lecture 4 multi-agent systems lecture 4 computer science wpi spring 2002 adina magda florea

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3 n Logic based representation –unique (almost) syntax  x  y loves(x,y) –formal (clear, well-defined) semantics Bel A loves(Bill, Mary)  shape(round)   color(green)  type(apple) n Rule based representation –situation-action or condition-conclusion rules + facts –subset of logic (Horn clauses) that emphasize implication if shape(round) and color(green) then type=apple n Frame-based representation –units, frames –subset of logic, represents relationship structured around objects in the universe apple01 shape: roundcolor: greentype: apple 1 Knowledge representation for agents Cognitive agents declarative representaton, AI What the agent knows/ believes

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Multi-Agent SystemsLecture 4Computer Science WPISpring 2002Adina Magda Floreaadina@wpi.edu

Formal models for representing agentsLecture outline

1Knowledge representation for agents

2FOL

3Modal logic

4Logics of knowledge and belief

5Dynamic logic, temporal logic

6BDI logics

7Commitments as change

8 Practical BDI interpreters - see Sect. 8.4 MA book

1 Knowledge representation for agents

Logic based representation

unique (almost) syntaxxy loves(x,y)

formal (clear, well-defined) semanticsBelAloves(Bill, Mary)

shape(round) color(green) type(apple)

Rule based representation

situation-action or condition-conclusion rules + facts

subset of logic (Horn clauses) that emphasize implication

if shape(round) and color(green) then type=apple

Frame-based representation

units, frames

subset of logic, represents relationship structured around objects in the universe

apple01

shape: roundcolor: greentype: apple

*

Cognitive agents

declarative representaton, AI

What

the agent

knows/

believes

Plan representation

represent actions

may be combined with any of the previous representations

partial representation of statesstack(x,y)

Precond: hold(x) clear(y)

Postcond: clear(x) hold(x)

on(x,y) armempty

BDI representations

combines most (all) of the above

A big diversity of techniques and formalisms to represent interactions:

communication

cooperation

coordination

No symbolic representation

*

When and what

to do

What the agent believes and

when and what to do

How to cope with other

agents in the environment

Reactive agents

Logic based representations

2 possible aims

to make MAS function according to the logic

to specify and validate the design

Conceptualization of the world / problemSyntax - wffsSemantics - significance, modelModel - the domain interpretation for which a formula is trueModel - linear or structured; index in a modelM |=S - " is true or satisfied in component S of the structure M"

Model theory

Generate new wffs that are necessarily true, given that the old wffs are true - entailmentKB |=

Proof theory

Derive new wffs based on axioms and inference rulesKB |-i

*

PrL, PL

*

Extend PrL, PL

Tropistic agents

(reactive)

Sentential logic

of beliefs

Uses beliefs atoms BA()

Index PL with agents

Modal logic

Modal operators

Logics of knowledge

and belief

Modal operators B and K

Dynamic logic

Modal operators

for actions

Temporal logic

Modal operators for time

Linear time

Branching time

CTL logic

Branching time

and action

BDI logic

Adds agents, B, D, I

Linear model

Structured models

Situation calculus

Adds states, actions

Symbol level

Knowledge level

2 First order logic

LP - the language of Propositional logic - the set of atomic propositions

Sin-1) implies that LP

Sin-2) p, q LP implies that pq LP, q LP

M0 = is the formal model for LPL - interpretation

Sem-1) M0 |= iff L, where

Sem-2) M0 |= pq iff M0 |= p and M0 |= q

Sem-3) M0 |= p iff M0 |=/ p

p=A BA - it rainsq=ABB - take umbrellar=AA

*

A B A B AB AA

T T T T T

T F F F T

F T T F T

F F T F T

Knowledge represents:

atomic propositions

Predicate logic

Knowledge represents:

Extensional knowledge

existence of objects: (x)(P(x)) is true exactly when P is true for at least one object of D, (x)(P(x))

facts about objects, not about properties of objects

p = (x) young(x) success(x)q = (x) young(x) success(x)

D = {Bill, Tom, Alice}MM |= p

x young(x) success(x)M |=/ q

Bill T T

Tom F T

Alice F F

*

*

Higher order logic

knowledge

propositional

first-order

Paul is a man

a

man(Paul)

Bill is a man

b

man(Bill)

men are mortal

c

((x) (man(x) ( mortal(x))

knowledge

first-order

second-order

smaller is transitive

(x) ((y) ((z) ((

3 Modal logic

LM - the language of Modal logic2 modal operators

p - p possible true p - p necessarily true

Sin-3) the rules of LP are in LM

Sin-4) p LP implies that p, p LM

Possible worldsThe structure of the model is given by relating different worlds via a binary accessibility relationM1 =W - a set of worlds

