁnite model satisﬁes the

condition C

= W for all agents a ∈ G (note G is

ﬁnite), we can construct an effective procedure gen-

erating all the ﬁnite Kripke models and checking if

ϕ is falsiﬁed at some point of a ﬁnite model. To-

gether with an effective procedure of enumerating all

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

176

the theorems of K

, we obtain the decision procedure

of Theoremhood of K

3 DYNAMIC OPERATORS FOR

CHANNEL COMMUNICATION

This section introduces two dynamic operators which

allows us to talk about agents’ belief changes in

terms of informing action. The ﬁrst dynamic oper-

ator (semi-private announcement) speciﬁes both the

sender and the receiver, but the second operator (in-

trospective announcement via channel) just speciﬁed

the sender agents and we need to calculate the re-

ceivers of the information via communication chan-

nels.

3.1 Semi-private Announcement

One of the most well-known dynamic operators is

public announcement operator (Plaza, 1989), but our

operator of this section differs from it by the follow-

ing requirement:

(R3) Our introducing operators are semi-private or

non-public announcements to some speciﬁc

agents. We assume that an agent a can send a mes-

sage to an agent b only when there is a channel

from a to b.

When an agent informs one of the other agents of

something, our basic assumption is that we need a

(context-dependent) channel between those agents.

The notion of channel was formalized as channel

propositions c

Let us denote our intended dynamic operator by

[ϕ↓

], whose reading is ‘after the agent a informs the

agent b of the message ϕ via channel’. Our intended

reading of [ϕ↓

]ψ is ‘after the agent a informs the

agent b to ϕ, ψ’. We provide the semantic clause for

[ϕ↓

]ψ on a model M = (W, (R

)

a∈G

, (C

)

a,b∈G

,V) is

given as follows:

M, w |= [ϕ↓

]ψ iff M

ϕ↓

, w |= ψ

where M

ϕ↓

= (W, (R

′

)

a∈G

, (C

)

a,b∈G

,V) and

′

)

c∈G

is deﬁned as: if c = b, for all x ∈ W, we set

′

(x) :=

(

(x) ∩ JϕK

if M,x |= B

ϕ∧ c

(x) otherwise.

If c 6= b, R

′

:= R

. Semantically speaking, [ϕ↓

] re-

stricts b’s attention to the ϕ’s worlds if there is a chan-

nel from the agent a to b and agent a believes ϕ. Oth-

erwise, the action [ϕ↓

] will not change b’s belief.

Table 2: Hilbert-style Axiomatization K

c[ ·↓

]

In addition to all the axioms and rules of K

, we add:

[ϕ↓

]p ↔ p,

[ϕ↓

↔ c

[ϕ↓

]

ψ ↔

[ϕ↓

]ψ,

[ϕ↓

](ψ∨ χ) ↔ [ϕ↓

]ψ∨ [ϕ↓

]χ,

[ϕ↓

ψ ↔ B

[ϕ↓

]ψ (c 6= b)

[ϕ↓

ψ ↔ ((c

∧ B

ϕ) → B

(ϕ → [ϕ↓

]ψ))∧

(¬(c

∧ B

ϕ) → B

[ϕ↓

]ψ)

(Nec

[ϕ↓

]

) From ψ, infer [ϕ↓

]ψ

Theorem 3. For all formulas ϕ in the expanded syn-

tax L with [ψ↓

], ϕ is a theorem in K

c[ ·↓

]

of Table 2

iff ϕ is valid on the class of all ﬁnite Kripke models.

Proof. By ⊢ ψ (or ⊢

ψ), we mean that ψ is a theorem

of the axiomatization K

(or, K

c[ ·↓

]

, respectively.)

