Luís Moniz Pereira

Centro de Inteligência Artificial - CENTRIAUniversidade Nova de Lisboa, Portugal

Pierangelo Dell’Acqua

Dept. of Science and Technology - ITNLinköping University, Sweden

PADL’03

Motivation

To provide control over the epistemic agents in a Multi-Agent System (eMAS) the need arises to:

- explicitly represent its organizational structure,- and its agent interactions.

We introduce a logical framework F, suitable for representing organizational structures of eMAS.

- we provide its declarative and procedural semantics.

- F having a formal semantics, it permits us to prove properties of eMAS structures.

Our agents

We have been proposing a LP approach to agents which can:Reason on their own or in collaborationReact to other agents and to the environmentUpdate their own knowledge, reactions, and goals Interact by updating the theory of any other agentDecide whether to accept an update subject to the

requesting agentCapture the representation of social evolution

Framework

This framework builds on the works:

Updating Agents P. Dell’Acqua & L. M. Pereira - MAS’99

Multi-dimensional Dynamic Logic Programming L. A. Leite & J. J. Alferes & L. M. Pereira - CLIMA’01

and subsequent ones.

Updating agent’s cycle

Updating agent: a rational, reactive agent that dynamically changes its own knowledge and goals.

In its cycle, in some order, it:

makes observations reciprocally updates other agents with goals and rules thinks (rational) selects and executes actions (reactive)

Logic framework

Atomic formulae:A atom

not A default atom

generalized rule

Formulae:

every Li is an atom or a default atom

L0 L1 Ln

Integrity constraints

Action rules

Agents’ knowledge state sequences

Knowledge states represent dynamically evolving states of an agent’s knowledge. They undergo change due to updates (DLP).

Given the current knowledge state Ps , its successor knowledge state Ps+1 is produced as a result of the occurrence of a set of parallel updates.

Update actions do not modify the current or any of the previous knowledge states.

They affect only the successor state: the precondition of an action is evaluated in the current state, and its postcondition updates the successor state.

MDLP Motivating Example

Parliament issues law L1 at time t1 A local authority issues law L2 at time t2 > t1 Parliamentary laws override local laws, but not vice-versa:

More recent laws have precedence over older ones:

L2 L1

L1 L2

How to combine these two dimensions of knowledge precedence?

DLP with Multiple Dimensions (MDLP)

Multi-Dimensional Logic Programming

In MDLP knowledge is given by a set of programs.

Each program represents a different piece of updating knowledge assigned to a state.

States are organized by a DAG (Directed Acyclic Graph) representing their precedence relation.

MDLP determines the composite semantics at each state according to the DAG paths.

MDLP allows for combining knowledge updates that evolve along multiple dimensions.

MDLP for Agents

Flexibility, modularity, and compositionality of MDLP makes it suitable for representing the evolution of several agents’ combined knowledge

How to encode, in a DAG, the relationships among every agent’s

evolving knowledge, along its multiple dimensions ?

Two basic dimensions of a MAS

®

¹

°

º ¯

Hierarchy of agents

¯ ¯ 0 ¯ 1 . .. ¯ c

Temporal evolution of one agent

How to combine these dimensions into one DAG ?

Equal Role Representation

Assigns equal role to the two dimensions:

®'

...

®0

¹ 0

° 0

º 0 ¯ 0

0

®1

¹ 1

° 1

º1 ¯ 1

1

®c

¹ c

° c

º c ¯ c

c

Time Prevailing Representation

Assigns priority to the time dimension:

®'®0 ®1 ®c

.. .

®0

¹ 0

° 0

º0 ¯ 0

®0

®1

¹ 1

° 1

º 1 ¯ 1

®1

®c

¹ c

° c

º c ¯ c

®c

.. .

Hierarchy Prevailing Representation

Assigns priority to the hierarchy dimension:

®'®

¹

°

º ¯

°

° 0

° 1

...

° c

¯ ¯ 0 ¯ 1 .. . ¯ c

Inter- and Intra- Agent Relationships

The above representations refer to a community of agents

But they can be employed as well for relating the several sub-agents of an agent

®d

®b

®a

®e

A sub-agent Hierarchy

Intra- and Inter- Agent Example

Prevailing hierarchy for inter-agents

Prevailing time for sub-agents

®

...

.. .

®'®

¹

°

º ¯

°

° 0

° 1

...

° c

®d

®b

®a

®e

0 ®d

®b

®a

®e

1 ®d

®b

®a

®e

c

¯ ¯ 0 ¯ 1 .. . ¯ c

®0 ®1 ®c.. .

MDLPs revisited

Def. MDLP – Multi-Dimensional Logic Program

A MDLP is a pair (D,D), where:

D=(V,E,w) is aWDAG - Weighted directed acyclic graph

and,

D={Pv : vV} is a set of generalized logic programs indexed by the vertices of D.

Weighted directed acyclic graphs

Def. Weighted directed acyclic graph (WDAG)

A weighted directed acyclic graph is a tuple D=(V,E,w) :

- V is a set of vertices,- E is a set of edges,- w : E R+ maps edges into positive real numbers,- no cycle can be formed with the edges of E.

We write v1 v2 to indicate a path from v1 to v2.

This paper: MDLPs revisited

We generalize the definition of MDLP by assigning weights to the edges of a DAG.

In case of conflictual knowledge, incoming into a vertex v by two vertices v1 and v2, the weights of v1 and v2 may resolve the conflict.

If the weights are the same bothconclusions are false.

(Or, two alternative conclusionscan be made possible.)

v0.1

v1 v2

0.2

{a} {not a}

[ a ]

Path dominance

Def. Dominant path

Let a1 an be a path with vertices a1,a2,…,an.

a1 an is a dominant path if there is no other path b1,b2,…,bm such that:

- b1= a1, bm= an, and

- i, j such that ai= bj and w((ai-1,ai)) < w((bj-1,bj)).

