[PPT] - Multi-agent Abduction Using Doxastic Temporal Models Thomas PowerPoint Presentation

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Multi-agent Abduction Using Doxastic Temporal Models

Thomas Bolander DTU Compute, Technical University of Denmark Joint work with Sonja Smets, ILLC

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 1/22

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Sally-Anne in Dynamic Epistemic Logic

Bolander: Seeing is Believing—Formalising False-Belief Tasks in Dynamic Epistemic Logic, in Outstanding Contributions to Logic, Springer, 2018.

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 2/22

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Pepper passing false-belief tasks

http://www2.compute.dtu.dk/~tobo/forskerzonen_trimmed.mp4

Forskerens favorit: Robotten Pepper lærer at sætte sig i andres sted. https://videnskab.dk/teknologi-innovation/ forskerens-favorit-robotten-pepper-laerer-at-saette-sig-i-andres-sted

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 3/22

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The best of both worlds: Machine learning + explainable logical reasoning

Research in social artificial intelligence at DTU: A Pepper robot with social perspective-taking abilities. The robot solves cognitive tasks: false-belief tasks of arbitrary order. Humans can solve first-order at age 4, second-order at age 10 and third-order at age 20.

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 4/22

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Plausibility models

Agents: A. Propositions: P. (Multi-pointed) plausibility model: M = W , (i)i∈A, V , Wd, where

W is a set of possible worlds, marked .
Each i is a plausibility relation: a set of mutually disjoint

well-preorders covering W .

V is a valuation.
Wd ⊆ W is a set of designated worlds (one of which is the actual),

marked . w1 :t w2 :x S iff w1 S w2 iff S finds w1 more plausible than w2. ∼i := i ∪ i. w1 ∼i w2 means w1 and w2 are (epistemically) indistinguishable to agent i.

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 5/22

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Language and semantics

Language: φ ::= ¬φ | φ ∧ φ | Biφ | Kiφ | Cφ. Semantics:

M, w |

= Biφ iff φ holds in the most plausible worlds i cannot epistemically distinguish from w.

M, w |

= Ki iff φ holds in all worlds i cannot epistemically distinguish from w.

M |

= φ iff M, w | = φ for all w ∈ Wd. M = w1 :t w2 :x S M | = BSt ∧ ¬KSt

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 6/22

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Event models

(Multi-pointed) event model: E = E, (i)i∈A, pre, post, Ed, where

E is a set of possible events, marked .
Each i is a plausibility relation (as before).
For each e ∈ E, pre(e) is a precondition: a formula.
For each e ∈ E, post(e) is a simple postcondition: a conjunction of

literals.

Ed ⊆ E is a set of designated events (one of which is the actual),

marked . Events e are labelled by (pre(e), post(e)). e1 :(⊤, ⊤) e2 :(⊤, x ∧ ¬t) S

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 7/22

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Product update

The product update of a plausibility model M = W , (i)i∈A, V , Wd with an event model E = E, (i)i∈A, pre, post, Ed is the plausibility model M ⊗ E = W ′, (′

i)i∈A, V ′, W ′ d

where

W ′ = {(w, e) ∈ W × E | M, w |

= pre(e)}

′

i= {((w, e), (v, f )) ∈ W ′ × W ′ | (w ∼i v and e i f and f i

e) or (e i f and f i e and w i v)} (action-priority update).

V ′(p) = {(w, e) ∈ W ′ | post(e) |

= p or (w ∈ V (p) and post(e) | = ¬p)}.

W ′

d = {(w, e) ∈ W ′ | w ∈ Wd and e ∈ Ed}.

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 8/22

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DoTL models

A DoTL model is a plausibility model where the worlds are structured into histories over an alphabet of events. Formally: A DoTL model (model of Doxastic Temporal Logic) over a set of events Σ is a plausibility model D = H, (i)i∈A, V , Hd, where H ⊆ Σ⋆ is closed under non-empty prefixes. Elements of H are called histories. The frontier of a DoTL model is the plausibility model consisting of the maximal histories. Formally: The frontier of a DoTL model D = H, (i)i∈A, V , Hd is the plausibility model given by frontier(D) = Hmax, (i ↾ H2

max)i∈A, V ↾ Hmax, Hd ∩ Hmax, where

Hmax = {h ∈ H | there is no h′ ∈ H with |h′| > |h|}. The inclusive (product) update of a DoTL model D with an event model E extends D by updating its frontier with E. Formally: The inclusive (product) update of a DoTL model D = H, (i)i∈A, V , Hd

ver Σ and an event model E = E, (i)i∈A, pre, post, Ed is the DoTL

model D ⊗∪ E over Σ ∪ E given by the union of D and frontier(D) ⊗ E.

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 9/22

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Example: The Sally-Anne task

initial state: w0 : S, t Leave: leave :(⊤, ¬S) Swap: skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S Enter: enter :(⊤, S) PeekBasket: ¬t!:(¬t, ⊤)

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 10/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S)

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S) t leave

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter S, t S, x enter enter S

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter S, t S, x enter enter S PeekBasket ¬t!:(¬t, ⊤)

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter S, t S, x enter enter S PeekBasket ¬t!:(¬t, ⊤) t ¬t!