The soundness of the axioms is easy. One can also

check that the necessitation rule (Nec

[ϕ↓

]

) preserves

the validity on the class of all ﬁnite models. As for the

completeness part, we can reduce the completeness of

our dynamic extension to the static counterpart (i.e.,

Theorem 1) as follows. With the help of the axioms

of Table 2, we can deﬁne a mapping t sending a for-

mula ψ of the expanded syntax (we denote this by L

below) with the dynamic operators [ϕ↓

] to a formula

t(ψ) of the original syntax L , where we start rewrit-

ing the innermost occurrences of [ϕ↓

]. For example,

t([ϕ↓

(p ∨ c

)) := B

(p ∨ c

). For this mapping

t, we can show that ψ ↔ t(ψ) is valid on all ﬁnite

models and ⊢

ψ ↔ t(ψ). Then, we can proceed as

follows. Fix any formula ψ of L

such that ψ is valid

on all ﬁnite models. By the validity of ψ ↔ t(ψ) on

all ﬁnite models, we obtain that t(ψ) is valid on all

ﬁnite models. By Theorem 1, ⊢ t(ψ), which implies

⊢

t(ψ). Finally, it follows from ⊢

ψ ↔ t(ψ) that

⊢

ψ, as desired.

Example 2. In Example 1, we obtain the truth of

[p↓

p at w

, i.e., ‘after agent a informs agent b

of the message ϕ via channel, agent b comes to be-

lieve p’ in w

. Figure 2 is the updated model of M by

[p↓

]. On the other hand, agent a does not have any

channel to b in w

, and so, the accessible worlds from

will be unchanged even after the update of M by

[p↓

]. Therefore, [p↓

p is false at w

. Similarly,

agent a does not believe ¬p in w

, i.e., B

¬p fails in

, and so, the informing action [p↓

] will not change

the accessible worlds from w

LinearAlgebraicSemanticsforMulti-agentCommunication

177

Figure 2: Updated accessibility relation of agent b.

Table 3: Hilbert-style Axiomatization K

c[·↓

]

In addition to all the axioms and rules of K

, we add:

[ϕ↓

]p ↔ p,

[ϕ↓

↔ c

[ϕ↓

]

ψ ↔

[ϕ↓

]ψ,

[ϕ↓

](ψ∨ χ) ↔ [ϕ↓

]ψ∨ [ϕ↓

]χ,

[ϕ↓

ψ ↔ (

b∈H

∧ B

ϕ) → B

(ϕ → [ϕ↓

]ψ))

∧(¬(

b∈H

∧ B

ϕ)) → B

[ϕ↓

]ψ)

(Nec

[ϕ↓

]

) From ψ, infer [ϕ↓

]ψ

3.2 Introspective Announcement Via

Communication Channels

In the dynamic operator [ψ↓

], we speciﬁed a and b

as the sender and the receiver of the information ϕ,

respectively. Even so, we may consider the situa-

tion where more than one agents, say a and b, send

a piece of information to the other agents, and who

will receive the information may change, depending

on communication channels between agents. In this

sense, we do not specify the receivers in advance here.

Rather, we calculate the receivers of the information

from the senders and the communication channels.

We may expand our static syntax L with a dynamic

operator [ϕ↓

] (H ⊆ G) whose reading is ‘after a

group H of agents sends a piece ϕ of information via

communication channels’. Given a Kripke model M

= (W, (R

)

a∈G

, (C

)

a,b∈G

,V) and a world w ∈ W, we

deﬁne the semantics of [ϕ↓

]ψ by:

M, w |= [ϕ↓

]ψ iff M

ϕ↓

, w |= ψ,

where M

ϕ↓

= (W, (R

′

)

a∈G

, (C

)

a,b∈G

,V) and R

′

deﬁned as follows: for all w ∈ W, if there is some

b ∈ H such that w ∈ C

and M, w |= B

ϕ, we put

′

(w) := R

(w) ∩ JϕK

Otherwise, we put R

′

(w) := R

(w).

By the similar argument to Theorem 3, we can

prove the completeness theorem for K

c[ ·↓

]

over the

class of all the ﬁnite Kripke models.

Theorem 4. For all formulas ϕ in the expanded syn-

tax L with [ψ↓

], ϕ is a theorem in K

c[ ·↓

]

of Table 3

iff ϕ is valid on the class of all ﬁnite Kripke models.

