Combining agents into
societies
Luís Moniz Pereira
Centro de Inteligência Artificial – CENTRIA
Universidade Nova de Lisboa
DEIS, Università di Bologna, 22 Marzo 2004
Summary
Goal and motivation
Overview of MDLP (Multi-Dimensional LP)
Combining inter- and intra-agent societal
viewpoints
An architecture for evolving Multi-Agent
viewpoints
A logical framework for modelling societies
Future work and conclusion
Goal
Explore the applicability of MDLP to represent
multiple agents’ view of societal knowledge
dynamics and evolution
The representation is the core of the
agent
architecture
and
system
MINERVA.
It was designed with the aim of providing
a common agent framework based on the
strengths of Logic Programming.
Motivation - 1
The notion of agency has claimed a major role in
modern AI research
LP and non-monotonic reasoning are appropriate
for rational agents:
Utmost efficiency is not always crucial
Clear specification and correctness are crucial
LP provides a general, encompassing, rigorous
declarative and procedural framework for rational
functionalities
Motivation - 2
 Till recently, LP could be seen as good for representing
static non-contradictory knowledge.
 In the agency paradigm we need to consider:
Ways of integrating knowledge from different sources evolving in
time
Knowledge expressing state transitions
Knowledge about environment and societal evolution, and each
agent’s own behavioural evolution
 LP declaratively describes states well.
 But LP must describe state transitions too.
MDLP overview
DLP synopsis
MDLP motivation
MDLP semantics
Multiple representational dimensions in a multiagent system
Representation prevalence
Overview conclusions
Dynamic LP
 DLP was introduced to express LP’s linear evolution in
dynamic environments, via updates
 DLP gives semantics to sequences of GLPs
 Each program represents a distinct state of knowledge,
where states may specify:
different time points, different hierarchical instances, different
viewpoints, etc.
 Different states may have mutually contradictory or
overlapping information, and DLP determines the
semantics for each state sequence
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:
L2
L1
 More recent laws have precedence over older ones:
L1
L2
How to combine these two dimensions of
knowledge precedence?
 DLP with Multiple Dimensions (MDLP)
MDLP
 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
Generalized Logic Programs
To represent negative info in LP updates, we
need LPs allowing not in heads
Programs are sets of generalized LP rules:
A
 B1,…, Bk, not C1,…,not Cm
not A  B1,…, Bk, not C1,…,not Cm
The semantics is a generalization of SMs
MDLP - definition
Definition:
A Multi-Dimensional Dynamic Logic Program, P, is a pair
(PD, D)
where:
D= (V, E) is an acyclic digraph
PD= { PV : v  V} is a set of generalized logic
programs indexed by the vertices of D
MDLP - semantics 1
Definition:
Let P=(PD,D) be a MDLP.
An interpretation Ms is a stable model of the multi-
dimensional update at state sV iff, where Ps=

is
Ms= least( [Ps – Reject(s, Ms)]  Defaults (Ps, Ms) )
j1
j2
j3
s
Pi :
j1
MDLP - semantics 2
j2
j3
s
Ms= least( [Ps – Reject(s, Ms)]  Defaults (Ps, Ms) )
where:
Reject(s, Ms) =
{r  Pi | r’  Pj , ijs, head(r)=not head(r’)  Ms |= body(r’)}
Defaults (Ps, Ms) =
{not A | r  Ps : head(r)=A  Ms |= body(r)}
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 multiple dimensions ?
Two basic dimensions of a
multi-agent system
Hierarchy of agents
º
¯
¹
Temporal evolution of one agent
¯
¯0
¯1
...
¯c
°
®
How to combine these dimensions into one DAG ?
Equal Role Representation
Assigns equal role to the two dimensions:
0
1
º0
¯0
º1
¹0
°0
c
¯1
ºc
¹1
°1
®0
...
¯c
¹c
°c
®1
®c
®'
Equal Role - 2
0
1
º0
¯0
º1
¹0
°0
c
¯1
...
