Updating Agents

1
A logical framework for modelling eMAS

• Pierangelo DellAcqua
• Dept. of Science and Technology - ITN
• Linköping University, Sweden

• Luís Moniz Pereira
• Centro de Inteligência Artificial - CENTRIA
• Universidade Nova de Lisboa, Portugal

PADL03
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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.

3
Our agents

• We have been proposing a LP approach to agents
  which can
• Reason on their own or in collaboration
• React to other agents and to the environment
• Update their own knowledge, reactions, and goals
• Interact by updating the theory of any other
  agent
• Decide whether to accept an update subject to the
  requesting agent
• Capture the representation of social evolution

4
Framework

• This framework builds on the works
• Updating Agents
• P. DellAcqua L. M. Pereira - MAS99
• Multi-dimensional Dynamic Logic Programming
• L. A. Leite J. J. Alferes L. M. Pereira
  - CLIMA01
• and subsequent ones.

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Updating agents cycle

• Updating agent a rational, reactive agent that
  dynamically changes its own knowledge and goals.
• In its cycle, in some order, it

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Logic framework

• Atomic formulae

A atom not A default atom
generalized rule
Integrity constraints Action rules
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Agents knowledge state sequences

• Knowledge states represent dynamically evolving
  states of an agents knowledge. They undergo
  change due to updates (DLP).

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

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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)

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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.

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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 its
multiple dimensions ?
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Two basic dimensions of a MAS
How to combine these dimensions into one DAG ?
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Equal Role Representation

• Assigns equal role to the two dimensions

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Time Prevailing Representation

• Assigns priority to the time dimension

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Hierarchy Prevailing Representation

• Assigns priority to the hierarchy dimension

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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

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Intra- and Inter- Agent Example

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

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MDLPs revisited

• Def. MDLP Multi-Dimensional Logic Program
• A MDLP ? is a pair (?D,D), where
• D(V,E,w) is a
• WDAG - Weighted directed acyclic graph
• and,
• ?DPv v?V is a set of generalized logic
  programs indexed by the vertices of D.

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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.
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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 both
• conclusions are false.
• (Or, two alternative conclusions
• can be made possible.)

a
v
0.1
a
not a
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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)).

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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
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Example formalizing agents

• Epistemic agents can be formalized via MDLPs.

• 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

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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
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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.

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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

• Procedural semantics based on a syntactic
  transformation.

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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.

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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.

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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.

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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
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Example semantics
Model of agent B Mv3 phoneRing, secretary,
answerPhone
Model of agent C Mv5 phoneRing, secretary,
not answerPhone
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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.

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The End
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Prevalence

• 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 (1lt i ?
  n).
• 2. 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.

1.
2.
a1 ? ai
a1 ? bj
an
an
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Links

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

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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
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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

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Stable models of MDLP

• Def. Stable models of MDLP
• Let ?(?D,D) be a MDLP, 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
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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.