Example 3. In Example 1, let H = { a} be a group

of senders. Then, when we focus on the world w

, we

can calculate the receivers by the calculation just be-

fore this example and specify the receivers as {a, b},

since there is a channel from a to b in w

and a be-

lieves p in w

. So, we obtain the truth of [p↓

at w

, i.e., ‘after the group of agent H sends a piece

p of information via communication channel, agent

b comes to believe p’ in w

. Moreover, the updated

model of M by [p↓

] is the same as Figure 2.

However, when we change the group of senders

to H

′

= { b}, agent b does not believe p in w

(i.e.,

p is false in w

), and so, the accessible worlds from

will be unchanged even after the update of M by

[p↓

′

]. Therefore, [p↓

′

p is still false at w

4 MATRIX REPRESENTATION

OF KRIPKE SEMANTICS

A usual Kripke frame (W, R) (for a single agent) can

be regarded as a directed graph, i.e., a set W of possi-

ble worlds corresponds to a set of nodes, and a set R

of accessibility relation corresponds to a set of edges.

Generally speaking, such set of edges can be written

as a boolean matrix. Therefore, the accessibility rela-

tion (= a belief state of an agent) can be represented in

a matrix. In this case, the accessibility from possible

world i to j can be mapped to the (i, j)-element of the

matrix. In what follows, we use M(m × n) to mean

the set of all m× n-boolean matrix.

Let us provide a matrix representation of our no-

tions of frame and model. First, we start with frames.

Given any Kripke frame F = (W, (R

)

a∈G

, (C

)

a,b∈G

)

with #W = n, we write W = { w

, w

, . . . , w

} and de-

ﬁne matrix representations of C

and R

as follows.

In accordance with C

⊆ W (a, b ∈ G), C

is a

matrix in M(n × 1), i.e., a column vector where the

k’s component is 1 if w

∈ C

, otherwise 0. In gen-

eral, given any relation R ⊆ W ×W, R

is a matrix in

M(n× n) such that

(i, j) =

(

1 if (w

, w

) ∈ R

0 otherwise

Now we move to deﬁne a matrix representation of

a model M = (W, (R

)

a∈G

, (C

)

a,b∈G

,V). Here we

assume that the number #Prop of propositional letters

is m and #W of possible worlds is n. Our matrix repre-

sentation of V(p) is similar to a channel relation C

That is, V(p)

is a matrix in M(n × 1) (= a column

vector) where the k’s component is 1 if w

∈ V(p),

otherwise 0.

Now we can rewrite Kripke semantics to our syn-

tax in terms of matrix. We inductively associate

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each formula ϕ of L with a column vector kϕk

∈

M(n× 1) as follows:

kpk

:= V(p)

:= C

k¬ϕk

kϕk

ϕk

:= R

kϕk

kϕ∨ ψk

:= kϕk

+ kψk

where, for X ∈ M(n × n),

X means the boolean com-

plementation of X. For the dual

of B

, it is easy

to see that k

ϕk

= R

kϕk

. If the underlying

model is clear from the context, we drop the subscript

‘M’ from kϕk

. We use kϕk

to means the i-th

component kϕk(i) of the column vector kϕk

, i.e.,

the truth value of the formula ϕ at w

of M.

Example 4. kB

in Example 1 is calculated as:

kpk





1 1 1

0 1 0

0 0 1





























This result coincides with our explanation in Exam-

ple 1 (recall also Figure 1).

This match up can be captured by the following

proposition.