¹1
°1
®0
ºc
¯c
¹c
°c
®1
®c
®'
In legal reasoning:
Lex Superior : rules issued by a higher authority
override those of a lower one
Lex Posterior : more recent rules override older ones
It potentiates contradiction:
There are many pairs of unrelated programs
Time Prevailing
Representation
Assigns priority to the time dimension:
®0
®1
º0
¯0
º1
¹0
°0
®c
¯1
...
ºc
¹1
¯c
¹c
°1
°c
®0
®1
®0
®1
®c
...
®c
®'
Time Prevailing - 2
®0
®1
º0
¯0
º1
¹0
°0
®c
¯1
...
ºc
¹1
¯c
¹c
°1
°c
®0
®1
®0
®1
®c
...
®c
 Useful in very dynamic situations, where competence is
distributed, i.e. agents normally provide rules about 
literals
 Drawback:
It requires all agents to be fully trusted, since all newer rules
override older ones irrespective of their mutual hierarchical
position
®'
Hierarchy Prevailing
Representation
Assigns priority to the hierarchy dimension:
¯
¯0
¯1
...
¯c
°
°0
º
°1
...
°c
¯
¹
°
®
®'
Hierarchy
Prevailing - 2
¯
¯0
¯1
...
¯c
°
°0
º
°1
...
°c
¯
¹
°
®
®'
Useful when some agents are untrustworthy
Drawback:
One has to consider the whole history of all higher
ranked agents in order to accept/reject a rule from a
lower ranked agent
However, techniques are being developed to reduce the size of a
MDLP (garbage collection).
Inter- and Intra- Agent
Relationships
The above representations refer to a
community of agents
But they can be used as well for relating the
several sub-agents of an agent
A sub-agent Hierarchy
®a
®b
®e
®d
Intra- and Inter- Agent
Example
¯
¯0
¯1
...
¯c
°
°0
Prevailing
hierarchy
for inter-agents
¯
°1
¹
...
Prevailing time
for sub-agents
º
°
°c
®
®'
®
®0
®1
®a
®a
®b
0
®e
®d
®b
1
®c
...
...
®a
®e
®d
®b
...
c
®e
®d
Current work of overview
A MINERVA agent:
Is based on a modular design
It has a common internal KB (a MDLP), concurrently
manipulated by its specialized sub-agents
Every agent is composed of specialized subagents that execute special tasks, e.g.
reactivity
planning
scheduling
belief revision
goal management
learning
preference evaluation
strategy
MDLP overview conclusions
 We’ve explored MDLP to combine knowledge from
several agents and multiple dimensions
 Depending on the situation, and relationships among
agents, we’ve envisaged several classes of DAGs for
their encoding
 Based on this work, and on a language (LUPS) for
specifying updates by means of transitions, we’ve
launched into the design of an agent architecture
MINERVA
Evolving multi-agent viewpoints
– one more overview
 Our agents
 Framework references
 Mutually updating agents
 MDLP synopsis
 Agent language: projects and updates
 Agent knowledge state and agent cycle
 Example
 An implemented example architecture
 Future work
Our agents
We propose a LP approach to agents that can:
 Reason and React to other agents
 Update their own knowledge, reactions, and
goals
 Interact by updating the theory of another agent
 Decide whether to accept an update depending
on the requesting agent
 Capture the representation of social evolution
Updating agents
Updating agent: a rational, reactive agent that can
dynamically change its own knowledge and goals
makes observations
reciprocally updates other agents with goals and rules
thinks (rational)
selects and executes an action (reactive)
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.
New contribution
1 To extend the framework of MDLP with integrity
constraints and active rules.
2 To incorporate the framework of MDLP into a
multi-agent architecture.
3 To make the DAG of each agent updatable.
DAG
A directed acyclic graph DAG is a pair D = (V, E)
where V is a set of vertices and E is a set of directed
edges.
