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 A 1,…,A n be agents and F a propositinal sentence. When A 1  ≠ F (†) F is not proved by A 1 but F may be proved by other agents A 2,…,A n. ⇒ 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 {A 1,…, A n } with agents A i (1 ≦ i ≦ n), a multi-agent logic program (MLP) is defined as a set { P 1,…, P n } where P i is the program of A i. P i is an extended logic program which consists of rules of the form: L 0 ← L 1,…, L m, not L m+1,…, not L n where L i 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. P S is a set of ground rules s.t. L 0 ← L 1,…, L m is in P S iff L 0 ← L 1,…, L m, not L m+1,…, not L n is in ground(P) and { L m+1,…, L n }∩ S ＝ φ. 2. S = Cn( P S ). 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 A i is called rational if it has a consistent program P i. We assume an MAS {A 1,…, A n } where each agent A i is rational.

Semantics of MLP Given an MLP {P 1,…, P n }, the program Π i is defined as (i) P i ⊆ Π i (ii) Π i is a maximal consistent subset of P 1 ∪ ・・・ ∪ P n A set S of ground literals is called a belief set of an agent A i if S=T∩Lit(P i ) where T is an answer set of Π i.

Belief Sets An agent has multiple belief sets in general. Belief sets are consistent and minimal under set inclusion. Given an MAS { A 1,…, A n }, an agent A i (1 ≦ i ≦ n) concludes a propositional sentence F (written A i |=F ) if F is true in every belief set of A i.

Example Suppose an MLP { P 1, P 2 } such that P 1 : 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 )←. P 2 : state( f456, full ) ←.

Example (cont.) The agent A 1 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 { P 1, P 2, P 3 } such that P 1 : go_cinema ← interesting, not crowded ￢ go_cinema ← ￢ interesting P 2 : interesting ← P 3 : ￢ interesting ← The agent A 1 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 S H if S H is a consistent answer set of P ∪ H where H ⊆ A. ・ A belief set S H is maximal (with respect to A) if there is no belief set T K such that H ⊂ K.

Abductive Characterization of MLP Given an MLP {P 1,…, P n }, an agent A i has a belief set S iff S=T H ∩Lit(P i ) where T H is a maximal belief set of the abductive logic program 〈 P i ; P 1 ∪・・・∪ P i － 1 ∪ P i+1 ∪・・・∪ P n 〉.

Problem Solving in MLP We consider an MLP {P 1,…, P n } where each P i 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 { P 1, P 2 } with P 1 : 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 )←. P 2 : state( f456, full ) ←.

Example (cont.) Goal: ← travel( Date, Flight# ). P 1 ： 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# ). P 1 ： date(d1)←, date(d2) ← G: ← not scheduled(d1), reserve(d1,Flight# ). P 1 ： scheduled(d1)← fail

Example (cont.) ! Backtrack G: ← date( Date ), not scheduled( Date ), reserve( Date, Flight# ). P 1 ： date(d1)←, date(d2) ← G: ← not scheduled(d2), reserve(d2, Flight# ). G: ← reserve(d2, Flight# ).

Example (cont.) G: ← reserve(d2, Flight# ). P 1 : reserve( Date, Flight# ) ← flight( Date, Flight# ), not state( Flight#, full ). G: ← flight( d2, Flight# ), not state( Flight#, full ). P 1 : flight( d2, f456 )←. flight( d2, f789 )←. G: ← not state( f456, full ). ! Suspend the goal and ask P 2 whether state( f456, full ) holds or not.

Example (cont.) G: ← not state( f456, full ). P 2 : state( f456, full ) ← fail ! Backtrack G: ← flight( d2, Flight# ), not state( Flight#, full ). P 1 : flight( d2, f456 )←. flight( d2, f789 )←. G: ← not state( f789, full ).

Example (cont.) G: ← not state( f789, full ). ! Suspend the goal and ask P 2 whether state( f789, full ) holds or not. The goal ← state( f789, full ) fails in P 2 then G succeeds in P 1. As a result, the initial goal ← travel( Date, Flight# ) has the unique solution Date=d2 and Flight# = f789.

Correctness Let {P 1,…, P n } be an MLP where each P i is a stratified normal logic program. If an agent A i solves a goal G with an answer substitution θ, then A i |= Gθ, i.e., Gθ is true in the belief set of A i.

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.