Knowledge and awareness Hans van Ditmarsch LORIA CNRS / Universit - PowerPoint PPT Presentation

Knowledge and awareness Hans van Ditmarsch LORIA – CNRS / Universit´ e de Lorraine

Mus´ ee de l’Ecole de Nancy

Awareness and knowledge ◮ Knowledge is about uncertainty, awareness is about incompleteness. ◮ Knowledge and awareness combined in a single framework: ◮ Levesque, A logic of implicit and explicit belief ◮ Fagin, Halpern, Belief, unawareness, and limited reasoning ◮ Heifetz, Meier, Schipper, Interactive unawareness ◮ HvD, Tim French, Fernando Velazquez and Yi Wang use the structures for propositional awareness proposed by Fagin & Halpern to formalize the multi-agent dynamics of awareness implicit in the work by Heifetz et al. ◮ Conceptual innovations: ◮ awareness bisimulation ( the proper notion of structural equiv.) ◮ speculative knowledge (it is not explicit and also not implicit) ◮ dynamics of knowledge and awareness

Being unaware of a propositional variable Hans does not know whether coffee is served after his talk. Hans is unaware of it that wine is not served after his talk. Agent i may be uncertain about the value of fact / variable p . ¬ p p i i i Agent i may be unaware of the value of another prop. q . There are many ways in which agent i can become aware of q , e.g.: ¬ pq p ¬ q ¬ pq ¬ pq pq i i i i i i i i i i i i i ¬ pq pq ¬ p ¬ q p ¬ q ¬ p ¬ q p ¬ q i i i i i i i i i

Being unaware of a propositional variable ¬ p p i i i For i , all four structures below look like the structure above. The four below are bisimilar with respect to { p } . They are not bisimilar with respect to { p , q } . They are awareness bisimilar with respect to { p , q } . ¬ pq p ¬ q ¬ pq ¬ pq pq i i i i i i i i i i i i i ¬ pq pq ¬ p ¬ q p ¬ q ¬ p ¬ q p ¬ q i i i i i i i i i

Adding dynamics — agent i becomes aware of prop. var. q ¬ pq i becomes aware of q i ¬ p p i i i i i ¬ p ¬ q p ¬ q i i i Hans does not know whether coffee is served after his talk. Hans is unaware of it that wine is not served after his talk. Someone mentions that wine and coffee will not both be served. Hans is now uncertain about coffee and about wine.

Adding dynamics — agent i becomes aware of prop. var. q ¬ pq ¬ pq i i becomes aware of q i i i i i ¬ p ¬ q p ¬ q ¬ p ¬ q p ¬ q i i i i i i Hans does not know whether coffee is served after his talk. Hans is unaware of it that wine is not served after his talk. Someone mentions that wine and coffee will not both be served. Hans is now uncertain about coffee and about wine.

Keep dreaming — and back to basics Similarly to unawareness of propositions, one can model: ◮ unawareness of agents ◮ unawareness of actions Similarly to becoming aware, one can model: ◮ becoming unaware (forgetting)

Keep dreaming — and back to basics Similarly to unawareness of propositions, one can model: ◮ unawareness of agents ◮ unawareness of actions Similarly to becoming aware, one can model: ◮ becoming unaware (forgetting) Let us go back to basics first. We introduce structures , language , and semantics for logics with unawareness of propositions.

Structures An epistemic awareness model M = ( S , R , A , V ) for N and P : ◮ a domain S of (factual) states (or ‘worlds’), ◮ an accessibility function R : N → P ( S × S ), ◮ an awareness function A : N → S → P ( P ), ◮ a valuation function V : P → P ( S ). An epistemic awareness state is a pointed epistemic awareness model ( M , s ). For A ( i )( s ) write A i ( s ). The agent is aware of prop. variables in A i ( s ). The agent is unaware of prop. variables in P \ A i ( s ).

Structures An epistemic awareness model M = ( S , R , A , V ) for N and P : ◮ a domain S of (factual) states (or ‘worlds’), ◮ an accessibility function R : N → P ( S × S ), ◮ an awareness function A : N → S → P ( P ), ◮ a valuation function V : P → P ( S ). An epistemic awareness state is a pointed epistemic awareness model ( M , s ). For A ( i )( s ) write A i ( s ). The agent is aware of prop. variables in A i ( s ). The agent is unaware of prop. variables in P \ A i ( s ). Two examples: i i p p p i i p p ¬ p

Standard bisimulation — notation ( M , s ) ≃ P ′ ( M ′ , s ′ ) Given M = ( S , R , A , V ) and M ′ = ( S ′ , R ′ , A ′ , V ′ ). For all P ′ ⊆ P define R [ P ′ ] by ( s , s ′ ) ∈ R [ P ′ ] iff: atoms for all p ∈ P ′ , s ∈ V ( p ) iff s ′ ∈ V ′ ( p ); aware for all i ∈ N , A i ( s ) ∩ P ′ = A ′ i ( s ′ ) ∩ P ′ ; forth for all i ∈ N , if t ∈ S and R i ( s , t ) then there is a t ′ ∈ S ′ s.t. R ′ i ( s ′ , t ′ ) and ( t , t ′ ) ∈ R [ P ′ ]; back for all i ∈ N , if t ′ ∈ S ′ and R ′ i ( s ′ , t ′ ) then there is a t ∈ S such that R i ( s , t ) and ( t , t ′ ) ∈ R [ P ′ ].