Agent’s language
Atomic formulae:
objective atoms
A
i:C
iC updates
not A default atoms
Formulae:
generalized rules
A  L1 Ln
not A  L1 Ln
integrity
constraint
projects
Li is an atom, an update or
a negated update
Zj is a project
false  L1 Ln Z1 Zm
L1 Ln => Z
active rule
Projects and Updates
A project j:C denotes the intention of some agent i of
proposing the updating the theory of agent j with C.
iC denotes an update proposed by i of the current
theory of some agent j with C.
fredC
wilma:C
Agents’ knowledge states
 Knowledge states represent dynamically evolving states of
agents’ knowledge. They undergo change due to updates.
 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 only affect the successor state: the
precondition of the action is evaluated in the current state and
the postcondition updates the successor state.
Agent’s language
A project i:C can take one of the forms:
i : ( A  L1 Ln )
i : ( not A  L1 Ln )
i : ( false  L1 Ln Z1 Zm )
i : ( L1 Ln => Z )
i : ( ?- L1 Ln )
i : edge(u,v)
i : not edge(u,v)
Initial theory of an agent
A multi-dimensional abductive LP for an agent  is a tuple:
T =  D, PD, A, RD
- D = (V, E) is a DAG s.t. ´V (inspection point of ).
- PD = {PV | vV} is a set of generalized LPs.
- A is a set of atoms (abducibles).
- RD = {RV | vV} is a set of set of active rules.
The agent’s cycle
Every agent can be thought of as an abductive LP
equipped with a set of inputs represented as updates.
The abducibles are (names of) actions to be executed
as well as explanations of observations made.
Updates can be used to solve the goals of the agent as
well as to trigger new goals.
Happy story - example
DAG of Alfredo
alfredo´
inspection point of Alfredo
judge
mother
father
The goal of Alfredo is to be happy
alfredo
girlfriend
state 0
Happy story - example
alfredo´
judge
mother
father
alfredo
hasGirlfriend 
not happy => father : (?-happy)
not happy => mother : (?-happy)
getMarried  hasGirlfriend => girlfriend : propose
moveOut => alfredo : rentApartment
custody(judge,mother) => alfredo : edge(father,mother)
girlfriend
{moveOut, getMarried}
state 0
abducibles
Happy story - example
alfredo´
judge
mother
father
alfredo
hasGirlfriend 
not happy => father : (?-happy)
not happy => mother : (?-happy)
getMarried  hasGirlfriend => girlfriend : propose
moveOut => alfredo : rentApartment
custody(judge,mother) => alfredo : edge(father,mother)
girlfriend
{moveOut, getMarried}
state 0
Agent theory
The initial theory of an agent  is a multi-dimensional
abductive LP.
Let an updating program be a finite set of updates,
and S be a set of natural numbers. We call the
elements sS states.
An agent  at state s, written s , is a pair (T,U):
- T is the initial theory of .
- U={U1,…, Us} is a sequence of updating programs.
Multi-agent system
A multi-agent system M={1s ,…, ns } at state s
is a set of agents 1,…,n at state s.
M characterizes a fixed society of evolving agents.
The declarative semantics of M characterizes the
relationship among the agents in M, and how the
system evolves.
The declarative semantics is stable models based.
Happy story - 1st scenario
Suppose that at state 1, Alfredo receives
from the mother:
mother  (happy  moveOut)
mother  (false  moveOut  not getMarried)
mother  (false  not happy)
and from the father:
father  (happy  moveOut)
father  (not happy  getMarried)
Happy story - 1st scenario
alfredo´
false  not happy
happy  moveOut
false  moveOut  not getMarried
judge
mother
father
happy  moveOut
not happy  getMarried
alfredo
girlfriend
state 1
In this scenario, Alfredo cannot achieve his goal
without producing a contradiction. Not being able
to make a decision, Alfredo is not reactive at all.
