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/ believesTRANSCRIPT
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