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Formal models for representing agents Lecture outline

Multi-Agent Systems Lecture 4 University “Politehnica” of Bucarest 2003 - 2004 Adina Magda Florea adina@cs.pub.ro http://turing.cs.pub.ro/blia_2004. Formal models for representing agents Lecture outline. 1 Knowledge representation for agents 2 FOL 3 Modal logic

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Formal models for representing agents Lecture outline

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  1. Multi-Agent SystemsLecture 4University “Politehnica” of Bucarest2003 - 2004Adina Magda Floreaadina@cs.pub.rohttp://turing.cs.pub.ro/blia_2004

  2. 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

  3. 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(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: round color: green type: apple What the agent knows/ believes 3

  4. 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 4

  5. 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 - entailmentKB |=  Proof theory • Derive new wffs based on axioms and inference rules KB |-i 5

  6. 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 PrL, PL Symbol level Modal logic Modal operators Structured models Knowledge level Temporal logic Modal operators for time Linear time Branching time Dynamic logic Modal operators for actions Logics of knowledge and belief Modal operators B and K CTL logic Branching time and action BDI logic Adds agents, B, D, I 6

  7. 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 =<L> 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 7

  8. 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 8

  9. Higher order logic 9

  10. 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 =<W, L, R> W - a set of worlds L:W  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 10

  11. w1 p, q, r w2 p, q, r w0 p, q, r w3 p, q, r 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 11

  12. 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 =<W, L, R> 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 12

  13. w1 p, q, r w2 p, q, r w0 p, q, r w3 p, q, r 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 Inference rules (R1) Epistemic necessitation |-  inferK(a, ) (R2) Logical omniscience    and K(a, ) inferK(a, ) problematic 13

  14. 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 14

  15. 5 Dynamic logic, temporal logic Dynamic logic - the modal logic of action LD and LR Builds 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 - deterministic choice a* - 0 or more (finitely many) iterations of a <a>p - the execution of a will possibly make p true [a]p - the execution of a will necessarily make p true <a>, [a]  LR , p  LD M2 = <W, L, R> 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<a> p iff (w': Ra(w,w')  M2 |=W' p) Sem-10) M2 |=W[a] p iff (w': Ra(w,w')  M2 |=W' p) 15

  16. 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 (T) with a partial order, <, 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 < 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 sometime (eventually) be true in the future - eventually Gp - p will always be true in the future – always G = ? F Gp  F p 16

  17. 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 state 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 Modal operators for actions - Dynamic logic indexed over agentsx[a]px<a>p and Va:p - there is a under which p comes true E = ?A Ep  A p 17

  18. 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 18

  19. 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 19

  20. LF- set of moment formula LS - set of path-formula Ag - set of agents, V - set of variables T - set of time points,  - partial order, | | - inverse of model Semantics M4 = <W, T, , | |, R> - 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,tX p iff M4 |=s,t+1 p) 20

  21. Sem-19) M4 |=s,t x[a]p iff (t's: [s;t,t']|a|x  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 x<a>p iff (t's: [s;t,t']|a|x  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 |=tA p iff (s: sSt  M4 |=s,t p) s is a path, St - all paths starting at the present moment Sem-22) M4 |=t (V a : p) iff (x: xAg and M4 |=t p|ax) there is an agent, be it x, capable of executing a under which p comes true, if a is executed at t 21

  22. 6 BDI logic Modal operators Bel, Des, Int, (Kh, KW) L I based on LF and LS, set of agents Ag Sin18) if pLS and x Ag then xBelp, xIntp, xDesp, xKhpL I xDes(AFwin)  xInt(EFbuy)  xBel(AFwin) M5=<W, T, , | |, R, B, D, I> 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 22

  23. 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 23

  24. 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,tFp) 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 F - eventually G - always A - inevitable E - optional (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 24

  25. 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 AF(xIntp)) F - eventually G - always A - inevitable E - optional 25

  26. 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(AFp) A (xInt(AFp)  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 26

  27. Blindly committed agent (same as the prev slide) • maintains its intentions until it believes it has achieved them xInt(AFp) A (xInt(AFp)  xBelp) Single-minded committed agent • maintains its intentions as long as it belives they are still options xInt(AFp) A (xInt(AFp)  (xBelp  xBel(EFp))) Open-minded committed agent • maintains its intentions as long as these intentions are still its desires (goals) xInt(AFp) A (xInt(AFp)  (xBelp  xDes(EFp))) F - eventually G - always A - inevitable E - optional 27

  28. 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. 28

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