Happy story - 2nd scenario
Suppose that at state 1 Alfredo’s parents decide to get
divorced, and the judge gives custody to the mother.
judge  custody(judge,mother)
Happy story - 2nd scenario
alfredo´
custody(judge,mother)
judge
mother
father
alfredo
girlfriend
state 1
hasGirlfriend 
not happy => father : (?-happy)
not happy => mother : (?-happy)
getMarried  hasGirlfriend => girlfriend : propose
moveOut => alfredo : rentApartment
custody(judge,mother) => alfredo : edge(father,mother)
Happy story - 2nd scenario
alfredo´
Note that the internal update produces a
change in the DAG of Alfredo.
judge
mother
father
alfredo
girlfriend
state 2
Suppose that when asked by Alfredo, the
parents reply in the same way as in the
1st scenario.
Happy story - 2nd scenario
alfredo´
false  not happy
happy  moveOut
false  moveOut  not getMarried
judge
mother
father
happy  moveOut
not happy  getMarried
alfredo
girlfriend
state 2
Now, the advice of the mother prevails over and
rejects that of his father.
Happy story - 2nd scenario
alfredo´
Thus, Alfredo gets married, rents an apartment,
moves out and lives happily ever after.
judge
mother
father
alfredo
girlfriend
state 2
hasGirlfriend 
not happy => father : (?-happy)
not happy => mother : (?-happy)
getMarried  hasGirlfriend => girlfriend : propose
moveOut => alfredo : rentApartment
custody(judge,mother) => alfredo : edge(father,mother)
Syntactical transformation
The semantics of an agent  at state s, s=(T,U), is established by a
syntactical transformation  that maps s into an abductive LP:
 s = P,A,R
1. s  P´,A,R
P´ is a normal LP, A and R are a set of abducibles and active
rules.
2. Default negation can then be removed from P´ via the abdual
transformation (Alferes et al. ICLP99, TCLP04):
P´  P
P is a definite LP.
Agent architecture
 s = P,A,R
Java
CC
InterProlog
(Declarativa)
can abduce
Rational
P
XSB Prolog
InterProlog
(Declarativa)
Reactive
P+R
XSB Prolog
cannot abduce
Agent architecture
ActionH
External
Interface
UpdateH
Updates
 s = P,A,R
ext.project
int.project
CC
projects
Rational
P
Reactive
P+R
Future work of overview
At the agent level:
 How to combine logical theories of agents expressed over
graph structures.
 How to incorporate other rational abilities, e.g., learning.
At the multi-agent system level:
 Non synchronous, dynamic multi-agent system.
 How to formalize dynamic societies of agents.
 How to formalize the notion of organisational reflection.
A logical framework for modelling
eMAS
Motivation
 To provide control over the epistemic agents in a MultiAgent 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.
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 both
conclusions are false.
(Or, two alternative conclusions
can be made possible.)
v
0.2
v1
{a}
[a]
0.1
v2
{not a}
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.
WMDLPs
Def. WMDLP – Weighted Multi-Dimensional Logic Program
A WMDLP  is a pair (D,D), where:
D = (V,E,w) is a
WDAG - Weighted directed acyclic graph
and,
D = {Pv : v  V} is a set of generalized logic programs
indexed by the vertices of D.
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
a4
a3
Let w((a5,a4)) < w((a3,a4)) .
Then,
a5
a2
a1
a1, a2 , a3, a4 is a dominant path.
Prevalence
Def. Prevalence wrt. a vertex an
Let a1  an be a dominant path with vertices a1,a2,…,an. Then,
- every vertex ai prevails a1 wrt. an (1< i  n).
- 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.
a1  ai
an
a. n
..
a2
a1
a. n
..
ai
a1  bj
an
b. m
..
b1
ai-1
..
.
a1
Example: formalizing agents
 Epistemic agents can be formalized via WMDLPs.
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
Example: formalizing agents
A
B
C
A = (DA,DA)
DA = ({v1},{},wA)
Pv1 = {answerPhone  secretary  phoneRing}
A
v1
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
B
v3
v4
C
v5
v6
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 WMDLPs
• L is a set of links among the i
• and wL : L  R+.
