Download Abductive and Default Reasoning in Multi

Default and Cooperative Reasoning
in Multi-Agent Systems
Chiaki Sakama
Wakayama University, Japan
Programming Multi-Agent Systems based on Logic
Dagstuhl Seminar, November 2002
Incomplete Knowledge in
Multi-Agent System (MAS)
• An individual agent has incomplete
knowledge in an MAS.
• In AI a single agent performs default
reasoning when its knowledge is incomplete.
• In a multi-agent environment, caution is
requested to perform default reasoning based
on an agent’s incomplete knowledge.
Default Reasoning
by a Single Agent
Let A be an agent and F a propositional
sentence. When
A |≠ F
F is not proved by A and ～F is assumed by
default (negation as failure).
Default Reasoning in
Multi-Agent Environment
Let A1,…,An be agents and F a propositinal
sentence. When
A1 |≠ F (†)
F is not proved by A1 but F may be proved
by other agents A2,…,An .
⇒ It is unsafe to conclude ～F by default due to
the evidence of (†).
Default Reasoning v.s.
Cooperative Reasoning in MAS
• An agent can perform default reasoning
if it is based on incomplete belief wrt
an agent’s internal world.
• Else if an agent has incomplete knowledge
about its external world, it is more appropriate
to perform cooperative reasoning.
Purpose of this Research
• It is necessary to distinguish different types
of incomplete knowledge in an agent.
• We consider a multi-agent system based on
logic and provide a framework of
default/cooperative reasoning in an MAS.
Problem Setting
• An MAS consists of a finite number of
agents.
• Every agent has the same underlying
language and shared ontology.
• An agent has a knowledge base written by
logic programming.
Multi-Agent Logic Program
(MLP)
• Given an MAS {A1 ,…, An } with agents Ai
(1≦i≦n), a multi-agent logic program (MLP)
is defined as a set { P1 ,…, Pn } where Pi is
the program of Ai .
• Pi is an extended logic program which
consists of rules of the form:
L0 ← L1 ,…, Lm , not Lm+1 ,…, not Ln
where Li is a literal and not represents
negation as failure.
Terms / Notations
• Any predicate appearing in the head of no rule
in a program is called external, otherwise, it is
called internal . A literal with an
external/internal predicate is called an
external/internal literal．
• ground(P): ground instantiation of a program P.
• Lit(P): The set of all ground literals appearing
in ground(P).
• Cn(P) = { L | L is a ground literal s.t. P |= L }.
Answer Set Semantics
Let P be a program and S a set of ground
literals satisfying the conditions:
1. PS is a set of ground rules s.t.
L0← L1 ,…, Lm is in PS iff
L0 ← L1 ,…, Lm , not Lm+1 ,…, not Ln is in
ground(P) and { Lm+1 ,…, Ln }∩ S ＝φ.
2. S = Cn( PS ).
Then, S is called an answer set of P.
Rational Agents
• A program P is consistent if P has a
consistent answer set.
• An agent Ai is called rational if it has a
consistent program Pi.
• We assume an MAS {A1 ,…, An } where
each agent Ai is rational.
Semantics of MLP
Given an MLP {P1,…, Pn}, the program Πi is
defined as
(i) Pi ⊆Πi
(ii) Πi is a maximal consistent subset of
P1 ∪ ・・・ ∪ Pn
A set S of ground literals is called a belief set
of an agent Ai if S=T∩Lit(Pi ) where T is an
answer set of Πi .
Belief Sets
• An agent has multiple belief sets in general.
• Belief sets are consistent and minimal under
set inclusion.
• Given an MAS { A1 ,…, An }, an agent Ai
(1≦i≦n) concludes a propositional sentence
F (written Ai |=F ) if F is true in every
belief set of Ai .
Example
Suppose an MLP { P1, P2 } such that
P1: travel( Date, Flight# ) ← date( Date ),
not scheduled( Date ),
reserve( Date, Flight# ).
reserve( Date, Flight# ) ←
flight( Date, Flight# ),
not state( Flight#, full ).
date( d1 )←. date( d2 )←. scheduled(d1)←.
flight( d1, f123 )←. flight( d2, f456 )←.
flight( d2, f789 )←.
P2: state( f456, full ) ← .
Example (cont.)
The agent A1 has the single belief set
{ travel( d2, f789 ) , reserve( d2, f789 ),
date( d1 ), date( d2 ), scheduled(d1),
flight( d1, f123 ), flight( d2, f456 ),
flight( d2, f789 ), state( f456, full ) }
Example
Suppose an MLP { P1, P2 , P3 } such that
P1 : go_cinema ← interesting, not crowded
￢ go_cinema ← ￢ interesting
P2: interesting ←
P3: ￢ interesting ←
The agent A1 has two belief sets:
{ go_cinema, interesting }
{ ￢ go_cinema, ￢ interesting }
Abductive Logic Programs
・An abductive logic program is a tuple 〈P,A〉
where P is a program and A is a set of
hypotheses (possibly containing rules).
