Awareness and forgetting of facts and agents Hans van Ditmarsch - - PowerPoint PPT Presentation

Awareness and forgetting of facts and agents

Hans van Ditmarsch University of Sevilla, Spain & University of Otago, New Zealand Email: hans@cs.otago.ac.nz Tim French University of Western Australia, Perth, Australia Email: tim@csse.uwa.edu.au

Knowledge and awareness

◮ Difference between knowledge and awareness? ◮ You are unaware of a proposition iff you do not know that it is

the case, and you also do not know that it is not the case.

◮ becoming aware / forgetting

is related to program refinement / program abstraction

Becoming aware of a new fact

Agent i is uncertain of the value of fact (prop. variable) p. ¬p p i i i One way in which agent i becomes aware of another fact q. ¬p¬q p¬q ¬pq i i i i i i But what about an initial value for q?

Two types of facts, and forgetting

Distinguish two types of facts:

◮ the agent is aware of the relevant facts ◮ the agent is unaware of the irrelevant facts — between ( and )

¬p¬q p¬q ¬pq i i i i i i ¬p(q) p(q) i i i agent i becomes aware of fact q agent i forgets fact q

Becoming aware of other agents

¬p(q) p(q) ¬p(q) i j ij ij ij ¬p(q) p(q) i i(j) i(j) agent i becomes aware of agent j agent i forgets agent j

Agent i becomes aware of and forgets about agent j. On the right it holds that: If j knows that p is false, then j is uncertain if i knows that.

Implicit knowledge and explicit knowledge

No relation between implicit knowledge and explicit knowledge:

¬p¬q p¬q ¬pq i i i i i i ¬p(q) p(q) i i i agent i becomes aware of fact q

Implicit knowledge becomes explicit knowledge:

¬p¬q p¬q ¬pq i i i i i i ¬p(¬q) p(¬q) ¬p(q) i i i i i i agent i becomes aware of fact q

Logics for awareness change

◮ Logic of public global awareness ◮ Logic of individual global awareness ◮ Logic of individual local awareness ◮ Quantifying over all possible ways to become aware,

no specific awareness change

Structures

An epistemic awareness model M = (S, R, A, V ) for N and P consists of a domain S of (factual) states (or ‘worlds’), an accessibility function R : N → P(S × S), an awareness function A : N → S → P(P ∪ N) and a valuation function V : P → P(S). Given an agent i and a state s, a fact in Ai(s) is called relevant, and a fact in P \ Ai(s) is called irrelevant. Similarly, an agent in Ai(s) is called visible, and an agent in N \ Ai(s) is called invisible.

Structures — restrictions for the awareness function

◮ public global awareness:

the value of A is the same for all agents and for all states.

◮ individual global awareness:

the awareness is the same in all states, but maybe different between agents.

◮ individual local awareness:

the awareness may be different for all agents and in all states.

◮ no uncertain awareness:

if (s, t), (s, u) ∈ Ri, then Ai(t) = Ai(u). (for equivalence relations: Ri is a refinement of the partition induced by Ai.)

Logic of public global awareness — LPGA

The language L0 of public global awareness is defined as ϕ ::= p | ϕ ∧ ϕ | ¬ϕ | Kiϕ | ∃pϕ | ∃iϕ | Aϕ Notational abbreviations: ⊤ = ∃p(p ∨ ¬p) ˙ Kiϕ = Aϕ ∧ Kiϕ ˙ ∃pϕ = ¬Ap ∧ ∃p(ϕ ∧ Ap) ˙ ∃iϕ = ¬AKi⊤ ∧ ∃i(ϕ ∧ AKi⊤) ˙ pϕ = Ap ∧ ∃p(ϕ ∧ ¬Ap) ˙ iϕ = AKi⊤ ∧ ∃i(ϕ ∧ ¬AKi⊤) ˙ Kiϕ agent i (explicitly) knows ϕ ˙ ∃pϕ after the agents become aware of fact p, ϕ ˙ ∃iϕ after the agents become aware of agent i, ϕ ˙ pϕ after the agents forget fact p, ϕ ˙ iϕ after the agents forget agent i, ϕ

Logic of public global awareness — semantics

(M, s) | = p iff s ∈ V (p) (M, s) | = ϕ ∧ ψ iff (M, s) | = ϕ and (M, s) | = ψ (M, s) | = ¬ϕ iff (M, s) | = ϕ (M, s) | = Kiϕ iff for all t : (s, t) ∈ Ri ⇒ (M, t) | = ϕ (M, s) | = ∃pϕ iff there is a (M′, s′) such that (M, s)↔p(M′, s′) and (M′, s′) | = ϕ (M, s) | = ∃iϕ iff there is a (M′, s′) such that (M, s)↔i(M′, s′) and (M′, s′) | = ϕ (M, s) | = Aϕ iff var(ϕ) ⊆ A(S)

Public global awareness — example

¬p¬q p¬q ¬pq i i i i i i ¬p(q) p(q) i i i agent i becomes aware of fact q

The following hold throughout the initial model: Ap, ¬Aq, ˙ ∃q ˙ Ki¬(p ∨ q) The two models are bisimilar except for fact q.

Public global awareness — another example

¬p(q) p(q) ¬p(q) i j ij ij ij ¬p(q) p(q) i i(j) i(j) agent i becomes aware of agent j

In the initial model, in the (left) state where p is false and relevant and q is true and irrelevant, it is true that:

◮ ∃j(Kj¬p → ¬KjKiKj¬p ∧ ¬Kj¬KiKj¬p)

After the agents become aware of j, then if that agent knows that p is false he is uncertain if agent i knows that. The two models are bisimilar except for agent j.

