Combining agents into societies

1
Combining agents into societies

• Luís Moniz Pereira

Centro de Inteligência Artificial
CENTRIA Universidade Nova de Lisboa
DEIS, Università di Bologna, 22 Marzo 2004
2
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

3
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.

4
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

5
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 agents own behavioural
  evolution
• LP declaratively describes states well.
  
• But LP must describe state transitions too.

6
MDLP overview

• DLP synopsis
• MDLP motivation
• MDLP semantics
• Multiple representational dimensions in a
  multi-agent system
• Representation prevalence
• Overview conclusions

7
Dynamic LP

• DLP was introduced to express LPs 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

8
MDLP Motivating Example

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

• More recent laws have precedence over older ones

• How to combine these two dimensions of knowledge
  precedence?

• DLP with Multiple Dimensions (MDLP)

9
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

10
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

11
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

12
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 Pi
Ms least( Ps Reject(s, Ms) ? Defaults (Ps,
Ms) )
13
MDLP - semantics 2
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)
14
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 agents evolving knowledge along multiple
dimensions ?
15
Two basic dimensions of a multi-agent system
How to combine these dimensions into one DAG ?
16
Equal Role Representation

• Assigns equal role to the two dimensions

17
Equal Role - 2

• 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

18
Time Prevailing Representation

• Assigns priority to the time dimension

19
Time Prevailing - 2

• Useful in very dynamic situations, where
  competence is distributed, i.e. ¹ agents normally
  provide rules about ¹ literals
• Drawback
• It requires all agents to be fully trusted, since
  all newer rules override older ones irrespective
  of their mutual hierarchical position

20
Hierarchy Prevailing Representation

• Assigns priority to the hierarchy dimension

21
Hierarchy Prevailing - 2

• 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).

22
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

23
Intra- and Inter- Agent Example

• Prevailing hierarchy for inter-agents
• Prevailing time for sub-agents

24
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
  sub-agents that execute special tasks, e.g.
• reactivity
• planning
• scheduling
• belief revision

• goal management
• learning
• preference evaluation
• strategy

25
MDLP overview conclusions

• Weve explored MDLP to combine knowledge from
  several agents and multiple dimensions
• Depending on the situation, and relationships
  among agents, weve envisaged several classes of
  DAGs for their encoding
• Based on this work, and on a language (LUPS) for
  specifying updates by means of transitions, weve
  launched into the design of an agent architecture
  MINERVA

26
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

27
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

28
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)

29
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.

30
New contribution

• To extend the framework of MDLP with integrity
  constraints and active rules.
• To incorporate the framework of MDLP into a
  multi-agent architecture.
• To make the DAG of each agent updatable.

31
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.

32
Agents language

• Atomic formulae

A objective atoms not A default atoms
iC projects i?C updates
Formulae
generalized rules
Li is an atom, an update or a negated update
A L1 Ù...Ù Ln not A L1 Ù...Ù Ln
Zj is a project
integrity constraint
false L1 Ù...Ù Ln Ù Z1 Ù...Ù Zm
active rule
L1 Ù...Ù Ln ? Z
33
Projects and Updates
A project jC 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
wilmaC
34
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 Ps1 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.

35
Agents language
A project iC can take one of the forms
i ( A L1 Ù...Ù Ln )
i ( not A L1 Ù...Ù Ln )
i ( false L1 Ù...Ù Ln Ù Z1 Ù...Ù Zm )
i ( L1 Ù...Ù Ln ? Z )
i ( ?- L1 Ù...Ù Ln )
i edge(u,v)
i not edge(u,v)
36
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.

37
The agents 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.

38
Happy story - example
DAG of Alfredo
inspection point of Alfredo
The goal of Alfredo is to be happy
39
Happy story - example
alfredo
judge
hasGirlfriend not happy ? father
(?-happy) not happy ? mother (?-happy) getMarrie
d Ù hasGirlfriend ? girlfriend propose moveOut
? alfredo rentApartment custody(judge,mother) ?
alfredo edge(father,mother) moveOut,
getMarried
mother
father
alfredo
girlfriend
abducibles
state 0
40
Happy story - example
alfredo
judge
hasGirlfriend not happy ? father
(?-happy) not happy ? mother (?-happy) getMarrie
d Ù hasGirlfriend ? girlfriend propose moveOut
? alfredo rentApartment custody(judge,mother) ?
alfredo edge(father,mother) moveOut,
getMarried
mother
father
alfredo
girlfriend
state 0
41
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 ?. - UU1,,
Us is a sequence of updating programs.
42
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.
43
Happy story - 1st scenario

• Suppose that at state 1, Alfredo receives
• from the mother

mother ? (happy moveOut) mother ? (false
moveOut Ù not getMarried) mother ? (false not
happy)
and from the father
father ? (happy moveOut) father ? (not happy
getMarried)
44
Happy story - 1st scenario
alfredo
false not happy
happy moveOut false moveOut Ù not getMarried
judge
mother
father
happy moveOut not happy getMarried
alfredo
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.
girlfriend
state 1
45
Happy story - 2nd scenario

