Multi-Agent%20Systems%20Lecture%203%20University%20

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Multi-Agent SystemsLecture 3University
Politehnica of Bucarest2004 - 2005Adina
Magda Floreaadina_at_cs.pub.rohttp//turing.cs.pub
.ro/blia_2005
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Formal models for representing agentsLecture
outline

• 1 Knowledge representation for agents
• 2 FOL
• 3 Modal logic
• 4 Logics of knowledge and belief
• 5 Dynamic logic, temporal logic
• 6 BDI logics
• 7 Commitments as change

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1 Knowledge representation for agents
Cognitive agents declarative representaton, AI

• Logic based representation
• unique (almost) syntax ?x?y loves(x,y)
• formal (clear, well-defined) semantics BelAloves(B
  ill, 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
  typeapple
• Frame-based representation
• units, frames
• subset of logic, represents relationship
  structured around objects in the universe
• apple01
• shape round color green type apple

What the agent knows/ believes
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• Plan representation
• represent actions
• may be combined with any of the previous
  representations
• partial representation of states stack(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
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• Logic based representations
• 2 possible aims
• to make MAS function according to the logic
• to specify and validate the design
• Conceptualization of the world / problem
• Syntax - wffs
• Semantics - significance, model
• Model - the domain interpretation for which a
  formula is true
• Model - linear or structured
• M 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 - entailment KB
  ?
• Proof theory
• Derive new wffs based on axioms and inference
  rules KB -i ?

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• PrL, PL

Linear model
Extend PrL, PL
Sentential logic of beliefs Uses beliefs atoms
BA(?) Index PL with agents
Tropistic agents (reactive)
Situation calculus Adds states, actions
Symbol level
Modal logic Modal operators
Structured models
Knowledge level
Dynamic logic Modal operators for actions
Temporal logic Modal operators for time Linear
time Branching time
Logics of knowledge and belief Modal operators B
and K
CTL logic Branching time and action
BDI logic Adds agents, B, D, I
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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 p?q ? LP, ?q ? LP
• M0 ltLgt is the formal model for LP
• L? ? - interpretation
• Sem-1) M0 ? iff ??L, where ? ??
• Sem-2) M0 p?q iff M0 p and M0 q
• Sem-3) M0 ?p iff M0 / p
• p A ? B A - it rains
• q A ? B B - take umbrella
• r A ? ?A

Knowledge represents atomic propositions
A B A ? B A?B A??A T T T
T T T F F F
T F T T F T F F
T F T
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• 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 M M p
• x young(x) success(x) M / q
• Bill T T
• Tom F T
• Alice F F

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• (from Lecture 2)
• (a) Deduction rules
• At(0,0) ? Free(0,1) ? Exit(east) ?
  Do(move_east)
• Facts and rules about the environment
• At(0,0)
• Wall(1,1)
• ?x ?y Wall(x,y) ? ?Free(x,y)
• (b) Use situation calculus describe change in
  FOPL
• Define a function Result(Action,State)
  NewState
• At((0,0), S0) ? Free(0,1) ? Exit(east) ?
• At((0,1), Result(move_east,S0))

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• c) Sentential logics of beliefs
• Uses beliefs atoms BA(?)
• Index PL with agents
• Inference rule attachment
• BA(p1) ? q1 p1 ? p2 ? .. pn -A pn1
• BA(p2) ? q2
• BA(pn) ? qn
• ? BA(pn1) ? qn1
• q1 ? q2 ? .. ? qn ? qn1

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Higher order logic
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3 Modal logic

• LM - the language of Modal logic
• 2 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 worlds
• The structure of the model is given by relating
  different worlds via a binary accessibility
  relation
• M1 ltW, L, Rgt W - a set of worlds
• LW ? P(?) - set of formula true in a world, R
  ? W X W
• ? p p - it rains in NY
• ? q q - the sun will rise tomorrow

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• Sem-4) M1 W ? iff ??L(w), where ? ??
• Sem-5) M1 W p?q 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
• ? ? q
• 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

