©Agent Technology, 2008, Ai Lab NJU Agent Technology Agent model and theory.

©Agent Technology, 2008, Ai Lab NJU

Agent Technology

Agent model and

theory

Nov., 2008©Gao Yang, Ai Lab NJU2

Agent model and theory

6.1 Logical Background– 6.1.1 Basic Concepts– 6.1.2 Propositional and Predicate Logic– 6.1.3 Modal Logic– 6.1.4 Dynamic Logic– 6.1.5 Temporal Logic

6.2 Cognitive Primitives– 6.2.1 Knowledge and Beliefs– 6.2.2 Desires and Goals– 6.2.3 Intentions


Chapter 4: Agent model and theory

– 6.2.4 Commitments– 6.2.5 Know-how

6.3 Belief Revision– 6.3.1 AGM Framework– 6.3.2 Epistemic entrenchment

6.4 Social Primitives– 6.4.1 Team and Organizational Structure– 6.4.2 Mutual Beliefs and Joint Intentions– 6.4.3 Social Commitments– 6.4.4 Group Know-how and Intentions


6.1 Logical Background

6.1.1 Basic Concepts– Why need formal method?

As internal specification languages to be used by the agent in its reasoning or action;

As external metalanguages to be used by the designer to specify, design, and verify certain behavioral properties of agents situated in a dynamic environment.

– Differents between these languages One would like to have the same logical language server

both of the above purpose. The internal language should be computationally efficie

nt. The external language should be more expressive.



6.1.1 Basic Concepts– Three aspects to logic

Well-formed formulas: some statements; Proof-theory: is also called the syntax; Model-theory: is also called the semantics.

– Purpose of the semantics

iM p

M p



6.1.2 Propositional and Predicate Logic– How to use this logic in agent

Is simplest Represent factual information, often about the agents’

environment.

– Example 6.1 The facts “it rains” and ‘road is wet”; Atomic propositions

– Rain– Wet-road

Implication that “if it rains, then the road is wet” can be captured by the propositional formula

– Rain -> wet-road



6.1.2 Propositional and Predicate Logic

– The language of propositional logic Assume a set is given atomic propositions;

SYN-1.

SYN-2.

– The formal model Let L identifies the set of atomic propositions that are

true.

SEM-1.

SEM-2.

SEM-3.

implies that pL

pL

, implies that ,p pp q L p q p L

0 iff ,where M L

0 0 0 iff and M p q M p M q

0 0 iff M p M p

0defM L



6.1.2 Propositional and Predicate Logic

– Imply is true if p is false irrespective of q

– Predicate logic Do not use predicate logic in the specification language.

Use it in metalanguage, which is used in semantic condi

tions.

Universal quantifier

Existential quantifier

p q



6.1.3 Modal Logic

– Objective To investigate different modes of truth, such as

possibly true and necessarily true.

In agents’ study, it is used to give meaning to concepts

such as belief and knowledge.

– Modal language Classical propositional logic is extended with two modal

operators: for possibility and for necessity.

SYN-3.

SYN-6.

ML

the rules for pL

implies that , p Mp L p p L



6.1.3 Modal Logic

– Example 6.2 “it is possible that it rains” as

“it is necessary that the sun rises in the east” as

– Model is the set of the worlds

gives the set of formulas true at a world

is an accessibility relation

SEM-6.

rain

sun rises in the east

1 , ,defM W L R W

: 2L W

R W W

1 iff ,where wM L w



SEM-5.

SEM-6.

SEM-6.

SEM-8.

– Algebric properties of the accessibility relation R is reflexive iff

R is serial iff

R is transitive iff

R is symmetric iff

R is euclidean iff

1 1 1 iff w w wM p q M p and M q

1 1 iff w wM p M p

1 1 iff : , &w wM p w R w w M p

1 1 iff : ,w wM p w R w w M p

: ,w w w R

: : ,w w w w R

1 2 1 2 2 1, : , ,w w w w R w w R

1 2 3 1 2 2 3 1 3, , : , & , ,w w w w w R w w R w w R

1 2 3 1 2 1 3 2 3, , : , & , ,w w w w w R w w R w w R



6.1.4 Dynamic Logic

– What is dynamic logic Can be thought of as the modal logic of action

The necessity and possibility of dynamic logic are

based upon the kinds of actions available

Can be used in a number of areas of DAI.

– Model Language and its sublanguage Sublanguage define action regular expressions

is a set of atomic action symbols

SYN-5.

SYN-6.

DL RL

RL

B

the rules for applied to p DL L

implies that RL B



SYN-6.

SYN-8.

SYN-9.

Notes:

– a;b means doing a and b in sequence.

– a+b means doing either a or b.

– p? is an action based on confirming the truth value of p

roposition p.

– a* means 0 or more (but finitely many) iterations of a.

– Example 6.3 “If q then a else b endif”

, implies that a;b, a+b , * R Ra b L a L

implies that ? D Rp L p L

and implies that a , R D Ra L p L p a p L

?; ?;q a q b



– Model Here W, L defined as model logic;

Is a transition relation.

RP-1.

RP-2.

RP-3.

RP-6.

SEM-9.