• Suppose that at state 1 Alfredos parents decide
  to get divorced, and the judge gives custody to
  the mother.

judge ? custody(judge,mother)
46
Happy story - 2nd scenario
alfredo
custody(judge,mother)
judge
hasGirlfriend not happy ? father
(?-happy) not happy ? mother (?-happy) getMarrie
d Ù hasGirlfriend ? girlfriend propose moveOut
? alfredo rentApartment custody(judge,mother) ?
alfredo edge(father,mother)
mother
father
alfredo
girlfriend
state 1
47
Happy story - 2nd scenario
alfredo
Note that the internal update produces a change
in the DAG of Alfredo.
judge
mother
father
Suppose that when asked by Alfredo, the parents
reply in the same way as in the 1st scenario.
alfredo
girlfriend
state 2
48
Happy story - 2nd scenario
alfredo
false not happy
happy moveOut false moveOut Ù not getMarried
judge
mother
father
happy moveOut not happy getMarried
alfredo
Now, the advice of the mother prevails over and
rejects that of his father.
girlfriend
state 2
49
Happy story - 2nd scenario
Thus, Alfredo gets married, rents an apartment,
moves out and lives happily ever after.
alfredo
judge
hasGirlfriend not happy ? father
(?-happy) not happy ? mother (?-happy) getMarrie
d Ù hasGirlfriend ? girlfriend propose moveOut
? alfredo rentApartment custody(judge,mother) ?
alfredo edge(father,mother)
mother
father
alfredo
girlfriend
state 2
50
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.
51
Agent architecture
? ??s ?P,A,R?
Java
CC
InterProlog (Declarativa)
InterProlog (Declarativa)
Rational P
Reactive PR
can abduce
cannot abduce
XSB Prolog
XSB Prolog
52
Agent architecture
? ??s ?P,A,R?
53
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.

54

A logical framework for modelling eMAS
55
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.

56
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.)

a
v
0.1
a
not a
57
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.
58
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.

59
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)) lt
  w((bj-1,bj)).

60
Example path dominance
a4
Let w((a5,a4)) lt w((a3,a4)) . Then, a1, a2 , a3,
a4 is a dominant path.
a3
a5
a2
a1
61
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 (1lt i ? n).
• - if there exists a path b1 ? ai with vertices
  b1,,bm,ai and
• w((ai-1,ai)) lt w((bm,ai)), then every vertex
  bj prevails a1 wrt. an.

a1 ? ai
a1 ? bj
an
an
62
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

63
Example formalizing agents
?A (?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
64
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.

65
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 Pv1
answerPhone Pv2 not answerPhone if
then MsanswerPhone
66
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.

67
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.

68
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.

69
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.

70
Example formalizing groups
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
?G (?DG,DG) DG (v2,,wG) Pv2
G
F
71
Example semantics
Model of agent B Mv3 phoneRing, secretary,
answerPhone
Model of agent C Mv5 phoneRing, secretary,
not answerPhone
72
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.

73
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 lt role, agent gt
• actor lt role, actor gt
• The notion of actor is important to define
  situations where an agent plays some role by
  virtue of playing another role.

74
Adding roles to agents

• Actors can be expressed in our framework in a
  modular, flexible way.
• role
• agent
• By assigning different weights w1 and w2,
  different types of behaviour can be modelled
• w1 gt w2 the actor will obey the norms of its
  role.
• w2 gt w1 the personality of the actor will
  prevail its role.
• w1 w2 the actor will operate in accordance to
  both its personality and role.

i
w1
w2
75
Adding roles to agents

• An actor can fulfill several roles depending on
  the context.

Two agents playing the same role. One agent
playing two distinct roles.
76
Adding roles to agents
An actor
An actor playing two roles
Hierarchy of actors
77
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.

78
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
i2
i1
i3
Context
79
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.

80
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.
81
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.

82
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

83
Agent societies based on confidence factors

• CDAGs can be formalized via WDAGs as follows

CDAG
WDAG
Suppose, for any edge, that w(e) ( wi(e)
wt(e) ) / 2
84
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.

85
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

86
The End !
MORE ...
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Progress in Artificial Intelligence, Procs. 11th
Portuguese Int. Conf. on Artificial Intelligence
(EPIA'03), pp.379-393,  Springer, LNAI,  Beja,
Portugal,  December 2003.
87
Links

• Def. Link
• Given two WDAGs, D1 and D2, a link is an edge
  between a vertice of D1 and a vertice D2.

88
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 e?Ei wL(e) if e?L
89
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) and
• - ?D ?i ?Di

90
Stable models of WMDLP

• Def. Stable models of WMDLP
• Let ?(?D,D) be a WMDLP, where D(V,E,w) and
  ?DPv 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),
X Q - Reject(s,M) Default(X,M) not
A ?? (ABody) in X and M Body