Logic of individual global awareness — LIGA

The language L of individual awareness is defined as ϕ ::= p | ϕ ∧ ϕ | ¬ϕ | Kiϕ | ∃ipϕ | ∃iiϕ | Aiϕ Abbreviations for explicit knowledge and awareness: ˙ Kiϕ = Aiϕ ∧ Kiϕ ˙ ∃ipϕ = ¬Aip ∧ ∃ip(ϕ ∧ Aiϕ) ˙ ∃ijϕ = ¬AiKj⊤ ∧ ∃ij(ϕ ∧ AiKj⊤) (M, s) | = ∃ipϕ iff there is a (M′, s′) such that (M, s)↔i(M′, s′), (M, s)↔p(M′, s′), and (M′, s′) | = ϕ (M, s) | = ∃ijϕ iff there is a (M′, s′) such that (M, s)↔i(M′, s′), (M, s)↔j(M′, s′), and (M′, s′) | = ϕ (M, s) | = Aiϕ iff var(ϕ) ⊆ Ai(S)

Individual global awareness — example

Let’s skip that one!

Awareness bisimulation — example

In the actual state s agent i is aware of agent j and of fact p, and state t is i-accessible from the actual state. In state t, agent j is aware of p and q. That agent j is also aware of q should leave agent i indifferent, as she was not aware of q in the actual state. Therefore, in case agent i were to become aware of q in state s, she should consider it possible that j is unaware of q in that i-accessible state t. Under conditions of public or individual global awareness this is not a variation we care to consider: if j is aware

• f q in t, then he is already aware of q in the actual state s.

Clearly, we do not want to change the value of atoms of which agents are aware in the actual state.

Bisimulation — definition

A non-empty relation R ⊆ S × S′ is a bisimulation, iff for all s ∈ S and s′ ∈ S′ with (s, s′) ∈ R: atoms s ∈ V (p) iff s′ ∈ V ′(p) for all p ∈ P; aware for all i ∈ N, Ai(s) = A′

i(s′);

forth for all i ∈ N and t ∈ S, if Ri(s, t) then there is a t′ ∈ S′ such that Ri(s′, t′) and (t, t′) ∈ R; back for all i ∈ N and t′ ∈ S′, if Ri(s′, t′) then there is a t ∈ S such that Ri(s, t) and (t, t′) ∈ R.

◮ (M, s)↔(M′, s′): there is a bisimulation between M and M′

linking s and s′.

◮ A bisimulation except for fact p satisfies atoms for P − p,

and aware to the extent that Ai(s) − p = Ai(s′) − p.

◮ (M, s)↔p(M′, s′): there is a bisimulation except for fact p.

Awareness bisimulation — definition

A non-empty relation RA ⊆ S × S′ is an awareness bisimulation between (M, u) and (M′, u′), notation (M, u)↔A(M′, u′), iff (u, u′) ∈ RA and RA =

j∈N(u) RA j [A(u)]. We continue by

defining RA

j [A′′] for any A′′ : N → P(P ∪ N). Let such a A′′ be

given, s ∈ S, and s′ ∈ S′, then (s, s′) ∈ RA

j [A′′] iff:

atoms s ∈ V (p) iff s′ ∈ V ′(p) for all p ∈ A′′

j ;

aware for all i ∈ A′′

j , Ai(s) ∩ A′′ j = A′ i(s′) ∩ A′′ j ;

forth for all i ∈ A′′

j and t ∈ S, if Ri(s, t) then there is a

t′ ∈ S′ s.t. Ri(s′, t′) and (t, t′) ∈ RA

j [A′′ ∩ A′(t)];

back for all i ∈ A′′

j and t′ ∈ S′, if Ri(s′, t′) then there is a

t ∈ S such that Ri(s, t) and (t, t′) ∈ RA

j [A′′ ∩ A′(t)].

In the back and forth clauses, the relation RA

j [A′′ ∩ A′(t)] is

inductively assumed to be already defined.

Awareness bisimulation RA versus bisimulation R

◮ R is a refinement of RA ◮ Public global awareness: R|A(S) = RA ◮ Individual global awareness: a more complex relation, but this

is also a boundary case.

Logic of individual local awareness — LILA

Basic construct for becoming aware is ∃A

i pϕ, with an upper index

to distinguish it from the previous ∃ipϕ, where the A expresses that it is interpreted using RA. Its semantics is: (M, s) | = ∃A

i pϕ

iff there is a (M′, s′) s.t. (M, s)↔A(M′, s′) and (M′, s′)Ai +p | = ϕ This says that (there is a way in which) the agent i becomes aware

• f atom p in the current state if there is a model similar to the

current one in all its observable aspects except that fact p is added to the awareness set for that agent in all states accessible for that agent from actual state s (in accordance with ‘no uncertain awareness’).

Many issues of ongoing and further research

◮ Precise sense in which ‘public global’ and ‘individual global’

are boundary cases of ‘individual local’.

◮ Axiomatization, model checking (aye, bisimulation quantified

logics...)

◮ Logics for awareness change and information change,

such as announcements addressing an issue.

Many issues of ongoing and further research

◮ Precise sense in which ‘public global’ and ‘individual global’

are boundary cases of ‘individual local’.

◮ Axiomatization, model checking (aye, bisimulation quantified

logics...)

◮ Logics for awareness change and information change,

such as announcements addressing an issue. (‘I am playing cello tomorrow’)