Proposition 5. Given any ﬁnite model M and any

formula ϕ of L, we can show that (JϕK

)

= kϕk

5 MATRIX REPRESENTATION

OF DYNAMIC OPERATORS

Given a Kripke model M with a domain W

= { w

, . . . , w

}, we may easily rewrite semantic

clauses of [ϕ↓

] and [H↓

] in terms of matrix such

as: k[ϕ↓

]ψk

:= kψk

ϕ↓

and k[H ↓

]ψk

kψk

ϕ↓

where k[ϕ↓

]ψk

and k[H↓

]ψk

are ma-

trices in M(n×1). However, it is not so clear if we can

capture processes of updating M to M

ϕ↓

and M

ϕ↓

in terms of operations over matrices. (Van Benthem

and Liu, 2007) propose a general framework of up-

dating agents’ accessibility relations in terms of pro-

gram term of propositional dynamic logic. With the

help of their ideas, this section provides matrix rep-

resentations of our two dynamic operators [ϕ↓

] and

[H↓

]. First, we expand our syntax of static logic of

agents’ belief with terms of (iteration free) proposi-

tional dynamic logic, and then we explain the main

idea of (Van Benthem and Liu, 2007) in Section 5.1.

Finally, we rewrite their semantic idea in terms of ma-

trix in Section 5.2.

In order to handle multiple agents G, (Fitting, 2003)

employed the notion of P (G)-valued matrix. However, we

keep ourselves to the boolean matrices in this paper.

5.1 Propositional Dynamic Logic of

Relation Changers

The syntax of PDL-extension of L is deﬁned by si-

multaneous induction on a program term π and a for-

mula ϕ:

π ::= R

|(π ∪ π)| (π;π)|ϕ? (a ∈ G)

ϕ ::= p|c

|¬ϕ|ϕ ∨ ϕ|[π]ϕ (p ∈ Prop, a, b ∈ G)

Here we regard R

as an atomic program (for agent

a). [R

] corresponds to the previous belief operator

. So, in what follows, we also write B

for [R

], if

no confusion arises from the context. Then, we may

read the program terms as follows: (π ∪ π

′

) is to read

‘do π or π

′

, non-deterministically”; (π;π

′

) is to read

“do π followed by π

′

”; ϕ? is to read “proceed if ϕ true,

else fail”. As is well-known, we can introduce some

standard programming constructs by deﬁnitional ab-

breviation. For example,

if ϕ then π else π

′

:= (ϕ?;π) ∪ ((¬ϕ)?;π

′

Given a model M = (W, (R

)

a∈G

, (C

)

a,b∈G

,V), we

deﬁne the semantics of our PDL-extension by:

:= R

Jπ∪ π

′

:= JπK

∪ Jπ

′

Jπ;π

′

:= JπK

◦ Jπ

′

Jϕ?K

:= {(w, v)| w = v and w ∈ JϕK

}

JpK

:= V(p)

:= C

J¬ϕK

:= W \ JϕK

Jϕ∨ ψK

:= JϕK

∪ JψK

J[π]ϕK

:= {w ∈ W |JπK

(w) ⊆ JϕK

where R◦ S is the relational composition of R with S,

i.e., (w, v) ∈ R◦ S iff (w, u) ∈ R and (u,v) ∈ S for some

u ∈ W, and JπK

(w) := {v ∈ W |(w, v) ∈ JπK

Note that J[R

]ϕK

is the same meaning as the truth

set { w ∈ W |M, w |= ϕ} of the previous Kripke se-

mantics.

Recall that, in the semantics of [ϕ↓

] and [ϕ↓

]

(H ⊆ G), we keep the domain of a model, channel re-

lations, and a valuation for proposition letters but re-

deﬁne the accessibility relation (R

)

a∈G

. In this sense,

we may say that those operations are relation chang-

ers. (Van Benthem and Liu, 2007) observed that, if

relation changing operations are written in terms of

program terms generated from atomic programs by

the composition ;, the union ∪ and the test ϕ?, then

we can automatically generate the set of reduction ax-

ioms (as in Tables 2 and 3) to assure semantic com-

pleteness of propositional dynamic logic with relation

changing operations. Let us suppose that our relation

changer for a relation R

= JR

is written in terms

of a program term π

(a ∈ G). Then, we may de-

note by [(R

:= π

)

a∈G

] our dynamic operator which

LinearAlgebraicSemanticsforMulti-agentCommunication

179

changes an original relation R

into a new relation R

′

via π

for all agents a ∈ G. Then, our key equivalence

for generating the reduction axioms takes the follow-

ing form:

[(R

:= π

)

a∈G

][R

]ϕ ↔ [π

][(R

:= π

)

a∈G

]ϕ.

where we generalize van Benthem and Liu’s equiva-

lence for a single agent to multi-agents.