L:W P() - set of formula true in a world, R W X W

pp - it rains in NY qq - the sun will rise tomorrow

*

Sem-4) M1 |=W iff L(w), where

Sem-5) M1 |=W pq iff M1 |=W p and M1 |=W q

Sem-6) M1 |=W p iff M1 |=/W p

Sem-7) M1 |=W p iff (w': R(w,w') M1 |=W' p)

Sem-8) M1 |=W p iff (w': R(w,w') M1 |=W' p)

in w0 ? p, ? q, ? r

? p

The accessibility relation

- reflexive iff (w: (w,w)R) p p

- serial iff (w: (w': (w,w')R)) p p

- transitive iff (w1,w2,w3: (w1,w2)R (w2, w3)R (w1,w3)R)

p p

- symmetric iff (w1,w2: (w1,w2)R (w2,w1)R)p p

- euclidian iff (w1,w2,w3: (w1,w2)R (w1, w3)R (w2,w3)R)

p p

*

w0

p, q, r

w1

p, q, r

w2

p, q, r

w3

p, q, r

*

FOL augmented with two modal operators

K(a,) - a knows

B(a,) - a believes

Associate with each agent a set of possible worldsMk =W - a set of worlds

L:W P() - set of formula true in a world, R A x W X W

An agent knows/believes a propositions in a given world if the proposition holds in all worlds accessible to the agent from the given world

B(Bill, father-of(Zeus, Cronos))

? B(Bill, father-of(Jupiter,Saturn))

referential opaque operators

The difference between B and K is given by their properties

4 Logics of knowledge and belief

Properties of knowledge

(A1) Distribution axiom K(a, ) K(a, ) K(a, )

(A2) Knowledge axiom K(a, ) - satisfied if R is reflexive

(A3) Positive introspection axiom K(a, ) K(a, K(a, ))

- satisfied if R is transitive

(A4) Negative introspection axiomK(a, ) K(a, K(a, ))

- satisfied if R is euclidian

*

Properties of beliefs

(A1) - OK, (A2) - no, (A3) - yes,

(A4) - maybe but more problematic

Inference rules

(R1) Epistemic necessitation

|- infer K(a, )

(R2) Logical omniscience

and K(a, ) infer K(a, )

problematic

in w0 ?K(a,p), ?K(a, r), ?K(a,q)

w0

p, q, r

w1

p, q, r

w2

p, q, r

w3

p, q, r

Two-wise men problem - Genesereth, Nilsson

(1) A and B know that each can see the other's forehead. Thus, for example:

(1a) If A does not have a white spot, B will know that A does not have a white spot

(1b) A knows (1a)

(2) A and B each know that at least one of them have a white spot, and they each know that the other knows that. In particular

(2a) A knows that B knows that either A or B has a white spot

(3) B says that he does not know whether he has a white spot, and A thereby knows that B does not know

*

1. KA(White(A) KB( White(A)) (1b)

2. KA(KB(White(A) White(B))) (2a)

3. KA(KB(White(B))) (3)

4. White(A) KB(White(A))1, A2

5. KB(White(A) White(B))2, A2

6. KB(White(A)) KB(White(B))5, A1

7. White(A) KB(White(B))4, 6

8. KB(White(B)) White(A)contrapositive of 7

9. KA(White(A))3, 8, R2

Proof

5 Dynamic logic, temporal logic

Dynamic logic - the modal logic of action

LD and LRBuilds on LP , A - set of action symbols

a;b - do a and b in sequence

a+b - do either a or b - nondeterministic choice

p? - an action based on the truth value of p

a* - 0 or more (finitely many) iterations of a

p - the execution of a will possibly make p true

[a]p - the execution of a will necessarily make p true

, [a] LR , p LD

M2 = W - a set of worlds

L:W P() - set of formula true in a world, R A X W X W

R - accessibility relation based on LR - a world is accessible by executing an action a

Sem-9) M2 |=W p iff (w': Ra(w,w') M2 |=W' p)

Sem-10) M2 |=W [a] p iff (w': Ra(w,w') M2 |=W' p)

*

Temporal logic - the modal logic of time

Linear vs. branching; the branching can be in the past, in the future of bothTime is viewed as a set of moments with a strict partial order,

Branching temporal and action logic - CTL

Temporal structure with a branching time future and a single past - time treeSituation - a world w at a particular time point t, wtState formulas - evaluated at a specific time point in a time treePath formulas - evaluated over a specific p

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