Semantics of F
 Declarative semantics of F is stable model based.
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:
s
Pv1 = {answerPhone}
Pv2 = {not answerPhone}
if v2  v1 then Ms={answerPhone}
s
v2
v1
Proof theory
The operational semantics for WMDLPs is based on a syntactic
transformation (P, s).
• (P, s) extends the syntactic transformation for MDLPs.
(P, s) is based on the (strong) prevalence relation Cs .
• Given a WMDLP P and a state s, the transformation (P, s)
produces a generalized logic program.
Correctness of the transformation.
• The stable models of (P, s) coincide with the stable models of
P at state 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 WMDLPs, 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
G = (DG,DG)
DG =
({v2},{},wG)
Pv2 = {}
F
G
v2
A
v1
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
C
v5
B
v3
v4
v6
Example: semantics
G
v2
0.5
0.5
A
v1
C
v5
0.6
0.7
B
v3
v6
v1  v6
v5
0.6
v4
v 4  v1
v3
Model of agent B: Mv3 = {phoneRing, secretary, answerPhone}
Model of agent C: Mv5 = {phoneRing, secretary, not answerPhone}
Adding roles to agents
 A role is a set of obligations and rights that governs the behaviour of an agent
occupying a particular position in the society.
 The adoption of roles as tools for description and modelling in multi-agent
systems has several benefits:
• Formal roles allow for generic models of agents with unknown internal states
to derive information to predict agent behaviour.
• The use of roles promotes flexibility since different modes of interaction
become possible among agents.
• Roles can adapt and evolve within the course of interactions to reflect the
learning process of the agents.
This allows for dynamic systems where the modes of interactions change.
Adding roles to agents
 When an agent plays a role, the overall behaviour of the agent obeys the
personality of the agent as well as its role.
We call actor an agent playing a role:
actor := < role, agent >
actor := < role, actor >
The notion of actor is important to define situations where an agent plays some
role by virtue of playing another role.
Adding roles to agents
 Actors can be expressed in our framework in a modular, flexible way.
i
w1
role
w2
agent
By assigning different weights w1 and w2, different types of behaviour can be
modelled:
• w1 > w2: the actor will obey the norms of its role.
• w2 > w1: the personality of the actor will prevail its role.
• w1 = w2: the actor will operate in accordance to both its personality and role.
Adding roles to agents
 An actor can fulfill several roles depending on the context.
i
w1
w2
i2
w3
i
w4
Two agents playing the same role.
i2
w1
w2
w3
w4
One agent playing two distinct roles.
Adding roles to agents
i2
i
w1
w5
i2
w2
w1
w1
w2
w2
i
i
w3
w3
An actor
w4
An actor playing
two roles
w3
w4
Hierarchy of actors
Engineering social agent societies
 Roles are associated with a default context that defines the different social
relationships and specifies the behaviour of the roles amongst each other.
 Agents may interact in several, different contexts. Therefore, there is a need
to consider different abstraction levels of contexts.
• More specific contexts can overturn the orderings between the roles of
more general contexts, and establish a social relation among them.
Engineering social agent societies
Def. Context
Let Ag a set of agents and R a set of roles.
A context is a pair (Ac,T) where Ac is a set of actors defined over Ag and R,
and T is a theory defining the normative relations of the context.
T
i1
i2
Context
i3
Engineering social agent societies
Def. Social agent society
An agent society  is modelled as a tuple:
 = (Ag,R,C)
where Ag is a set of epistemic agents, R is a set of roles,
and C is a set of contexts over Ag and R.
 Modelling agent societies by means of the notion of contexts is general and
flexible: several organizational structures can be expressed in terms of
contexts.
Engineering social agent societies
Agent society
Agents may form subgroups inside a greater society of agents. These
subgroups usually inherit the constraints of the greater society, override
some of them and add their own constraints.