Example 6. 1. Semi-private Announcement: In the

semantics of [ϕ↓

], we have rewritten the ac-

cessibility relations (R

)

a∈G

into the new ones

′

)

a∈G

. We may reformulate the semantics in

terms of binary relations.

• Let c = b. Then, R

′

:= (R

∩ (Jc

∧ B

ϕK ×

JϕK)) ∪ (R

∩ (J¬(c

∧ B

ϕ)K ×W)).

• Let c 6= b. Then, R

′

:= R

Then, the corresponding relation changer agent b

to [ϕ↓

] is the following. When c = b,

:= ((c

∧B

ϕ)?;R

;ϕ?)∪(¬(c

∧B

ϕ)?;R

If we employ the previous deﬁnitional abbrevia-

tion, we may write π

as:

:= if c

∧ B

ϕ then R

;ϕ? else R

When c 6= b, the relation changer for agent c for

[ϕ↓

] is: π

:= R

. Then, we may regard [ϕ↓

] as

[(R

:= π

)

a∈G

2. Introspective Announcement via Communication

Channel: Let a be any agent. The correspond-

ing relation changer to [ϕ↓

] is the following pro-

gram term π

′

:= (ψ?;R

;ϕ?) ∪ (¬ψ?;R

), where

ψ :=

a∈H

∧ B

ϕ). By the previous deﬁni-

tional abbreviation, we may write π

′

as:

′

:= if



a∈H

∧ B

ϕ)



then R

;ϕ? else R

Then, we may regard [ϕ↓

] as [(R

:= π

′

)

a∈G

5.2 Relation Changers in Matrix Form

Given two relations R

, R

⊆ W × W. Relational

union and composition ﬁt well with matrix addition

and multiplication as follows:

∪ R

)

= R

+ R

, (R

◦ R

)

= R

Let ϕ be a formula of static logic of agents’ belief.

Since Jϕ?K

= {(w, v)|w = v and M, w |= ϕ} is also

a relation on W, we may provide a matrix representa-

tion Jϕ?K

. By deﬁnition of R

, we obtain:

Jϕ?K

(i, j) =

(

1 if i = j and M, w

|= ϕ,

0 otherwise.

Therefore, Jϕ?K

is the matrix from which diagonal

components we may read off the information of truth

set of JϕK

of the formula ϕ. For test program, we

note the following proposition.

Proposition 7. Let ϕ and ψ be formulas. Then,

J(ϕ∧ ψ)?K = Jϕ?K ◦ Jψ?K. Therefore, J(ϕ∧ ψ)?K

Jϕ?K

Jψ?K

Example 8. Let us see whether our matrix repre-

sentation of model update for semi-private announce-

ment works on our running example (Example 1). As

is the same as in Example 2, we consider the update

by [p↓

]. There are channel between agent a and b,

and agent a believes that p at w

. By Proposition 7,

the ﬁrst part of a matrix calculation of R

becomes:

J(c

∧ B

p)?K

Jp?K

= Jc

p?K

Jp?K





1 0 0

0 1 0

0 0 0









0 0 0

0 1 0

0 0 0









1 1 1









0 0 0

0 1 0

0 0 0









0 0 0

0 1 0

0 0 0





Then calculate also the remaining part of R

′

i.e.,J¬(c

∧ B

p)?K

, we combine both results to

obtain updated relation R

′

of agent b as:

′

= J(c

∧ B

p)?K

Jp?K

+ J¬(c

∧ B

p)?K





0 0 0

0 1 0

0 0 0









1 1 1

0 0 0

1 1 1









1 1 1

0 1 0

1 1 1





This coincides with the result of Example 2 (see Fig-

ure 2)

6 IMPLEMENTATION

This section introduces two algorithms. One of them

calculates the truth value of a formula B

p and the

other one calculates the relation updates by [p↓

]. For

both algorithms, we assume that an input model M =

(W, (R

)

a∈G

, (C

)

a,b∈G

,V) is represented in terms of

boolean matrix.

