A LOCAL-GLOBAL MODEL FOR MULTIAGENT SYSTEMS

Sheaves on the Category MAS

Thomas Soboll

and Ulrike Golas

Fachbereich Computerwissenschaften, Universit¨at Salzburg, Salzburg, Austria

Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin, Berlin, Germany

Keywords:

Multiagent systems, Sheaves, Dynamic cooperation structures.

Abstract:

In multiagent systems, each agent has its own local view of the environment. Nevertheless, agents try to

cooperate to reach a common global goal. In this paper, we use a suitable Grothendieck topology and sheaves

to model the agents’ local data and their communication.

1 INTRODUCTION

Multiagent Systems (MASs) provide autonomous,

distributed, and ﬂexible problem solving capabilities

for a wide ﬁeld of problem areas. The present contri-

bution elaborates how sheaf theory can provide the

uniﬁcation and abstraction to integrate cooperation

structure, agents’ local knowledge, and communica-

tion in a single model. The idea is to un-couple struc-

tural information and the agent’s knowledge. Struc-

tural information comprises of all kinds of relations

and cooperations between agents and is encoded in so

called base diagrams. An agent’s gathered knowledge

is then encoded in a sheaf over base diagrams.

This paper is organized as follows. In Section 2,

we describe the category MAS, introduce our running

example and some notions in sheafs. In Section 3, we

apply the construction of sheaves to MAS .

2 PRELIMINARIES

In this section, we introduce the category MAS , our

running example and some results for sheaves.

2.1 Base Diagrams

The category MAS (Pfalzgraf and Soboll, 2007) has

as objects base diagrams representing the current co-

operation structure of the underlying MAS, describ-

ing the agents, their properties and relationships. The

morphisms in this category are maps respecting the

structure of base diagrams. This is necessary to pre-

serve the relational information encoded in there. For

the remainder of this paper we use the following run-

ning example describing a set of agents that cooperate

to weld two cubes. The cubes are delivered to the as-

sembly agent by agents equipped with a gripper.

Agent Properties: Ap = {1, 2, 3}, agent is equipped

with (1) a welding device, (2) a gripper, (3) agent can

act as relay agent for communication. Arrow types:

At = {c, d, r} (c) communication channel, (d) deliv-

ery channel (dotted lines), (r) request channel.

In Fig. 1, the left hand side shows the actual

robots, while the underlying base diagram is depicted

on the right. Of the four robots, b, c, and d have a

gripper (2), b is also a relay agent (3), and a is an as-

sembly robot with a welding device (1). Agent b has a

delivery channel to agent a and agent a has open com-

munication channels to b and c. A MAS morphism F

is depicted in Fig. 2. F is the obvious inclusion map,

where the communication channels, the arrow types,

as well as the object types are preserved.

Actions: The action types Act = {idle, wf r, weld, ed}

deﬁne of possible actions: Type 1 (welding agent) can

execute weld or idle. Type 2 (gripper agent) can exe-

cute wf r (wait for resources, if no cubes are available)

or idle, and it can set an outgoing delivery channel to

ed (execute delivery) or idle. Type 3 (relay agent)

can only execute idle, but may act as a relay agent for

communication and requests.

2,3

Figure 1: Example of a base diagram.

331

Soboll T. and Golas U..

A LOCAL-GLOBAL MODEL FOR MULTIAGENT SYSTEMS - Sheaves on the Category MAS.

DOI: 10.5220/0003742103310334

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 331-334

ISBN: 978-989-8425-96-6

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2,3

1,3

2,3

Figure 2: MAS morphism.

2.2 Applied Notions and Notations

Here we summarize notions, notations, and results

from sheaf-theory (MacLane and Moerdijk, 1994;

Kashiwara and Schapira, 2006). We will use these

and want to introduce them here in an informal,

(hopefully) intuitive and motivating manner.

Given some domain of (distributed) entities, like

agents, a sheaf is a mathematical device providing the

means to collate local information stored or gathered

by each entity/agent in the system to a global view, if

the junks of local information agree in overlapping ar-

eas. A presheaf is a very similar thing, but presheaves

do not require the local observations to be collate-able

to a unique global view, whereas sheaves do.

To be able to formalize the notion of overlapping

areas we need some notions of intersection, union

and covering, which are provided by a Grothendieck

Topology (GT) of base diagrams. Given some base

diagram B, we construct a subcategory Sub(B) which

is a collection of sub-diagrams of B together with as-

sociated inclusions. In this category we deﬁne what it

means that a selection of sub-base diagrams covers B.