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4 Logics of knowledge and belief

• FOL augmented with two modal operators
• K(a,?) - a knows ?
• B(a,?) - a believes ?
• Associate with each agent a set of possible
  worlds
• Mk ltW, L, Rgt W - a set of worlds
• LW ? 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

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• 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 axiom ?K(a, ?) ? K(a,
  ?K(a, ?))
• - satisfied if R is euclidian

in w0 ?K(a,p), ?K(a, ?r), ?K(a,q)
Properties of beliefs (A1) - OK, (A2) - no, (A3)
- yes, (A4) - maybe but more problematic Inferenc
e rules (R1) Epistemic necessitation
- ? infer K(a, ?) (R2) Logical omniscience
? ? ? and K(a, ?) infer K(a,
?) problematic
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• 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)
Proof
4. ?White(A) ? KB(?White(A)) 1, A2 A2 K(a, ?)
? ? 5. KB(?White(A) ? White(B)) 2, A2
6. KB(?White(A)) ? KB(White(B)) 5, A1 A1 K(a,
?) ? K(a, ? ? ?) ? K(a, ?) 7. ?White(A) ?
KB(White(B)) 4, 6
8. ?KB(White(B)) ? White(A) contrapositive of
7 9. KA(White(A)) 3, 8, R2
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• 5 Dynamic logic, temporal logic
• Dynamic logic - the modal logic of action
• LD and LR Builds on LP , A - set of action
  symbols
• ab - do a and b in sequence
• ab - do either a or b - nondeterministic
  choice
• p? - an action based on the truth value of p -
  deterministic choice
• a - 0 or more (finitely many) iterations of a
• ltagtp - the execution of a will possibly make p
  true
• ap - the execution of a will necessarily make p
  true
• ltagt, a ? LR , p ? LD
• M2 ltW, L, Rgt W - a set of worlds
• LW ? 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 ltagt p iff (?w' Ra(w,w') ? M2
  W' p)
• Sem-10) M2 W a p iff (?w' Ra(w,w') ? M2
  W' p)

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• Temporal logic - the modal logic of time
• Linear vs. branching the branching can be in the
  past, in the future of both
• Time is viewed as a set of moments with a strict
  partial order, lt, which denotes temporal
  precedence.
• Every moment is associated with a possible state
  of the world, identified by the propositions that
  hold at that moment
• In a branching logic of time, a path at a given
  moment is any maximal set of moments containing
  the given moment and all the moments in the
  future along some particular branch of lt
• Modal operators of temporal logic
• p U q - p is true until q becomes true - until
• Xp - p is true in the next moment - next
• Pp - p was true in a past moment - past
• Fp - p will eventually be true in the future -
  eventually
• Gp - p will always be true in the future always
• F ? U
• G ? F

Fp ? true U p
Gp ? ?F ?p
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• Branching temporal and action logic - CTL
• Temporal structure with a branching time future
  and a single past - time tree
• Situation - a world w at a particular time point
  t, wt
• State formulas - evaluated at a specific time
  point in a time tree
• Path formulas - evaluated over a specific path in
  a time tree
• Modal operators over both state and path formulas
• Temporal logic Fp - p will sometime be true in
  the future - eventually
• Gp - p will always be true in the future -
  always
• Xp - p is true in the next moment - next
• p U q - p is true until q becomes true - until
• Modal operators over path formulas - Branching
  temporal
• Ap - at a particular time moment, p is true in
  all paths emanating from that point - inevitable
  p
• Ep - at a particular time moment, p is true in
  some path emanating from that point - optional p
• Dynamic logic indexed over agents xap xltagtp
• Other modal operators
• Vap - there is a under which p comes true

