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

Becoming aware of other agents ij ¬ p ( q ) agent i becomes aware of agent j i ( j ) ¬ p ( q ) p ( q ) i ( j ) j i agent i forgets agent j ¬ p ( q ) p ( q ) ij i ij 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: ¬ pq agent i becomes aware of fact q i ¬ p ( q ) p ( q ) i i i i i ¬ p ¬ q p ¬ q i i i Implicit knowledge becomes explicit knowledge: ¬ pq ¬ p ( q ) i i agent i becomes aware of fact q i i i i ¬ p ¬ q p ¬ q ¬ p ( ¬ q ) p ( ¬ q ) i i i i i i

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 A i ( s ) is called relevant , and a fact in P \ A i ( s ) is called irrelevant . Similarly, an agent in A i ( s ) is called visible , and an agent in N \ A i ( 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 ) ∈ R i , then A i ( t ) = A i ( u ). (for equivalence relations: R i is a refinement of the partition induced by A i .)

Logic of public global awareness — LPGA The language L 0 of public global awareness is defined as ϕ ::= p | ϕ ∧ ϕ | ¬ ϕ | K i ϕ | ∃ p ϕ | ∃ i ϕ | A ϕ Notational abbreviations: ⊤ = ∃ p ( p ∨ ¬ p ) ˙ = A ϕ ∧ K i ϕ K i ϕ ˙ ∃ p ϕ = ¬ Ap ∧ ∃ p ( ϕ ∧ Ap ) ˙ ∃ i ϕ = ¬ AK i ⊤ ∧ ∃ i ( ϕ ∧ AK i ⊤ ) ˙ � p ϕ = Ap ∧ ∃ p ( ϕ ∧ ¬ Ap ) ˙ � i ϕ = AK i ⊤ ∧ ∃ i ( ϕ ∧ ¬ AK i ⊤ ) ˙ K i ϕ agent i (explicitly) knows ϕ ˙ ∃ p ϕ after the agents become aware of fact p , ϕ ˙ ∃ i ϕ after the agents become aware of agent i , ϕ ˙ after the agents forget fact p , ϕ � p ϕ ˙ after the agents forget agent i , ϕ � 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 ) | = K i ϕ iff for all t : ( s , t ) ∈ R i ⇒ ( M , t ) | = ϕ there is a ( M ′ , s ′ ) such that ( M , s ) | = ∃ p ϕ iff ( M , s ) ↔ p ( M ′ , s ′ ) and ( M ′ , s ′ ) | = ϕ there is a ( M ′ , s ′ ) such that ( M , s ) | = ∃ i ϕ iff ( M , s ) ↔ i ( M ′ , s ′ ) and ( M ′ , s ′ ) | = ϕ ( M , s ) | = A ϕ iff var ( ϕ ) ⊆ A ( S )

Public global awareness — example ¬ pq i agent i becomes aware of fact q ¬ p ( q ) p ( q ) i i i i i ¬ p ¬ q p ¬ q i i i The following hold throughout the initial model: Ap , ¬ Aq , ˙ ∃ q ˙ K i ¬ ( p ∨ q ) The two models are bisimilar except for fact q .

Public global awareness — another example ij ¬ p ( q ) agent i becomes aware of agent j i ( j ) ¬ p ( q ) p ( q ) i ( j ) j i ¬ p ( q ) p ( q ) ij i ij 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 ( K j ¬ p → ¬ K j K i K j ¬ p ∧ ¬ K j ¬ K i K j ¬ 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 | K i ϕ | ∃ i p ϕ | ∃ i i ϕ | A i ϕ Abbreviations for explicit knowledge and awareness: ˙ K i ϕ = A i ϕ ∧ K i ϕ ˙ ∃ i p ϕ = ¬ A i p ∧ ∃ i p ( ϕ ∧ A i ϕ ) ˙ ∃ i j ϕ = ¬ A i K j ⊤ ∧ ∃ i j ( ϕ ∧ A i K j ⊤ ) there is a ( M ′ , s ′ ) such that ( M , s ) | = ∃ i p ϕ iff ( M , s ) ↔ i ( M ′ , s ′ ) , ( M , s ) ↔ p ( M ′ , s ′ ) , and ( M ′ , s ′ ) | = ϕ there is a ( M ′ , s ′ ) such that ( M , s ) | = ∃ i j ϕ iff ( M , s ) ↔ i ( M ′ , s ′ ) , ( M , s ) ↔ j ( M ′ , s ′ ) , and ( M ′ , s ′ ) | = ϕ ( M , s ) | = A i ϕ iff var ( ϕ ) ⊆ A i ( 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 of 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 , A i ( s ) = A ′ i ( s ′ ); forth for all i ∈ N and t ∈ S , if R i ( s , t ) then there is a t ′ ∈ S ′ such that R i ( s ′ , t ′ ) and ( t , t ′ ) ∈ R ; back for all i ∈ N and t ′ ∈ S ′ , if R i ( s ′ , t ′ ) then there is a t ∈ S such that R i ( 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 A i ( s ) − p = A i ( s ′ ) − p . ◮ ( M , s ) ↔ p ( M ′ , s ′ ): there is a bisimulation except for fact p .

Awareness bisimulation — definition A non-empty relation R A ⊆ S × S ′ is an awareness bisimulation between ( M , u ) and ( M ′ , u ′ ), notation ( M , u ) ↔ A ( M ′ , u ′ ), iff ( u , u ′ ) ∈ R A and R A = � j ∈ N ( u ) R A j [ A ( u )]. We continue by j [ A ′′ ] for any A ′′ : N → P ( P ∪ N ). Let such a A ′′ be defining R A given, s ∈ S , and s ′ ∈ S ′ , then ( s , s ′ ) ∈ R A j [ A ′′ ] iff: atoms s ∈ V ( p ) iff s ′ ∈ V ′ ( p ) for all p ∈ A ′′ j ; aware for all i ∈ A ′′ j , A i ( s ) ∩ A ′′ j = A ′ i ( s ′ ) ∩ A ′′ j ; forth for all i ∈ A ′′ j and t ∈ S , if R i ( s , t ) then there is a t ′ ∈ S ′ s.t. R i ( s ′ , t ′ ) and ( t , t ′ ) ∈ R A j [ A ′′ ∩ A ′ ( t )]; j and t ′ ∈ S ′ , if R i ( s ′ , t ′ ) then there is a back for all i ∈ A ′′ j [ A ′′ ∩ A ′ ( t )]. t ∈ S such that R i ( s , t ) and ( t , t ′ ) ∈ R A j [ A ′′ ∩ A ′ ( t )] is In the back and forth clauses, the relation R A inductively assumed to be already defined.

Awareness bisimulation R A versus bisimulation R ◮ R is a refinement of R A ◮ Public global awareness: R |A ( S ) = R A ◮ 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 p ϕ , with an upper index to distinguish it from the previous ∃ i p ϕ , where the A expresses that it is interpreted using R A . Its semantics is: = ∃ A ( M , s ) | i p ϕ iff there is a ( M ′ , s ′ ) s.t. ( M , s ) ↔ A ( M ′ , s ′ ) and ( M ′ , s ′ ) A i + p | = ϕ This says that (there is a way in which) the agent i becomes aware of 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’).