Informally, this is the case if the union of a selection

of sub-base diagrams results in B, using a GT.

Example 1. In Fig. 3, a subcategorySub(B) is shown,

where the base diagram B is depicted as the right-

most object. Bold arrows deﬁne the morphisms in

Sub(B). We can observe that the set of inclusion mor-

phisms { 11, 12}, {7, 8}, and {9, 10} cover B, S

, and

respectively. On the other hand, the inclusions

{4, 5} do not cover S

because the arrow is missing.

Given a sheaf F on Sub(B), holding the observa-

tion gathered by the agents in B, for every subsystem

S of Sub(B), F(S) holds all the information gathered

or stored in S. We can perform a restriction of F to

S denoted by F|

, which is again a sheaf deﬁned on

Sub(S). A sub-sheaf of F on Sub(B) is simply a sheaf

′

on Sub(B) such that the information stored in F

′

a subset of the information in F for every subsystem.

For a presheaf of observations, where for some or

all observations there is no unique collation, we can

perform sheaﬁﬁcation. This operation provides for

any presheaf P the “best” sheaf F you can get from

P. F is obtained by identifying things that have the

same restrictions and then adding in all the things that

can be patched together (Mumford, 1999).

∅

2,3

Figure 3: Example of a subcategory Sub(B).

A very important notion is the gluing of sheaves.

The main idea is that for sheaves, i.e.knowledge on

different subsystems, where we explicitly allow inter-

sections, we can collate the observations to a single

sheaf if the corresponding “local” sheaves agree in

the overlaps. This means that the restrictions of the

“local” sheaves of the different subsystems to the in-

tersection of the subsystems need to be equal.

Example 2. Given the discrete topology on a set of

agents Ag, for any subset U ⊂ Ag the actual action

assignments f : U → Act of the agents can be deter-

mined locally. For V ⊂ U, the restriction of f to V,

denoted as f|

:V → Act is the action assignment for

the agents in V, this is a passage from global to local.

3 SHEAVES ON MAS

In this section, we apply the sheaf concepts to our

base diagrams. Note that we allow in our running ex-

ample that some arrows (here of type d) get actions

assigned (ed and idle). Such arrows will be called ac-

tion arrows (aA). For the other arrow types we do not

introduce actions, because they do not inﬂuence the

agent’s knowledge in its local view.

We deﬁne the (pre)sheaves representing the

agent’s knowledge as a functor P : Sub(B)

→ SET .

For all objects C of Sub(B), P(C) consists of a fam-

ily of maps deﬁned by P(C) = { f

: Ag(C) ∪Aa(C) →

Act}, where Ag(C) are the agents in C, Aa(C) are the

action arrows in C, Act is the set of actions and each

map f

∈ P(C) assigns to every agent and action ar-

row a single action of the set Act. Loosely speaking,

each f

represents a possible world compatible with

the agents sensor and/or communication information.

3.1 Agent View

Each agent has sensors to allocate information in its

environment, where the reading of each sensor results

in a certain base diagram. We assume that an agent

is capable of sensing the types of the agents and their

identity and has knowledge about actions associated

to these types. A suitable combination of all sensor

information of an agent leads to its local view.

ICAART 2012 - International Conference on Agents and Artificial Intelligence

332

Sub(E

)

∅

2,3

Figure 4: Env./Subcat. for agent a.

Deﬁnition 1 (Agent Environment). Given an agent

a with n sensors, each sensor i ∈ 1, 2, ..., n samples

an environment Ea

, where at least a is present. The

agent environment E

of a is the union of all environ-

ments Ea

, ..., Ea

Example 3. Suppose agent a is equipped with two

sensors which sample the environments Ea

and Ea

The agent environment E

is obtained by the union of

and Ea

(see Sub(E

) in Fig. 4).

Table 1: Example: Type sensor readings.

a c t

a b 99K

f idle wfr j idle wfr ed

g idle idle k idle idle ed

h weld wfr l weld wfr ed

i weld idle m weld idle ed

o idle wfr idle

p idle idle idle

q weld wfr idle

r weld idle idle

We distinguish type-sensors and action-sensors.

This distinction is necessary to apply adequate “glu-

ing” operations to collate the agent’s knowledge.