SEM-10.

2 , ,defM W L

W W B

, iff , ,R w w w w

; , iff : , & ,a b a bR w w w R w w R w w

, iff , or ,a b a bR w w R w w R w w

*

0 0 1

, iff

, , : & & : 0 ,

a

n n a i i

R w w

w w w w w w i i n R w w

2 2 iff : , &w a wM a p w R w w M p

2 2 iff : ,w a wM a p w R w w M p



6.1.5 Temporal Logic

– Several variants about temporal logic Linear versus Branching

Discrete versus Dense

Moment-Based versus Period-Based

– Some terms in temporal logic Moments: associated with possible state of the world,

has a strict partial order.

Path: set of moments containing the given moments.



t 0

t 1

t 2

t 3

t 4

q

q. . .

q. . .

q. . .

q. . .q. . .r . . .

. . .

. . .

. . .

. . .

r eal i t y

a| | c

a| | d

b| | c

b| | d

An exampl e st r uct ur e of t i me



– Linear-time temporal logic language SYN-10. SYN-11. Notes:

– means at a moment t on a path, q holds at a

future moment t’ on the given path, and p holds on

all moments between t and t’.

– means p holds sometime in the future on the

given path and abbr. with .

– means p always holds in the future on the given

path

– means p holds in the next moment.

– means q held in a past moment.

LL

the rules for pL

, implies that , , L p qp q L p q P L

p q

pF

pG

p

qP

true p



– Model Here T is the set of moments, < the temporal ordering

relation, and gives the denotations of the atomic

propositions.

SEM-11.

SEM-12.

SEM-13.

Note: M3 is linear, < is a total ordering.

3 , ,defM T

3 3 iff : and pt tM P t t t M p

3 3 1 iff pt t

M M p

3

3 3

iff

: and and :

t

t t

M p q

t t t M q t t t t M p


6.1 Logical Background (optional)

– Branching temporal and Action logic Builds on top of and , especially uses the ideas of

the well-known language CTL*.

captures the essential properties of actions and time

that are of value in specifying agents. SYN-12. SYN-13. SYN-16. SYN-15.

SYN-16. SYN-16.

BL

LL DL

BL

the rules for pL , implies that , :B p Bp q L P a p L

B sL L

, , , implies that

, , , , ,s

s

p q L x a

p q p p q p x a p x a p L

A B

implies that ,s p p Bp L A R L

\ and implies that :s B sp L L a a p L X



– Notes about syntax The branching-time operator, A, denotes “in all paths at

the present moment.”

E, denotes “in some path at the present moment.”

R, denotes “in the real path at the present moment.”

The constructor (V a:p) means that “there is an action u

nder which p becomes true.”

– Examples 6.4 EFr , AF(q v r) , RFq hold at t0.

E<b>r, A[a]q, A[d](q v r), A[e]true hold at t0.

(V e: Ex<e>true ^ Ax[e]q) holds at t0.



– Model SEM-16.

SEM-15.

SEM-16.

SEM-16.

SEM-18.

SEM-19.

SEM-20.

SEM-21.

4 , , ,defM T R

4 iff , where t

M t

4 4 4 iff and t t t

M p q M p M q

4 4 iff t t

M p M p

4 4 ,A iff : tt S t

M p S S S M p

4 4 ,R iff

t R t tM p M p

4 4P iff : t and t t

M p t t M p

4 4, , 1X iff

S t S tM p M p

4 4: iff : and ,where a

t t bM a p b b M p p L B



SEM-22.

SEM-23.

SEM-26. SEM-25.

SEM-26.

SEM-26.

SEM-28.

4 4 4, , , iff and

S t S t S tM p q M p M q

4 4, , iff

S t S tM p M p

4 4, iff ,where

S t tM p M p p L

4 4, , iff : ; , &

x

S t S tM x a p t S S t t a M p

4 4, , iff : ; ,

x

S t S tM x a p t S S t t a M p

4 4, ,: iff : and ,where \

a

sS t S t bM a p b b M p p L L B

4 ,

4 4, ,

iff

: and and :

S t

S t S t

M p q

t t t M q t t t t M p


6.2 Cognitive Primitives

Origin

– Intentional stance

– Knowledge level

– Functional level

BDI logic

– Be used to reason about agent

– Their beliefs, intentions, and actions bring about

the satisfaction of their desires



Modal operators– Bel(belief), Des(desire), Kh(know-how) and Int(intention)

– SYN-18.

Semantics for

Example 6.5– Consider an agent who has the desire to win a lottery event

ually and intends to buy a lottery ticket sometime, but does not believe that will ever win the lottery.

IL

5 , , , , , ,defM T R B D I

win buy winDesAF IntEF BelAF

h t, , implies that Int , K , K , Dess Ip q L x x p x p x p x p L A



6.2.1 Knowledge and Beliefs– B, a belief accessibility relation, which behaves as a modal

necessity operator,

– Knowledge(know-that), is customarily defined as a true beli

ef.

– B is serial, symmetric, euclidean and reflecive.

– SEM-29.

– B depends on the given moments, and agent can change it

s beliefs over time.

