Awareness and forgetting of facts and agents

Transcription

1 Awareness and forgettng of facts and agents Hans van Dtmarsch Unversty of Sevlla, Span & Unversty of Otago, New Zealand Emal: Tm French Unversty of Western Australa, Perth, Australa Emal:

2 Knowledge and awareness Dfference between knowledge and awareness? You are unaware of a proposton ff you do not know that t s the case, and you also do not know that t s not the case. becomng aware / forgettng s related to program refnement / program abstracton

3 Becomng aware of a new fact Agent s uncertan of the value of fact (prop. varable) p. p p One way n whch agent becomes aware of another fact q. pq p q p q But what about an ntal value for q?

4 Two types of facts, and forgettng Dstngush two types of facts: the agent s aware of the relevant facts the agent s unaware of the rrelevant facts between ( and ) agent becomes aware of fact q pq p(q) p(q) agent forgets fact q p q p q

5 Becomng aware of other agents agent becomes aware of agent j j p(q) (j) p(q) p(q) (j) j agent forgets agent j j p(q) p(q) j Agent becomes aware of and forgets about agent j. On the rght t holds that: If j knows that p s false, then j s uncertan f knows that.

6 Implct knowledge and explct knowledge No relaton between mplct knowledge and explct knowledge: agent becomes aware of fact q pq p(q) p(q) p q p q Implct knowledge becomes explct knowledge: p(q) agent becomes aware of fact q pq p( q) p( q) p q p q

7 Logcs for awareness change Logc of publc global awareness Logc of ndvdual global awareness Logc of ndvdual local awareness Quantfyng over all possble ways to become aware, no specfc awareness change

8 Structures An epstemc awareness model M = (S, R, A, V ) for N and P conssts of a doman S of (factual) states (or worlds ), an accessblty functon R : N P(S S), an awareness functon A : N S P(P N) and a valuaton functon V : P P(S). Gven an agent and a state s, a fact n A (s) s called relevant, and a fact n P \ A (s) s called rrelevant. Smlarly, an agent n A (s) s called vsble, and an agent n N \ A (s) s called nvsble.

9 Structures restrctons for the awareness functon publc global awareness: the value of A s the same for all agents and for all states. ndvdual global awareness: the awareness s the same n all states, but maybe dfferent between agents. ndvdual local awareness: the awareness may be dfferent for all agents and n all states. no uncertan awareness: f (s,t),(s,u) R, then A (t) = A (u). (for equvalence relatons: R s a refnement of the partton nduced by A.)

10 Logc of publc global awareness LPGA The language L 0 of publc global awareness s defned as ϕ ::= p ϕ ϕ ϕ K ϕ pϕ ϕ Aϕ Notatonal abbrevatons: = p(p p) K ϕ = Aϕ K ϕ pϕ = Ap p(ϕ Ap) ϕ = AK (ϕ AK ) Ⅎpϕ = Ap p(ϕ Ap) Ⅎϕ = AK (ϕ AK ) K ϕ pϕ ϕ Ⅎpϕ Ⅎϕ agent (explctly) knows ϕ after the agents become aware of fact p, ϕ after the agents become aware of agent, ϕ after the agents forget fact p, ϕ after the agents forget agent, ϕ

11 Logc of publc global awareness semantcs (M,s) = p ff s V (p) (M,s) = ϕ ψ ff (M,s) = ϕ and (M,s) = ψ (M,s) = ϕ ff (M,s) = ϕ (M,s) = K ϕ ff for all t : (s,t) R (M,t) = ϕ (M,s) = pϕ ff there s a (M,s ) such that (M,s) p (M,s ) and (M,s ) = ϕ (M,s) = ϕ ff there s a (M,s ) such that (M,s) (M,s ) and (M,s ) = ϕ (M,s) = Aϕ ff var(ϕ) A(S)

12 Publc global awareness example agent becomes aware of fact q pq p(q) p(q) p q p q The followng hold throughout the ntal model: Ap, Aq, q K (p q) The two models are bsmlar except for fact q.

13 Publc global awareness another example agent becomes aware of agent j j p(q) (j) p(q) p(q) (j) j j p(q) p(q) j In the ntal model, n the (left) state where p s false and relevant and q s true and rrelevant, t s true that: j(k j p K j K K j p K j K K j p) After the agents become aware of j, then f that agent knows that p s false he s uncertan f agent knows that. The two models are bsmlar except for agent j.

