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 SettingProblem 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 Π⊆ i

(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 (1i 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( d

2, 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 P∪ i+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( d

2, 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# ).

P1 ：　　 date(d1)← , date(d2) ←

G: ← not scheduled(d1), reserve(d1,Flight# ).

P1 ：　　 scheduled(d1)←fail

Example (cont.)

! BacktrackG: ← 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 ! BacktrackG: ← 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.