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• s is true in each time point (situation) and on
  all path
• r is true in each time point on a single path
• p will eventually be true on a single path
• q will eventually be true on all path

s
p s q
F - eventually G - always A - inevitable E -
optional
AGs EGr AFq EFp
r s
r s
r s q
s q
s
r - Alice is in Italy p -Alice visits Paris s
Paris is the capital of France q - it is spring
time
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• Each situation has associated a set of accessible
  words - the worlds the agent believes to be
  possible. Each such world is a time tree.
• Within these worlds, the branching future
  represents the choices (options) available to the
  agent in selecting which action to perform
• Similar to a decision tree in games of chance

Decision nodes
Player 1
Dice

• Each arc emanating from
• a chance node corresponds
• to a possible world

Player 2
1/18
1/36
Chance nodes
Dice

• Each arc emanating from
• a decision node corresponds
• to a choice available in a
• possible world

Player 1
1/36
1/18
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• LB - set of moment formula
• LS - set of path-formula, X - set of agents, A -
  set of actions
• Semantics
• M4 ltW, T, lt, , Rgt - every t?T has associated
  a world wt?W
• Sem-14) M4 t ? iff t??, where ? ??
• ? is true in the set of moments for which ?
  holds
• Sem-15) M4 t p?q iff M4 t p and M4 t q
• Sem-16) M4 t ?p iff M4 /t p
• Sem-17) M4 s,t pUq iff (?t' t?t' and M4
  s,t' q and
• (?t" t ? t"? t' ? M4 s,t" p))
• p holds on a path starting in the current
  moment until q comes true
• Sem-18) M4 s,t X p iff M4 s,t1 p)
• Fp ? true Up
• Gp ? ?F ?p

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• Sem-19) M4 s,t xap iff (?t'?s
  st,t'?ax ? M4 s,t' p)
• p is true on all the set of moments t' on a
  given path s starting at the current moment t
  while agent x executes action a
• Sem-20) M4 s,t xltagtp iff (?t'?s
  st,t'?ax ? M4 s,t' p)
• p is true at a moment t' on a given path s
  starting at the current moment t while agent x
  executes action a
• Sem-21) M4 t A p iff (?s s?St ? M4 s,t p)
• s is a path, St - all paths starting at the
  present moment
• E ?A
• Sem-22) M4 t (V a p) iff (?b b?B and M4
  t pab)
• there is an action, be it b, under which p
  comes true, if executed at t
• Sem-23) M4 t R p iff M4 R(t),t p)
• R picks out at each moment the real path at
  that moment
• p holds in the real path at the present moment

Ep ? ?A ?p
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• 6 BDI logic
• Modal operators Bel, Des, Int, (Kw)
• L I based on LB and LS, set of agents A
• Sin18) if p?LS and x ?A then xBelp, xIntp,
  xDesp, xKwp?L I
• xDes(A Fwin) ? xInt(E Fbuy) ? ?xBel(A Fwin)
• M5ltW, T, lt, , R, B, D, Igt
• B - belief accessible relation - belief
  accessible worlds the worlds the agent believes
  possible
• Require the desires to be consistent therefore
  Desires ? Goals
• D - desire (goal) accessible relation
• Each situation has associated a set of goal
  -accessible worlds - realism
• Strong realism for each belief-accessible world
  w at a given time moment t, there must be a
  goal-accessible world that is a sub-world of w at
  time t
• I - intention accessible relation
• Intentions - similarly represented by sets of
  intention-accessible worlds. These are the worlds
  the agent has committed to realize.
• Corresponding to each goal-accessible world at
  some time t there must be an intention-accessible
  world that is a subworld of w at time t

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intention accessible world
belief accessible world
s
s
p s q
p s q
s
r s
s q
r s
r s q
r s q
r s
r s
goal accessible world
r - Alice is in Italy p -Alice visits Paris s
Paris is the capital of France q - it is spring
time
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• Sem-24) M5 t xBelp iff (?t' (t,t')?B(x,t) ?
  M5 t' p)
• an agent x has a belief p in a given moment t
  if and only if p is true in all belief accessible
  worlds of the agent in that moment
• Sem-25) M5 t xDesp iff (?t' (t,t')?D(x,t) ?
  M5 t' p)
• an agent x has a desire p in a given moment t
  if and only if p is true in all goal accessible
  worlds of the agent in that moment
• Sem-26) M5 t xIntp iff (?s s?I(x,t) ? M5
  s,t Fp)
• at each moment t, I assigns a set of paths
  that the agent x has selected or preferred, i.e.,
  if the agent has selected p as an intention, p
  will hold eventually in the future
• Properties of BDI and KW