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Sally-Anne DoTL model: inclusive updates

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter S, t S, x enter enter S PeekBasket ¬t!:(¬t, ⊤) t ¬t! Let h = (e1, . . . , en) be a most plausible history for agent i. The event en is called a surprise to agent i in h if (e1, . . . , en−1) is not a most plausible history for agent i. The event ¬t! is a surprise above.

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

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Language and semantics of DoTL models

We extend the previous language by ∗-free PDL (including inverses) on events/histories: φ ::= · · · | πφ π ::= e | π; π | π ∪ π | π−1 where e ∈ Σ. Semantics. D, h | = πφ iff for some h′ with hRπh′ we have D, h′ | = φ, where

Re = {(h, he) | he ∈ H}
Rπ1∪π2 = Rπ1 ∪ Rπ2
Rπ1;π2 = Rπ1 ◦ Rπ2
Rπ−1 = (Rπ)−1

D | = φ iff D, h | = φ for all h ∈ Hmax ∩ Hd.

Example. Any DoTL model D has full synchrony: There is a number m

such that D | = C

(∪i∈Σei)−m⊤ ∧ [(∪i∈Σei)−(m+1)]⊥
Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 12/22

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Subjective models

When DoTL model are constructed from the subjective view of an agent, it might not be possible to point out a single designated history. The associated subjective model

f agent i of a plausibility

model/event model/DoTL model is achieved by closing the set of designated points under the epistemic indistinguishability relation

f agent i.

The DoTL to the right shows the inclusive product updates from the subjective view. w0 : S, t Leave t leave Swap t x move skip S Enter S, t S, x enter enter S PeekBasket t ¬t!

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 13/22

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Restricted inclusive update

Inclusive update of D with E restricted to a subset I of the frontier: Mark frontier histories outside I with “cut”. Only update frontier histories in I with E. Formally: Let D be a DoTL model over Σ and E an event model with events E. Let I ⊆ Hmax be a set of frontier histories of

D. The inclusive update of D with E restricted to I is the DoTL

model D ⊗I

∪ E over Σ ∪ E given by the union of D and

(frontier(D) ↾ I) ⊗ E, and where all h ∈ Hmax − I are marked by “cut”. The most plausible update for agent i of D with E, denoted D ⊗i

∪ E, is

the inclusive update restricted to the most plausible histories of the frontier for agent i—and closed under more plausible histories for other

agents. Formally: For a DoTL model D, the histories considered most

plausible by agent i are the histories h′ ∈ Hmax satisfying D | = Bih′−1⊤. Let = (∪i∈A i)+. The most plausible update for agent i of D with E is the inclusive update of D with E restricted to {h ∈ Hmax | h h′ for some most plausible history h′ for agent i in D}. A restricted inclusive update is called degenerate if no designated histories are added (at the new level).

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 14/22

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Most plausible updates on the Sally-Anne example

w0 : S, t Leave leave :(⊤, ¬S)

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 15/22

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Most plausible updates on the Sally-Anne example

w0 : S, t Leave leave :(⊤, ¬S) t leave

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 15/22

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Most plausible updates on the Sally-Anne example

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 15/22

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Most plausible updates on the Sally-Anne example

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 15/22

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Most plausible updates on the Sally-Anne example

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 15/22

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Most plausible updates on the Sally-Anne example

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter cut S, t S, x enter enter S

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 15/22

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Most plausible updates on the Sally-Anne example

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter cut S, t S, x enter enter S PeekBasket ¬t!:(¬t, ⊤)

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 15/22

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Most plausible updates on the Sally-Anne example

w0 : S, t Leave leave :(⊤, ¬S) t leave Swap skip :(⊤, ⊤) move :(⊤, x ∧ ¬t) S t x move skip S enter :(⊤, S) Enter cut S, t S, x enter enter S PeekBasket ¬t!:(¬t, ⊤) t ¬t! Degenerate!

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 15/22

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Degenerate models, surprise and abduction

When an agent i does most plausible updates and ends up with a degenerate model, it corresponds to a surprise: There are no longer any histories consistent with the observed. In those cases, the agent needs to do abduction: Expand some of the cut histories (histories labelled “cut”). Single abduction step: Choose a cut history, remove the “cut” label, and expand it.

Example. Agent i might do most plausible updates on an initial

(subjective) plausibility model M with (subjective) event models E1, . . . , En: M ⊗i

∪ E1 ⊗i ∪ E2 ⊗i ∪ · · · ⊗i ∪ En. An abduction step is then to

pick a cut history h of length m and replace the DoTL model by: M ⊗i

∪ E1 ⊗i ∪ E2 ⊗ · · · ⊗ Em−1 ⊗{most plausible}∪{h} ∪

⊗i

∪Em ⊗ · · · ⊗ En.

Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 16/22

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