Algorithm 1: Calculation of kB

procedure BELIEF-OF

input M, w

∈ W, a ∈ G, p ∈ Prop

pk :=

V(p)

return True if kB

pk(i) > 0; False otherwise

end procedure

Here we comment just on Algorithm 2. In order

to update an accessibility relation of agent b, the al-

gorithm loops to ﬁnd agent b. If the algorithm ﬁnds

agent b, a model updating procedure (for a single

agent) will be started, otherwise it just put R

′

= R

At the beginning of the updating procedure, the algo-

rithm generates test matrices through

Test

function

where an input of this function is a column vector,

and it enumerates the elements of the input vector

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

180

Algorithm 2: Calculation of [p↓

procedure SEMI-PRIVATE-ANNOUNCEMENT

input M, a, b ∈ G, p ∈ Prop

for c ∈ G do

if c = b then

X := Test(C

)

Y := Test(kB

pk)

Z :=Test(V(p)

)

′

:= XYR

Z +

XYR

else

′

:= R

end if

end for

return M

′

= (W, (R

′

)

a∈G

, (C)

a,b∈G

,V)

end procedure

in the diagonal components of an output matrix, and

ﬁlls 0 in the non-diagonal components of the matrix.

Then, it calculates the updated accessibility relation

of agent b in terms of boolean matrix. Note that

k¬ϕ?k can be calculated as

kϕ?k. Finally, the algo-

rithm returns the updated model M

′

Implemented Program. We have implemented the

preceding algorithms in a single calculator with GUI

by Java

7. It is now available on our web site

The main features of the calculator are summarized

as follows. First, we may edit the numbers of both

agents and worlds, and also accessibility relations for

agents in terms of boolean matrix. Second, it also

implemented an algorithm checking if a given acces-

sibility relation satisﬁes frame properties such as re-

ﬂexivity, transitivity, etc. Third, the calculator can vi-

sualize both an accessibility relation of an agent and a

channel relation (communication channels) between

agents at a world, with the help of Graphviz.

7 CONCLUSION

The main contribution of this paper can be summa-

rized as follows. First, we introduced the static doxas-

tic logic with communication channels (where we al-

ways assume self-channel on all agents) with the com-

plete axiomatization K

that is also decidable (Theo-

rems 1 and 2). We also extended such static logic with

two dynamic operators [ϕ↓

] (semi-private announce-

ment) and [ϕ↓

] (introspective announcement) with

reduction axioms (so extensions of both of them en-

joy completeness results, Theorems 3 and 4). A key

feature of our dynamic operators are non-public, i.e.,

effects of announcements are restricted to some spec-

iﬁed agents determined by communication channels.

http://cirrus.jaist.ac.jp:8080/soft/bc

http://www.graphviz.org/

Second, we followed the idea by (Fitting, 2003) to re-

formulate Kripke semantics to our doxastic logic in

linear algebraic form, and employ the idea of PDL-

format by (Van Benthem and Liu, 2007) to provide

matrix representations to our two dynamic operators.

Finally, based on this linear algebraic reformulation,

we implemented the calculation system of agents’ be-

liefs and updates of Kripke models by [ϕ↓

]. An im-

plementation of [ϕ↓

] is a direction of further work.

ACKNOWLEDGEMENTS

We would like to thank anonymous reviewers for their

helpful comments. The works of the second and third

authors were partially supported by JSPS KAKENHI

Grant-in-Aid for Young Scientists (B) No. 24700146

and Scientiﬁc Research (C) No. 25330434, respec-

tively.

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