(A2) Knowledge axiom aKwp ? p (A3)
Positive introspection axiom aBelp ?
aBel(aBelp)) - satisfied if B is
transitive (A4) Negative introspection
axiom ?aBelp ? aBel(?aBelp)) -
satisfied if B is euclidian
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• Belief-goal compatibility
• If an agent adopts p as a goal, the agent
  believes that there is a path on which p will be
  true as it is an adopted desire but it needs not
  believe that it will ever reach that point
• xDesp ? (xBel (E G p)
• Goal-intention compatibility
• If an agent adopts p as an intention, it should
  have adopted it as a goal to be achieved
• xIntp ? xDesp
• Beliefs about intentions
• xIntp ? xBel(xIntp))
• No infinite deferral
• The agent should not procrastinate with respect
  to its intentions if the agent forms an
  intention, then sometimes in the future it will
  give up this intention
• xIntp ?A F(?xIntp))

F - eventually G - always A - inevitable E -
optional
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• 7 Commitments as change
• Desires (goals) and intentions are quite similar
  in their semantic structure. The difference in
  these modalities arises in their relationships
  with the other modalities and in terms of how
  they may evolve over time.
• An agent is treated as being committed to its
  intention but, cf. no infinite deferral, it will
  give up these intentions eventually - when?
• Different types of agents will have different
  commitment strategies.
• Blindly committed agent
• maintains its intentions until it believes it has
  achieved them
• xInt(A Fp) ?A (xInt(A Fp) ? xBelp) (exclusive
  ?)
• an agent can be committed to means (p is an
  action) or to ends (p is a formula)
• defined only for intentions toward actions or
  conditions that are true for all paths in the
  agent's intention accessible worlds.

F - eventually G - always A - inevitable E -
optional
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• Blindly committed agent (same as the prev slide)
• maintains its intentions until it believes it has
  achieved them
• xInt(A Fp) ?A (xInt(A Fp) ? xBelp)
• Single-minded committed agent
• maintains its intentions as long as it belives
  they are still options
• xInt(A Fp) ?A (xInt(A Fp) ? (xBelp ? ?xBel(E
  Fp)))
• Open-minded committed agent
• maintains its intentions as long as these
  intentions are still its desires (goals)
• xInt(A Fp) ?A (xInt(A Fp) ? (xBelp ? ?xDes(E
  Fp)))

F - eventually G - always A - inevitable E -
optional
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• References
• M. P. Singh, A.S. Rao. Formal methods in DAI
  Logic-based representation and reasoning. In
  Multiagent Systems - A Modern Approach to
  Distributed Artficial Intelligence, G. Weiss
  (Ed.), The MIT Press, 2001, p.331-355.
• M. Wooldrige. Reasoning about Rational Agents.
  The MIT Press, 2000, Chapter 2
• A.S. Rao, M.P. Georgeff. Modeling rational agents
  within a BDI-architecture. In Readings in Agents,
  M. Huhns M. Singh (Eds.), Morgan Kaufmann,
  1998, p.317-328.
• M.R. Genesereth, N.J. Nilsson. Logical
  Foundations of Artificial Intelligence. Morgan
  Kaufmann, 1987, Chapter 9.
• D. Kayser La représentation des connaissances.
  Hermès, 1997.
• J.Y. Halpern. Reasoning about knowledge A
  survey. In Handbook of Logic in Artificial
  Intelligence and Logic Programming, Vol.4, D.
  Gabbay, C.A. Hoare, J.A. Robinson (Eds.), Oxford
  University Press, 1995, p.1-34.
• A. Florea. Bazele logice ale inteligentei
  artificiale. UPB, 1995.

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