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 pq i i i ¬pq p¬q i i i ¬p¬q p¬q ¬pq i i i i i i ¬p¬q p¬q ¬pq pq i 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 pq i i i ¬pq p¬q i i i ¬p¬q p¬q ¬pq i i i i i i ¬p¬q p¬q ¬pq pq i i i i i i i i i i

Adding dynamics — agent i becomes aware of prop. var. q

¬p¬q p¬q ¬pq i i i i i i ¬p p i i i i becomes aware of q

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

¬p¬q p¬q ¬pq i i i i i i ¬p¬q p¬q ¬pq i i i i i i i becomes aware of q

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 Ai(s). The agent is aware of prop. variables in Ai(s). The agent is unaware of prop. variables in P \ Ai(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 Ai(s). The agent is aware of prop. variables in Ai(s). The agent is unaware of prop. variables in P \ Ai(s). Two examples: p p p i i p p ¬p i i

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, Ai(s) ∩ P′ = A′

i(s′) ∩ P′;

forth for all i ∈ N, if t ∈ S and Ri(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 Ri(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, Ai(s) ∩ P′ = A′

i(s′) ∩ P′;

forth for all i ∈ N, if t ∈ S and Ri(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 Ri(s, t) and (t, t′) ∈ R[P′]. The two epistemic states below are not {p} standard bisimilar. p p p i i p p ¬p i i

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, Ai(s) ∩ P′ = A′

i(s′) ∩ P′;

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

i(s′, t′) and (t, t′) ∈ R[Ai(s) ∩ P′];

back for all i ∈ N, if t′ ∈ S′ and R′

i(s′, t′) then there is a

t ∈ S such that Ri(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, Ai(s) ∩ P′ = A′

i(s′) ∩ P′;

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

i(s′, t′) and (t, t′) ∈ R[Ai(s) ∩ P′];

back for all i ∈ N, if t′ ∈ S′ and R′

i(s′, t′) then there is a

t ∈ S such that Ri(s, t) and (t, t′) ∈ R[A′

i(s′) ∩ P′].

The two epistemic states below are {p} awareness bisimilar. p p p i i p p ¬p i i

Example of awareness bisimilar epistemic awareness states

p p p i i p p ¬p i i

◮ 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

p p p i i p p ¬p i i

◮ 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 ip is true above and false below.

The models are not modally equivalent.

Example of awareness bisimilar epistemic awareness states

p p p i i p p ¬p i i

◮ 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 ip 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 pq i i i ¬pq p¬q i i i ¬p¬q p¬q ¬pq i i i i i i ¬p¬q p¬q ¬pq pq i i i i i i i i i i

Implicit knowledge i, explicit kn. K E

i , speculative kn. K S i

Language L(A, , K E, K S) ∋ ϕ ::= p | ¬ϕ | ϕ∧ϕ | Aiϕ | iϕ | K E

i ϕ | K S i ϕ

Semantics (M, s) | = Aiϕ iff var(ϕ) ⊆ Ai(s) (M, s) | = iϕ iff ∀t ∈ Ri(s), (M, t) | = ϕ (M, s) | = K E

i ϕ iff var(ϕ) ⊆ Ai(s) and ∀t ∈ Ri(s), (M, t) |

= ϕ (M, s) | = K S

i ϕ iff ∀t ∈ Ri(s), ∀(M′, t′)↔Ai(s)(M, t), (M′, t′) |

= ϕ

◮ L: the logic of implicit knowledge

(i and Ai)

◮ LE: the logic of explicit knowledge

(K E

i

and Ai)

◮ LS: the logic of speculative knowledge

(K S

i

and Ai) K S

i ϕ (speculative knowledge): You know ϕ iff in all accessible

states, in all (for you) awareness bisimilar states, ϕ is true.

Implicit knowledge i, explicit kn. K E

i , speculative kn. K S i

¬pq pq i i i

◮ i implicitly knows q ◮ i does not speculative know q (she considers poss. q is false) ◮ i does not explicitly know q (she is unaware of q) ◮ i implicitly knows q ∨ ¬q ◮ i speculatively knows q ∨ ¬q ◮ i does not explicitly know q ∨ ¬q (she is unaware of q) ◮ i is implicitly uncertain about p ◮ i is speculatively uncertain about p ◮ i is explicitly uncertain about p

Standard bisimulation ≃ and awareness bisimulation ↔

p p p i i p p ¬p i i K E

i ip is the abbreviation for iip ∧ Aiip, which is in L.

This is true above and false below. Therefore, the models are not modally equivalent in the logic of implicit knowledge. The models are {p} awareness bisimilar. Awareness bisimilarity implies modal equivalence in LE. Therefore, the models are modally equivalent in the logic of explicit knowledge.

Standard bisimulation ≃ and awareness bisimulation ↔

p p p i i p p ¬p i i K E

i ip is the abbreviation for iip ∧ Aiip, which is in L.