Standard bisimulation — notation ( M , s ) ≃ P ′ ( M ′ , s ′ ) Given M = ( S , R , A , V ) and M ′ = ( S ′ , R ′ , A ′ , V ′ ). For all P ′ ⊆ P define R [ P ′ ] by ( s , s ′ ) ∈ R [ P ′ ] iff: atoms for all p ∈ P ′ , s ∈ V ( p ) iff s ′ ∈ V ′ ( p ); aware for all i ∈ N , A i ( s ) ∩ P ′ = A ′ i ( s ′ ) ∩ P ′ ; forth for all i ∈ N , if t ∈ S and R i ( s , t ) then there is a t ′ ∈ S ′ s.t. R ′ i ( s ′ , t ′ ) and ( t , t ′ ) ∈ R [ P ′ ]; back for all i ∈ N , if t ′ ∈ S ′ and R ′ i ( s ′ , t ′ ) then there is a t ∈ S such that R i ( s , t ) and ( t , t ′ ) ∈ R [ P ′ ]. The two epistemic states below are not { p } standard bisimilar. i i p p p i i p p ¬ p

Awareness bisimulation — notation ( M , s ) ↔ P ′ ( M ′ , s ′ ) Given M = ( S , R , A , V ) and M ′ = ( S ′ , R ′ , A ′ , V ′ ). For all P ′ ⊆ P define R [ P ′ ] by ( s , s ′ ) ∈ R [ P ′ ] iff: atoms for all p ∈ P ′ , s ∈ V ( p ) iff s ′ ∈ V ′ ( p ); aware for all i ∈ N , A i ( s ) ∩ P ′ = A ′ i ( s ′ ) ∩ P ′ ; forth for all i ∈ N , if t ∈ S and R i ( s , t ) then there is a t ′ ∈ S ′ s.t. R ′ i ( s ′ , t ′ ) and ( t , t ′ ) ∈ R [ A i ( s ) ∩ P ′ ]; back for all i ∈ N , if t ′ ∈ S ′ and R ′ i ( s ′ , t ′ ) then there is a t ∈ S such that R i ( s , t ) and ( t , t ′ ) ∈ R [ A ′ i ( s ′ ) ∩ P ′ ].

Awareness bisimulation — notation ( M , s ) ↔ P ′ ( M ′ , s ′ ) Given M = ( S , R , A , V ) and M ′ = ( S ′ , R ′ , A ′ , V ′ ). For all P ′ ⊆ P define R [ P ′ ] by ( s , s ′ ) ∈ R [ P ′ ] iff: atoms for all p ∈ P ′ , s ∈ V ( p ) iff s ′ ∈ V ′ ( p ); aware for all i ∈ N , A i ( s ) ∩ P ′ = A ′ i ( s ′ ) ∩ P ′ ; forth for all i ∈ N , if t ∈ S and R i ( s , t ) then there is a t ′ ∈ S ′ s.t. R ′ i ( s ′ , t ′ ) and ( t , t ′ ) ∈ R [ A i ( s ) ∩ P ′ ]; back for all i ∈ N , if t ′ ∈ S ′ and R ′ i ( s ′ , t ′ ) then there is a t ∈ S such that R i ( s , t ) and ( t , t ′ ) ∈ R [ A ′ i ( s ′ ) ∩ P ′ ]. The two epistemic states below are { p } awareness bisimilar. i i p p p i i p p ¬ p

Example of awareness bisimilar epistemic awareness states i i p p p i i ¬ p p p ◮ The two models are { p } awareness bisimilar in the root state; ◮ Because: { p } awareness bisimilar in the accessible state; ◮ Because: {∅} awareness bisimilar in the accessible (leaf) state.

Example of awareness bisimilar epistemic awareness states i i p p p i i ¬ p p p ◮ The two models are { p } awareness bisimilar in the root state; ◮ Because: { p } awareness bisimilar in the accessible state; ◮ Because: {∅} awareness bisimilar in the accessible (leaf) state. Consider implicit knowledge � i and explicit knowledge K E i . A formula is implicitly known if it is true in all accessible states. A formula is explicitly known if (above &) the agent is aware of it. K E i � i p is true above and false below. The models are not modally equivalent.

Example of awareness bisimilar epistemic awareness states i i p p p i i ¬ p p p ◮ The two models are { p } awareness bisimilar in the root state; ◮ Because: { p } awareness bisimilar in the accessible state; ◮ Because: {∅} awareness bisimilar in the accessible (leaf) state. Consider implicit knowledge � i and explicit knowledge K E i . A formula is implicitly known if it is true in all accessible states. A formula is explicitly known if (above &) the agent is aware of it. K E i � i p is true above and false below. The models are not modally equivalent. Bisimilar states are not modally equivalent. Is this a problem?

Recalling a previous example ¬ p p i i i For i , all four structures below look like the structure above. The four below are { p } standard bisimilar. They are not { p , q } standard bisimilar. They are { p , q } awareness bisimilar. ¬ pq p ¬ q ¬ pq ¬ pq pq i i i i i i i i i i i i i ¬ pq pq ¬ p ¬ q p ¬ q ¬ p ¬ q p ¬ q i i i i i i i i i