Example: path dominance

Let w((a5,a4)) < w((a3,a4)).

Then,

a1, a2 , a3, a4 is a dominant path.

a1

a2

a3

a4

a5

Example: formalizing agents

Example:

Formalize three agents A, B, and C, where:

• B and C are secretaries of A

• B and C believe it is not their duty to answer phone calls

• A believes it the duty of a secretary to answer phone calls

Epistemic agents can be formalized via MDLPs.

Example: formalizing agentsA = (DA,DA)

DA = ({v1},{},wA)

Pv1 = {answerPhone secretary phoneRing}

B = (DB,DB)

DB = ({v3,v4},{(v4,v3)},wB)

wB((v4,v3)) = 0.6

Pv3 = {}

Pv4 = {phoneRing, secretary, not answerPhone}

C = (DC,DC)

DC = ({v5,v6},{(v6,v5)},wC)

wC((v6,v5)) = 0.6

Pv5 = {} and Pv6 = Pv4

A

B

C

v3

v4

Bv3

v5

v6

Cv5

v1

Av1

Logical framework F

Def. Logical framework F

A logical framework F is a tuple (A, L, wL) where:

• A={1,…,n} is a set of MDLPs

• L is a set of links among the i

• and wL : L R+.

Semantics of F

Declarative semantics of F is stable model based.

v2 v1

s

Procedural semantics based on a syntactic transformation.

Idea: The knowledge of a vertex v1 overrides the knowledge of a vertex v2 wrt. a vertex s iff v1 prevails v2 wrt. s.

Example: Pv1 = {answerPhone}

Pv2 = {not answerPhone}

if then Ms={answerPhone}

v1v2

s

Modelling eMAS

Multi-agent systems can be understood as computational societies whose members co-exist in a shared environment.

A number of organizational structures have been proposed:

- coalitions, groups, institutions, agent societies, etc.

In our approach, agents and organizational structures are formalized via MDLPs, and glued together via F.

Modelling eMAS: groups

A group is a system of agents constrained in their mutual interactions.

A group can be formalized in F in a flexible way:

- the agents’ behaviour can be restricted to different degrees.

- formalizing norms and regulations may enhance trustfulness of the group.

Example: formalizing groups

Secretaries example:

Formalize group G, of agents A, B, and C, where:

• B must operate (strictly) in accordance with A, while

• C has a certain degree of freedom.

Example: formalizing groups

G = (DG,DG)

DG = ({v2},{},wG)

Pv2 = {}

v2

G

G F

F = (A,L,wL)

A = {A,B,C,G )

L = {(v1,v2), (v2,v3), (v2,v5)}

wL((v1,v2)) = wL((v2,v5)) = 0.5

wL((v2,v3)) = 0.7

v3

v4

Bv3

v5

v6

Cv5

v1

Av1

Example: semantics

v2

G

v3

v4

Bv3

v5

v6

Cv5

v1

Av1

0.5

0.5

0.6

0.6

0.7

Model of agent B: Mv3 = {phoneRing, secretary, answerPhone}

Model of agent C: Mv5 = {phoneRing, secretary, not answerPhone}

v1 v6v5

v4 v1v3

Conclusions and future work

Novel logical framework to model structures of epistemic agents:

- declarative semantics is stable model based,

- procedural semantics based on a syntactical transformation.

To represent F within the theory of each agent:

- to empower the agents with the ability to reason about and modify the agents’ structure,

- to handle open societies where agents can enter/leave the system.

The End

Prevalence

a1

an

a2

...

a1

ai-1

an

ai

...

...bm

b1

...

Def. Prevalence wrt. a vertex an

Let a1 an be a dominant path with vertices a1,a2,…,an. Then,

1. every vertex ai prevails a1 wrt. an (1< i n).

2. if there exists a path b1 ai with vertices b1,…,bm,ai and

w((ai-1,ai)) < w((bm,ai)), then every vertex bj prevails a1 wrt. an.

1. 2.

a1 aian

a1 bjan

Links

Def. Link

Given two WDAGs, D1 and D2, a link is an edge

between a vertice of D1 and a vertice D2.

Joining WDAGs

Def. Link

Given two WDAGs D1 and D2, a link is an edge between

vertices of D1 and D2.

Def. WDAGs joining

Given n WDAGs Di = (Vi,Ei,wi), a set L of links, and a function

wL : L R+, the joining ({D1,…, Dn},L,wL) is the WDAG D=(V,E,w) obtained by the union of all the vertices and edges, and

w(e) = wi(e) if eEi

wL(e) if eL

Joined MDLP

Def. Joined MDLP

Let F=(A,L,wL) be a logical framework.

Assume that A={1,…,n} and each i=(Di,Di).

The joined MDLP induced by F is the WDAG =(D,D) where:

- D= ({D1,…, Dn},L,wL) and

- D= i Di

Stable models of MDLP

Def. Stable models of MDLP

Let =(D,D) be a MDLP, where D=(V,E,w) and D={Pv : vV}. Let s V.An interpretation M is a stable model of at s iff:

M = least( X Default(X, M) ) where:

Q = v s Pv

Reject(s,M) = { r Pv2 : r’ Pv1, head(r)=not head(r’), M |= body(r’), }

X = Q - Reject(s,M)Default(X,M) = {not A : (ABody) in X and M | Body }

v2 v1s

Stable models of F

Def. Stable models of F

Let F=(A,L,wL) be a logical framework and the joined MDLP induced by F.

M is a stable model of F at state s iff M is a stable model of at state s.