Type sensors collect type information, which allows

an agent to deduce the possible actions the agents in

its environment may execute. Action-sensors on the

other hand capture observed actions. The combina-

tion of the sampled information provides the building

block for local knowledge or the agent view.

Type sampling. For every agent i, we construct the

presheaf Pr

: Sub(E

)

→ SET representing the in-

formation gathered by sensors observing type infor-

mation. Initially, Pr

is the functor assigning to each

object in Sub(E

) the empty set and to the empty

base diagram the singleton containing the empty map

only (which is a terminal object in SET). For every

type-sensor, the sensor reading contains type informa-

tion and results in a set of maps of the environment

∈ Sub(E

). It is joined with the already available

information in Pr

) and propagated into the sub-

environments of E

by restriction. By sheaﬁﬁcation

of Pr

we construct the sheaf T

Example 4. Let Pr

: Sub(E

)

→ SET be the ini-

tial presheaf. Agent a has two type sensors t

, t

sam-

pling the environments Ea

, Ea

(see Fig. 4). The

reading of t

in Ea

is “agent a has type 1 and agent

c has type 2,3”. From this, agent a deduces four pos-

sible action assignment maps f, g, h, i in Ea

. The

reading of t

in Ea

is “agent a has type 1 and agent

b has type 2” and a delivery channel is recognized.

This leads to eight possible action assignment maps

j, k, l, m, o, p, q, r in Ea

(see Table 1).

We update Pr

(Ea

) to Pr

(Ea

) ∪ { f, g, h, i} and

(Ea

) to Pr

(Ea

) ∪ { j, k, l, m, o, p, q, r}. More-

over, Pr

(Ea

) with inclusions into Ea

and Ea

is updated. The sheaﬁﬁcation of Pr gives the

sheaf T

with T

(Ea) = { f j, fk, g j, gk, hl, hm, il, im,

fo, f p, go, gp, ho, hp, io, ip} (see Table 2).

Action Sampling. For a set of action-sensors {s

| j =

1, ..., k}, each sampling actions in a sub-environment

of E

of agent i, we interpret the sensor reading as

a sheaf Si

on Sub(E

Example 5. Assume agent a has two action sensors

and as

sampling Ea

and Ea

such that the read-

ing of as

results in the map a

: {a, c} → Act with

(a) = idle, a

the map a

: {a} → Act with a

(a) = idle. Note that

the restrictions of these maps result in sheaves Sa

and Sa

, respectively.

Collating Type and Action Information. We col-

late the type information in T

and the action informa-

tion in Si

, ..., Si

leading to the sheaf P

: Sub(E

)

→

SET by gluing sheaves.

Example 6. Given the sheaves T

, Sa

, and Sa

from

Ex. 4 and 5, we construct the sheaf P

by gluing the

maximal sub-sheaf T

′

. P

) = { f j, fk, fo, f p} (see

Table 2). Including additional information that the

agent has, e.g. that the delivery channel is in state ed,

leads to only four possible action assignment maps in

) = { f j, fk}; for the sake of simplicity we will

assume this for the remainder of the paper.

Assume that agent c is waiting for resources, a

local view P

evaluated at E

for c is then given by

) = { fh, fi, gh, gi} as depicted in the right hand

side of Table 2.

Deﬁnition 2 (Agent View). The agent view of agent

i is the sheaf P

: Sub(E

)

→ SET. If the agent view

exists, i.e. the sensor information of i’s different ac-

tion sensors agree in the overlaps, we call the agent

view (locally) consistent.

3.2 Communication

Each communication arrow means that agents com-

municate selected information. Restrictions and sub-

sheaveswill be used for selecting speciﬁc information

that shall be communicated. The collation of commu-

nication content is similar to the way sensor informa-

tion was collated to form the agent view.