Agent societies based on
confidence factors
A society whose agents have the ability to associate a confidence factor:
• to the information incoming from other agents,
• to the information outgoing to other agents, and
• to its own information.
Confidence factors can be used:
• to indicate the level of trust/confidence of an agent towards another agent,
• the relevance of the information of a source agent,
• the confidence that an agent has about its own information,
• the strength with which an agent supports its information towards another
agent.
Agent societies based on
confidence factors
The structure of agent societies based on confidence factors is
expressed by CDAGs.
A directed acyclic graph with confidence factors (CDAG) is a tuple
(V, E, ws, wi, wt, w) where:
• V is a set of vertices,
• E a set of edges containing the edge (v,v) for any vertex v V,
• ws: VR
self-confidence
• wi: E R
confidence given to the outgoing edge
• wt: E R
confidence given to the incoming edge
• w: E R
final weight of the edge
Agent societies based on
confidence factors
CDAGs can be formalized via WDAGs as follows:
a
0.7
0.3
0.6
b
0.9
0.6
a
0.8
0.7
b
c
CDAG
Suppose, for any edge, that
w(e) = ( wi(e) + wt(e) ) / 2
0.75
0.7
c
a+
0.9
0.6
0.9
c+
b+
WDAG
Future work
Other notions of prevalence can be accommodated in our
framework:
• A voting system – based on the incoming edges of a certain
node. Rules can be rejected because they are outweighed or
outvoted by opting for the best positive or negative average.
The logical framework can be represented within the theory
of the agent members of the society.
• Doing so will empower the agents with the ability to reason
about and to modify the structure of their own graph together
with the general group structure comprising the other agents.
Conclusion
 We have introduced a novel powerful and flexible logical
framework to model structures of epistemic agents:

The declarative semantics is stable models based

The procedural semantics relies on a sequence of syntactical
transformations into normal programs
The End !
MORE ...
J. A. Leite, J. J. Alferes, L. M. Pereira, Combining Societal Agents' Knowledge, in L. M.
Pereira, P. Quaresma (eds.), Procs. of the APPIA-GULP-PRODE'01 Joint Conf. on
Declarative Programming (AGP'01), pp. 313-327, Évora, Portugal, September 2001.
P. Dell'Acqua, J. A. Leite, L. M. Pereira, Evolving Multi-Agent Viewpoints - an
architecture, in P. Brazdil, A. Jorge (eds.), Progress in Artificial Intelligence, 10th
Portuguese Int. Conf. on Artificial Intelligence (EPIA'01), pp. 169-182, Springer, LNAI
2258, Porto, Portugal, December 2001.
P. Dell'Acqua, L. M. Pereira, A Logical Framework for Modelling eMAS, in V. Dahl, P.
Wadler (eds.), Procs. Practical Aspects of Declarative Languages (PADL'03), New
Orleans, Louisiana, USA, pp. 241-255, Springer, LNCS 2562, 2003.
P. Dell'Acqua, Mattias Engberg, L. M. Pereira, An Architecture for a Rational Reactive
Agent, in F. Moura-Pires, S. Abreu (eds.), Progress in Artificial Intelligence, Procs. 11th
Portuguese Int. Conf. on Artificial Intelligence (EPIA'03), pp.379-393, Springer,
LNAI, Beja, Portugal, December 2003.
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)
wL(e)
if eEi
if eL
Joined WMDLP
Def. Joined WMDLP
Let F=(A,L,wL ) be a logical framework.
Assume that A={1,…,n} and every i=(Di,Di).
The joined WMDLP induced by F is the WDAG =(D,D)
where:
- D= ({D1,…, Dn},L,wL)
- D= i Di
and
Stable models of WMDLP
Def. Stable models of WMDLP
Let =(D,D) be a WMDLP, where D=(V,E,w) and
D={Pv : vV}. 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’),v2  v1 }
s
X = Q - Reject(s,M)
Default(X,M) = {not A :  (ABody) in X and M |= Body }