5 5Bel iff : , B ,t t

M x p t t t x t M p



6.2.2 Desires and Goals– D, a desire accessibility relation, which represent the

desires of the agent.

– SEM-30.

– In the philosophical view Desires can be inconsistent

Agent need not know the means of achieving these desires

– The role of desires According to inputs, agent choose a subset of desires that are

both consistent and achievable

– Goals The consistent achievable desires are usually called goals.

5 5Des iff : , D ,t t

M x p t t t x t M p



6.2.3 Intentions– I, a intend accessibility relation, defined as the conditions

that inevitably hold on each of the selected paths.

– SEM-31.

– Example 6.6 Consider next figure, assume that –r and –p hold everywhere

other than as shown. Let the agent x at moment t0 prefer the

path S1 and S2. Then, we have that x intend q (because it

occurs eventually on both the preferred paths) and does not

intend r(because it never occurs on S2)

5 5 ,Int iff : I , F

t S tM x p S S x t M p

0 0

5 5Int , Intt t

M x p M x r



t 0

t 1

t 2

t 3

t 4

q

q. . .

q. . .r . . .

r . . .. . .

. . .

. . .

S2

a| | c

a| | d

b| | c

b| | d

I nt ent i ons

. . .

. . .

. . .

S1



Some useful conclusion– IC1 Satisfiability

This says that if p is intended by x, then it occurs eventually on some path.

– IC2 Temporal Consistency

This says that if an agent intends p and intends q, then it (implicitly) intends achieving them in some undetermined temporal order: p before q, q before p , or both simultaneously.

– IC3 Persistence does not entail success

Just because an agent persists with an intention does not mean that it will succeed.

Int EFx p p

Int Int Int F Fx p x q x p q

EG Int is satisfiablex p p



6.2.4 Commitments

– Goals and intentions Are quite similar, and difference arises in their

relationship with other modalities and how they evolve

over time.

Commitment can separate them.

– Commitment Be treated as constraining how intentions are revised

and updated.

– Handling commitment IC4 shows how commitment may be expressed in the

present framework.



– IC4 Persist while succeeding

This constraint requires that agents desist from revising their intentions as long as they are able to proceed properly. If an agent selects some paths, then at future moments on those paths, it selects from among the future components of those paths.

I , ; ,

I , I ,

xS x t and S t t a

S x t S x t and S S



6.2.5 Know-how– Motivation

Intentions have an obvious connection with actions – agents

act to satisfy their intentions.

But intentions do not ensure success.

A key ingredient is know-how.

– Example 6.7 Consider former figure, at t0, x may do either action a or

action b, since both can potentially lead to one of the

preferred paths being realized. However if the other agent

does action d, then no matter which action x chooses, x will

not succeed with its intentions, because none of its preferred

paths will be realized.



– Let be the set of tree, and is defined as follows

– SYN-19.

1 1

1

1.

2.

3. ,..., , ,..., ,

; ,...,

m m

m

T is the empty tree

T a implies that a

T have different radices and a

implies that a

, , I

I

x A and p L implies that

x p L



– SEM-32.

– SEM-33.

– SEM-36.

– SEM-35.

tt tM p iff M K p

t tt tM a p iff M K E a true A a K p

1

1

; ,...

:

mt

t i m i it

M a p iff

M K E a true A a p

:

ht

t

M xK p iff

M x p


6.3 Belief Revision

Beliefs– The bird caught in the trap is a swan– The bird caught in the trap comes from Sweden– Sweden is part of Europe– All European swans are white

Consequences– The bird caught in the trap is white

New information– The bird caught in the trap is black

Which sentence would you give up?


6.3.1 AGM Framework

Alchourron, Gardenfors, and Makinson (1985)

– Epistemic states: sets of formulas K.

– Epistemic attitudes: - α accepted

- α rejected

Otherwise - α undetermined

– Input: formula

– Change operations: expansion, contraction, and r

evision

K

K


6.3.1 AGM Framework

Belief sets

Three operations:

– Expansion

– Contraction

– Revision

(Levi identity)

(Levi identity)

For contraction and revision, rationality postulates.

K Cn K

K Cn K

K

*K

*K K

*K K K


6.3.1 AGM Framework

Contraction Postulates

1 ( )

2 ( )

3 ,

4 ,

5 , cov

6 ,

7

K K is a belief set closure

K K K inclusion

K If K then K K vacuity

K If not then K success

K If K then K K re ery

K If then K K equivalence

K K

8 ,

K K

K if K then K K


6.3.1 AGM Framework

Revision Postulates

*1

*2 *

*3 *

*4 * , *

*5 *

*6 , * *

*7 * *

*8 * , * *

K K * α = Cn K * α

K K

K K K

K If K then K K

K K K if and onlyif

K If then K K

K K K

K if K then K K


6.4 Social Primitives

6.6.1 Team and Organizational Structure



6.6.2 Mutual Beliefs and Joint Intentions



6.6.3 Social Commitments



6.6.4 Group Know-how and Intentions

©Agent Technology, 2008, Ai Lab NJU Agent Technology Agent model and theory.

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Transcript of ©Agent Technology, 2008, Ai Lab NJU Agent Technology Agent model and theory.