Deﬁnition 3 (Communication). Given the agent

views P

and P

on Sub(E

) and Sub(E

), respectively,

A LOCAL-GLOBAL MODEL FOR MULTIAGENT SYSTEMS - Sheaves on the Category MAS

333

Table 2: Examples: Value of the sheaf T

at E

, P

at E

(bold) and P

at E

) a b c 99K a b c 99K

fj idle wfr wfr ed hl weld wfr wfr ed

fk idle idle wfr ed hm weld idle wfr ed

gj idle wfr idle ed il weld wfr idle ed

gk idle idle idle ed im weld idle idle ed

fo idle wfr wfr idle ho weld wfr wfr idle

fp idle idle wfr idle hp weld idle wfr idle

go idle wfr idle idle io weld wfr idle idle

gp idle idle idle idle ip weld idle idle idle

) a c d

fh idle wfr idle

ﬁ weld wfr idle

gh idle wfr wfr

gi weld wfr wfr

and a communication arrow from agent k to agent i in

the environment of k. k selects a sub-sheaf K of a re-

striction E

′

of its environment and sends it to agent i, i

computes the union of the environments E

and E

′

de-

noted asC

i,k

and called communication-environment.

Agent i computes the maximal sub-sheaves K

′

and P

′

of the communicated sheaf K and its agent view P

respectively, such that K

′

and P

′

can be glued to the

sheaf Comm

i,k

: Sub(C

i,k

)

→ SET. Comm

i,k

holds

the information of agent i including the communi-

cated knowledge of agent k.

Deﬁnition 4 (Communication Consistency). Given

communication arrows from agents a

, ..., a

to an

agent i. We call the communication consistent if for

all sheaves Comm

i,a

, ..., Comm

i,a

there exist sub-

sheaves Comm

′

i,a

, ..., Comm

′

i,a

that can be glued to

form the sheaf Comm

on the union C

of all C

i,a1

i,an

holding all information available to agent i.

Table 3: Example: Value of sheaf Comm

at C

Comm

) a b c d 99K

fjfh idle wfr wfr idle ed

fjgh idle wfr wfr wfr ed

fkfh idle idle wfr idle ed

fkgh idle idle wfr wfr ed

Example 7. Assume agent c communicates its en-

tire agent view P

to agent a. The resulting sheaf

representing agent a’s knowledge including commu-

nication is Comm

= Comm

a,c

on C

depicted by

the dashed elements in Fig. 3, since c is the only

agent communicating to a. This is a sheaf in case

the local observations of a and c are not contra-

dictory. In Ex. 6, 6 we have deﬁned the sheaves

, P

with P

) = { f j, fk, fo, f p} and P

) =

{ fh, fi, gh, gi} . P

′

with P

′

) = { f j, fk, fo, f p}

and P

′

with P

′

) = { fh, gh} are compatible lead-

ing to the gluing sheaf Comm

: Sub(C

)

→ SET .

It represents the observation of a including the com-

municated data, where Comm

) := { f j fh, f jgh,

fk fh, fkgh, fofh, fogh, f p fh, f pgh} (see Table 3).

and Comm

on Sub(E

) and Sub(C

), respec-

tively, describe the knowledge available to agent i lo-

cally without and with communication. During the

construction of Comm

speciﬁc maps can be elimi-

nated (by building a subsheaf of P

) using the addi-

tional information available to the agent. The elimina-

tion of maps by building sub-sheaves reduces the set

of possible worlds, and hence this is in fact a method

to gain knowledge rather than to loose it.

4 CONCLUSIONS AND

OUTLOOK

We have demonstrated how sheaves can be applied to

model local-global dependencies within a Multiagent

System based on structural information of its base di-

agram and using a suitable Grothendieck topology for

MAS. Sheaves allow us to collate the local obser-

vations through communication for a “wider” view

of a single agent and to construct group knowledge.

The sheaf model implicitly checks for inconsistency

in overlapping observations.

Future work includes the integration of coopera-

tion rules describing rule-based changes of the base

diagrams. The possible worlds that are generated by

the sheaves represent an agent’s knowledge, where

based on this information it can decide whether and

how to execute speciﬁc cooperation rules. As a next

step, one could include additional information like

more speciﬁc resource data in the model by including

resources as agent properties or by deﬁning additional

sheaves representing the distribution of resources.

REFERENCES

Kashiwara, M. and Schapira, P. (2006). Categories and

Sheaves, volume 332 of Grundlehren der Mathema-

tischen Wissenschaften. Springer.

MacLane, S. and Moerdijk, I. (1994). Sheaves in Geome-

try and Logic: A First Introduction to Topos Theory.

Springer. Corrected ed.

Mumford, D. (1999). The Red Book of Varieties and

Schemes. Springer. 2nd exp. ed.

Pfalzgraf, J. and Soboll, T. (2007). On a General Notion of

Transformation for Multiagent Systems. In Proceed-

ings of IDPT ’07. SDPS.

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