・〈P,A〉 has a belief set SH if SH is a consistent
answer set of P ∪ H where H⊆A.
・A belief set SH is maximal (with respect to A) if
there is no belief set TK such that H⊂K .
Abductive Characterization of MLP
Given an MLP {P1 ,…, Pn} , an agent Ai has a
belief set S iff S=TH∩Lit(Pi ) where TH is a
maximal belief set of the abductive logic
program
〈 Pi ; P1∪・・・∪ Pi－1 ∪ Pi+1 ∪・・・∪ Pn 〉.
Problem Solving in MLP
• We consider an MLP {P1 ,…, Pn} where each
Pi is a stratified normal logic program.
• Given a query ← G, an agent solves the goal in
a top-down manner.
• Any internal literal in a subgoal is evaluated
within the agent.
• Any external literal in a subgoal is suspended
and the agent asks other agents whether it is
proved or not.
Simple Meta-Interpreter
solve(Agent, (Goal1,Goal2)):solve(Agent,Goal1), solve(Agent,Goal2).
solve(Agent, not(Goal)):- not(solve(Agent, Goal)).
solve(Agent, int(Fact)):- kb(Agent, Fact).
solve(Agent, int(Goal)):- kb(Agent, (Goal:-Subgoal)),
solve(Agent, Subgoal).
solve(Agent, ext(Fact)):- kb(AnyAgent, Fact).
solve(Agent, ext(Goal)):- kb(AnyAgent, (Goal:-Subgoal)),
solve(AnyAgent, Subgoal).
Example
Recall the MLP { P1, P2 } with
P1: travel( Date, Flight# ) ← date( Date ),
not scheduled( Date ),
reserve( Date, Flight# ).
reserve( Date, Flight# ) ←
flight( Date, Flight# ),
not state( Flight#, full ).
date( d1 )←. date( d2 )←. scheduled(d1)←.
flight( d1, f123 )←. flight( d2, f456 )←.
flight( d2, f789 )←.
P2: state( f456, full ) ← .
Example (cont.)
Goal: ← travel( Date, Flight# ).
P1： travel( Date, Flight# ) ←
date( Date ),
not scheduled( Date ),
reserve( Date, Flight# ).
G:
← date( Date ),
not scheduled( Date ),
reserve( Date, Flight# ).
Example (cont.)
G: ← date( Date ),
not scheduled( Date ),
reserve( Date, Flight# ).
date(d1)← , date(d2) ←
G: ← not scheduled(d1), reserve(d1,Flight# ).
P1 ：
P1 ：
scheduled(d1)←
fail
Example (cont.)
! Backtrack
G: ← date( Date ),
not scheduled( Date ),
reserve( Date, Flight# ).
P1 ： date(d1)← , date(d2) ←
G: ← not scheduled(d2), reserve(d2, Flight# ).
G: ← reserve(d2, Flight# ).
Example (cont.)
G: ← reserve(d2, Flight# ).
P1: reserve( Date, Flight# ) ←
flight( Date, Flight# ),
not state( Flight#, full ).
G: ← flight( d2, Flight# ),
not state( Flight#, full ).
P1: flight( d2, f456 )←. flight( d2, f789 )←.
G: ← not state( f456, full ).
! Suspend the goal and ask P2 whether
state( f456, full ) holds or not.
Example (cont.)
G: ← not state( f456, full ).
P2 : state( f456, full ) ←
fail
! Backtrack
G: ← flight( d2, Flight# ),
not state( Flight#, full ).
P1: flight( d2, f456 )←. flight( d2, f789 )←.
G: ← not state( f789, full ).
Example (cont.)
G: ← not state( f789, full ).
! Suspend the goal and ask P2 whether
state( f789, full ) holds or not.
The goal ← state( f789, full ) fails in P2 then
G succeeds in P1 .
As a result, the initial goal
← travel( Date, Flight# ) has the unique
solution Date=d2 and Flight# = f789.
Correctness
Let {P1 ,…, Pn} be an MLP where each Pi
is a stratified normal logic program.
If an agent Ai solves a goal G with an
answer substitution θ, then Ai |= Gθ,
i.e., Gθ is true in the belief set of Ai .
Further Issue
• The present system suspends the goal with
an external predicate and waits a response
from other agents.
• When an expected response is known,
speculative computation [Satoh et al, 2000]
would be useful to proceed computation
without waiting for responses.
Further Issue
• The present system asks every other agent
about the provability of external literals and
has no strategy for adopting responses.
• When the source of information is known,
it is effective to designate agents to be asked
or to discriminate agents based on their
reliability.
Summary
• A declarative semantics of default and
cooperative reasoning in an MAS is provided.
Belief sets characterize different types of
incompleteness of an agent in an MAS.
• A proof procedure for query-answering in an
MAS is provided. It is sound under the belief
set semantics when an MLP is stratified.