This is true above and false below. Therefore, the models are not modally equivalent in the logic of implicit knowledge. The models are {p} awareness bisimilar. Awareness bisimilarity implies modal equivalence in LE. Therefore, the models are modally equivalent in the logic of explicit knowledge. On image-finite models: (M, s) ≡

Q (M′, s′) iff (M, s) ≃Q (M′, s′).

On image-finite models: (M, s) ≡E

Q (M′, s′) iff (M, s)↔Q(M′, s′).

On image-finite models: (M, s) ≡S

Q (M′, s′) iff (M, s)↔Q(M′, s′).

Expressivity

A logic L is at least as expressive as another logic L′ if there are structures (M, s) and (M′, s′) and a formula ϕ in the language of L such that ϕ is true in (M, s) and false in (M′, s′), but the structures are modally equivalent in L′. L(K E) is at least as expressive as L(): K Ep is true in (the root

• f) M but false in M′; but M and M′ are modally equiv. in L().

M: p p M′: p p L() is at least as expressive as L(K E): p is true in (the root

• f) M but false in M′; but M and M′ are modally equiv. in L(K E).

M: p p M′: p ¬p

Expressivity results

The logic of implicit knowledge (lang. L(, A)) is more expressive than the logic of explicit knowledge (language L(K E, A)).

◮ L(, A), L(, K E), . . . are equally expressive.

K E

i ϕ

⇔ iϕ ∧ Aiϕ Aiϕ ⇔ K E

i (ϕ ∨ ¬ϕ) ◮ L(K E, A), L(K E), . . . are equally expressive.

Aiϕ ⇔ K E

i (ϕ ∨ ¬ϕ) ◮ L(K S, A) and L(K E, A) are equally expressive.

Proof using interpolation. There are more results below L(K E), as shown on previous slide.

Axiomatization results

The axiomatizations of the logic of implicit knowledge and of the logic of explicit knowledge are reported in

◮ Fagin & Halpern, Belief, unawareness, and limited reasoning ◮ Halpern, Alternative semantics for unawareness.

Some variations (multi S5 or multi K) are of interest. The axiomatization of the logic of speculative knowledge is novel. Compare: K E

i ϕ → Aiϕ

From ϕ infer Aiϕ → K E

i ϕ

with K S

i ϕ → (¬Aip → K S i ϕ[p\ψ])

p ∈ var(ψ) From ϕ infer K S

i ϕ

Back to dynamics — agent i becomes aware of prop. var. q

¬p¬q p¬q ¬pq i i i i i i ¬p¬q p¬q ¬pq i i i i i i i becomes aware of q

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.

Back to dynamics — agent i becomes aware of prop. var. q

¬p¬q p¬q ¬pq i i i i i i ¬p¬q p¬q ¬pq i i i i i i i becomes aware of q

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. The agent was not aware of ‘wine’, but after the announcement she is: ¬Aiq ∧ [A+q]Aiq. Here, [A+q] is a dynamic modal operator and A+q a singleton epistemic awareness action (i.e., action model).

Dynamics of knowledge and dynamics of awareness

◮ dynamic modalities for change of knowledge

announcements, private announcements, . . .

◮ dynamic modalities for change of awareness

publicly becoming aware, privately becoming aware, . . .

◮ dynamic modalities for change of knowledge and awareness

addressing a novel issue, . . . In the presence of factual change, any change of knowledge or awareness can be done! Given finite (M, s) and (M′, s′), there is an epistemic awareness action (M, s) s.t. (M, s) ⊗ (M, s) ≃ (M′, s′).

The dynamic logic of explicit knowledge is more expressive

p p p i i p p ¬p i i A+p ⇒ p p p i i p p ¬p i i Left: K E

i ip is true above and false below.

Left: indistinguishable in the logic of explicit knowledge. Right: K E

i K E i p is true above and false below.

Therefore, left: [A+p]K E

i K E i p is true above and false below.

Left: distinguishable in the dynamic logic of explicit knowledge. Some expressivity results involving dynamics: – dynamic logic of explicit knowl. > the logic of explicit knowledge. – logic of implicit knowledge > the logic of explicit knowledge – dynamic logic of implicit knowl. = dyn. logic of explicit knowl.

References, applications, generalizations

(Self-)References:

◮ van Ditmarsch, French, Velazquez, Wang. Knowledge,

awareness, and bisimulation. Proc. of 14th TARK, 2013.

◮ van Ditmarsch, French, Velazquez. Action models for

knowledge and awareness. Proc. of 11th AAMAS, 2012.

◮ van Ditmarsch, French. Becoming aware of propositional

• variables. Proc. of ICLA 2011, LNAI 6521, 2011.

Applications:

◮ Economics: equilibria for incomplete information games ◮ Artificial Intelligence: memory-bounded reasoning ◮ Logic: lowering bounds for expected/average complexity

Generalizations:

◮ unawareness of agents ◮ unawareness of actions ◮ becoming unaware (forgetting)