1. Artificial Intelligence Chapter 23. Multiple Agents

2. (C) 2000, 2001 SNU CSE Biointelligence Lab
Outline
Interacting Agents
Models of Other Agents
A Modal Logic of Knowledge
Additional Readings and Discussion
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23.1 Interacting Agents
Agents’ objectives
To predict what another agent will do :
Need methods to model another
To affect what another agent will do :
Need methods to communicate with another
Focus
Distributed artificial intelligence (DAI)
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23.2 Models of Other Agents
Varieties of models
Need of model : to predict the behavior of other agents and processes
Model focused : high level model ( e.g. T-R model)
The model and its apparatus for using that model to select actions : cognitive structure
Cognitive structure often includes an agent’s goals and intentions
Our focus of cognitive structure : its model of its environment and of the cognitive structures of others agents
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23.2 Models of Other Agents
Model strategies
Iconic based
Attempts to simulate relevant aspects of the environment
Feature based
Attempts to describe the environment
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23.2 Models of Other Agents
Simulation Strategies
Often useful but suffer from the difficulty representing ignorance or uncertainty
Simulation Databases
Build a hypothetical DB of formulas presumed to be the same formulas that our agent thinks actually populate that other agent’s world model
It has the same deficiencies as iconic models
Uncertainty : whether our agent has a formula or other agent has the formula
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23.2 Models of Other Agents
The Intentional Stance
Capable of describing another agent’s knowledge and beliefs about the world
Describing another agent’s knowledge and beliefs : taking intentional stance
3 possibilities for constructing intentional-stance
1. Reify the other agent’s beliefs [McCarthy 1979] : Bel( On(A,B) )
2 . Our agent assume that the other agent actually represents its beliefs about the world by predicate-calculus formulas in its DB
: Bel( Sam, On(A,B) )
3. Use modal operators
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8. 23.3 A Modal Logic of Knowledge
Modal Operators
Modal operator
to construct a formula whose intended meaning is that a certain agent knows a certain proposition
e.g) K( Sam, On(A,B) )  K(α,φ) or K α(φ)
knowledge and belief
Whereas an agent can believe a false proposition, it cannot know anything that is false.
logic of knowledge is simpler than logic of belief
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9. 23.3 A Modal Logic of Knowledge
Modal first-order language
using the operator K
syntax
1. All of the wffs of ordinary first-order predicate calculus are also wwf of the modal language
2. If φ is a closed wff of the modal language, and if α is a ground term, then K(α, φ) is a wff of the modal language.
3. As usual, if φ and ψ are wffs, then so are any expressions that can be constructed from φ and ψ by the usual propositional connectives.
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10. 23.3 A Modal Logic of Knowledge
As examples,
K[Agent1, K[Agent2, On(A,B))] : Agent1 knows that Agent2 knows that A is on B.
K(Agent1, On(A,B))  K(Agnet1, On(A,C)) : Either Agent1 knows that A is on B or it knows that A is on C
K[Agent1, On(A,B)  On(A,C)] : Agent1 knows that either A is on B or that A is on C.
K(Agent1, On(A,B))  K(Agent1, ¬On(A,B)) : Agent1 knows whether or not A is on B.
¬K(Agent1, On(A,B)) : Agent1 doesn’t know that A is on B.
(x)K(Agent1, On(x,B)) : illegal wwf
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11. 23.3 A Modal Logic of Knowledge
Knowledge Axioms
,  : compositional semantics
Semantics of K is not compositional.
truth value of Kα [φ] is not depend on α and φ compositionally
φ  ψ, Kα(φ)  Kα(ψ) for all α : not necessary since α might not know that φ is equivalent to ψ.
axiom schemas
distribution axiom
[Kα(φ) Kα(φ ψ)] Kα(ψ) … (1)
(  Kα(φ  ψ) [Kα(φ) Kα(ψ)] … (2) )
knowledge axiom
Kα(φ)  φ …(3) : An agent cannot possibly know something that is false.
positive-introspection axiom
Kα(φ)  Kα(Kα(φ)) … (4)
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12. 23.3 A Modal Logic of Knowledge
negative-introspection axiom
Kα(φ)  Kα(¬Kα(φ)) … (5)
epistemic necessitation
├ φ infer Kα(φ) … (6)
logically omniscienct
φ ├ ψ and from Kα(φ) infer Kα(ψ) … (7)
(  ├ (φψ) infer Kα(φ) Kα(ψ) … (8)
from logical omniscience, K(α, (φψ))  K(α, φ) K(α, ψ) … (9)
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13. 23.3 A Modal Logic of Knowledge
Reasoning about Other Agents’ Knowledge
Our agent can carry out proofs of some statements about the knowledge of other agents using only the axioms of knowledge, epistemic necessitation, and its own reasoning ability (modus ponens, resolution).
e.g) Wise-Man puzzle
assumption : among three wise men, at least one has a white spot on his forehead. Each wise man can see the others’ foreheads but not his own. Two of them said, “I don’t know whether I have a white spot”.
proof of K(A, White(A)) (where, A is the third man.)
1. KA[¬White(A)  KB(¬White(A))] (given)
2. KA[KB(¬White(A)  White(B))] (given)
3. KA(¬KB(White(B))) (given)
4. ¬White(A)  KB(¬White(A)) (1, and axiom 3)
5. KB[¬White(A)  White(B)) (2, and axiom 2)
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14. 23.3 A Modal Logic of Knowledge
6. KB(¬White(A))  KB(White(B)) (5. and axiom 2)
7. ¬White(A)  KB(White(B)) (resolution on the clause forms of 4. and 6.)
8. ¬KB(White(B))  White(A) (contrapositive of 7.)
9. KA[¬KB(White(B))  White(A)] (1.- 5., 8., rule 7)
10. KA(¬KB(White(B)))  KA(White(A)) (axiom 2)
11. KA(White(A)) (modus ponens using 3. and 10.)
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15. 23.3 A Modal Logic of Knowledge
Predicting Actions of Other Agents
in order to predict what another agent A1,
If A1 is not too complex, our agent may assume that A1’s action are controlled by a T-R program. Suppose the conditions in that program are i, for i=1, …, k. To predict A1’s future actions, our agent needs to reason about how A1 will evaluate these conditions.
It is often appropriate for our agent to take an intentional stance toward A1 and attempt to establish whether or not KA1(i) for
i=1, …, k
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16. Additional Readings and Discussion
[Shoham 1993] : “agent-oriented programming”
[Minsky 1986] : Society of Mind
[Hintikka 1962] : modal logic of knowledge
[Kripke 1963 : possible-worlds semantics for modal logics
[Moore 1985b] : possible-worlds semantics within first-order logic
[Levesque 1984b, Fagin & Halpern 1985, Konoliege 1986]
[Cohen & Levesque 1990] : modal logic for the relationship between intention and commitment
[Bond & Gasser 1988] : DAI paper collection
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