14 Logc of ndvdual global awareness LIGA The language L of ndvdual awareness s defned as ϕ ::= p ϕ ϕ ϕ K ϕ pϕ ϕ A ϕ Abbrevatons for explct knowledge and awareness: K ϕ = A ϕ K ϕ pϕ = A p p(ϕ A ϕ) jϕ = A K j j(ϕ A K j ) (M,s) = pϕ ff there s a (M,s ) such that (M,s) (M,s ),(M,s) p (M,s ), and (M,s ) = ϕ (M,s) = jϕ ff there s a (M,s ) such that (M,s) (M,s ),(M,s) j (M,s ), and (M,s ) = ϕ (M,s) = A ϕ ff var(ϕ) A (S)

15 Indvdual global awareness example Let s skp that one!

16 Awareness bsmulaton example In the actual state s agent s aware of agent j and of fact p, and state t s -accessble from the actual state. In state t, agent j s aware of p and q. That agent j s also aware of q should leave agent ndfferent, as she was not aware of q n the actual state. Therefore, n case agent were to become aware of q n state s, she should consder t possble that j s unaware of q n that -accessble state t. Under condtons of publc or ndvdual global awareness ths s not a varaton we care to consder: f j s aware of q n t, then he s already aware of q n the actual state s. Clearly, we do not want to change the value of atoms of whch agents are aware n the actual state.

17 Bsmulaton defnton A non-empty relaton R S S s a bsmulaton, ff for all s S and s S wth (s,s ) R: atoms s V (p) ff s V (p) for all p P; aware for all N, A (s) = A (s ); forth for all N and t S, f R (s,t) then there s a t S such that R (s,t ) and (t,t ) R; back for all N and t S, f R (s,t ) then there s a t S such that R (s,t) and (t,t ) R. (M,s) (M,s ): there s a bsmulaton between M and M lnkng s and s. A bsmulaton except for fact p satsfes atoms for P p, and aware to the extent that A (s) p = A (s ) p. (M,s) p (M,s ): there s a bsmulaton except for fact p.

18 Awareness bsmulaton defnton A non-empty relaton R A S S s an awareness bsmulaton between (M,u) and (M,u ), notaton (M,u) A (M,u ), ff (u,u ) R A and R A = j N(u) RA j [A(u)]. We contnue by defnng R A j [A ] for any A : N P(P N). Let such a A be gven, s S, and s S, then (s,s ) R A j [A ] ff: atoms s V (p) ff s V (p) for all p A j ; aware for all A j, A (s) A j = A (s ) A j ; forth for all A j and t S, f R (s,t) then there s a t S s.t. R (s,t ) and (t,t ) R A j [A A (t)]; back for all A j and t S, f R (s,t ) then there s a t S such that R (s,t) and (t,t ) R A j [A A (t)]. In the back and forth clauses, the relaton R A j [A A (t)] s nductvely assumed to be already defned.

19 Awareness bsmulaton R A versus bsmulaton R R s a refnement of R A Publc global awareness: R A(S) = R A Indvdual global awareness: a more complex relaton, but ths s also a boundary case.

20 Logc of ndvdual local awareness LILA Basc construct for becomng aware s A pϕ, wth an upper ndex to dstngush t from the prevous pϕ, where the A expresses that t s nterpreted usng R A. Its semantcs s: (M,s) = A pϕ ff there s a (M,s ) s.t. (M,s) A (M,s ) and (M,s ) A +p = ϕ Ths says that (there s a way n whch) the agent becomes aware of atom p n the current state f there s a model smlar to the current one n all ts observable aspects except that fact p s added to the awareness set for that agent n all states accessble for that agent from actual state s (n accordance wth no uncertan awareness ).

21 Many ssues of ongong and further research Precse sense n whch publc global and ndvdual global are boundary cases of ndvdual local. Axomatzaton, model checkng (aye, bsmulaton quantfed logcs...) Logcs for awareness change and nformaton change, such as announcements addressng an ssue.

22 Many ssues of ongong and further research Precse sense n whch publc global and ndvdual global are boundary cases of ndvdual local. Axomatzaton, model checkng (aye, bsmulaton quantfed logcs...) Logcs for awareness change and nformaton change, such as announcements addressng an ssue. ( I am playng cello tomorrow )