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

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

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

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

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

Plausibility models Agents : A . Propositions : P . (Multi-pointed) plausibility model : M = � W , ( � i ) i ∈A , V , W d � , 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. • W d ⊆ W is a set of designated worlds (one of which is the actual), marked . w 1 : t w 2 : x S iff w 1 � S w 2 iff S finds w 1 more plausible than w 2 . ∼ i := � i ∪ � i . w 1 ∼ i w 2 means w 1 and w 2 are (epistemically) indistinguishable to agent i . Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 5/22

Language and semantics Language : φ ::= ¬ φ | φ ∧ φ | B i φ | K i φ | C φ. Semantics : • M , w | = B i φ iff φ holds in the most plausible worlds i cannot epistemically distinguish from w . • M , w | = K i iff φ holds in all worlds i cannot epistemically distinguish from w . • M | = φ iff M , w | = φ for all w ∈ W d . w 1 : t w 2 : x S M = M | = B S t ∧ ¬ K S t Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 6/22

Event models (Multi-pointed) event model : E = � E , ( � i ) i ∈A , pre , post , E d � , 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. • E d ⊆ E is a set of designated events (one of which is the actual), marked . Events e are labelled by ( pre ( e ) , post ( e )). e 1 :( ⊤ , ⊤ ) e 2 :( ⊤ , x ∧ ¬ t ) S Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 7/22

Product update The product update of a plausibility model M = � W , ( � i ) i ∈A , V , W d � with an event model E = � E , ( � i ) i ∈A , pre , post , E d � 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 ) } . d = { ( w , e ) ∈ W ′ | w ∈ W d and e ∈ E d } . • W ′ Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 8/22

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 , H d � , 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 , H d � is the plausibility model given by frontier ( D ) = � H max , ( � i ↾ H 2 max ) i ∈A , V ↾ H max , H d ∩ H max � , where H max = { 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 , H d � over Σ and an event model E = � E , ( � i ) i ∈A , pre , post , E d � 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

Example: The Sally-Anne task initial state: w 0 : S , t Leave: leave :( ⊤ , ¬ S ) S skip :( ⊤ , ⊤ ) move :( ⊤ , x ∧ ¬ t ) Swap: Enter: enter :( ⊤ , S ) PeekBasket: ¬ t !:( ¬ t , ⊤ ) Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 10/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave :( ⊤ , ¬ S ) Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave leave :( ⊤ , ¬ S ) t Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave leave :( ⊤ , ¬ S ) t S Swap skip :( ⊤ , ⊤ ) move :( ⊤ , x ∧ ¬ t ) Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave leave :( ⊤ , ¬ S ) t S skip move Swap skip :( ⊤ , ⊤ ) move :( ⊤ , x ∧ ¬ t ) S x t Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave leave :( ⊤ , ¬ S ) t S skip move Swap skip :( ⊤ , ⊤ ) move :( ⊤ , x ∧ ¬ t ) S x t Enter enter :( ⊤ , S ) Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave leave :( ⊤ , ¬ S ) t S skip move Swap skip :( ⊤ , ⊤ ) move :( ⊤ , x ∧ ¬ t ) S x t enter enter Enter enter :( ⊤ , S ) S S , t S , x Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave leave :( ⊤ , ¬ S ) t S skip move Swap skip :( ⊤ , ⊤ ) move :( ⊤ , x ∧ ¬ t ) S x t enter enter Enter enter :( ⊤ , S ) S S , t S , x PeekBasket ¬ t !:( ¬ t , ⊤ ) Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave leave :( ⊤ , ¬ S ) t S skip move Swap skip :( ⊤ , ⊤ ) move :( ⊤ , x ∧ ¬ t ) S x t enter enter Enter enter :( ⊤ , S ) S S , t S , x PeekBasket ¬ t ! ¬ t !:( ¬ t , ⊤ ) t Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 11/22

Sally-Anne DoTL model: inclusive updates w 0 : S , t Leave leave leave :( ⊤ , ¬ S ) t S skip move Swap skip :( ⊤ , ⊤ ) move :( ⊤ , x ∧ ¬ t ) S x t enter enter Enter enter :( ⊤ , S ) S S , t S , x PeekBasket ¬ t ! ¬ t !:( ¬ t , ⊤ ) t Let h = ( e 1 , . . . , e n ) be a most plausible history for agent i . The event e n is called a surprise to agent i in h if ( e 1 , . . . , e n − 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

Language and semantics of DoTL models We extend the previous language by ∗ -free PDL (including inverses) on events/histories: φ ::= · · · | � π � φ π ::= e | π ; π | π ∪ π | π − 1 where e ∈ Σ. Semantics . = � π � φ iff for some h ′ with hR π h ′ we have D , h ′ | D , h | = φ , where • R e = { ( h , he ) | he ∈ H } • R π 1 ∪ π 2 = R π 1 ∪ R π 2 • R π 1 ; π 2 = R π 1 ◦ R π 2 • R π − 1 = ( R π ) − 1 D | = φ iff D , h | = φ for all h ∈ H max ∩ H d . Example . Any DoTL model D has full synchrony: There is a number m � � � ( ∪ i ∈ Σ e i ) − m �⊤ ∧ [( ∪ i ∈ Σ e i ) − ( m +1) ] ⊥ such that D | = C Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 12/22

Subjective models When DoTL model are constructed w 0 : S , t from the subjective view of an agent, it might not be possible to point out a single designated history. Leave leave The associated subjective model t of agent i of a plausibility skip move model/event model/DoTL model is Swap achieved by closing the set of S x t designated points under the epistemic indistinguishability relation enter enter Enter of agent i . S S , t S , x The DoTL to the right shows the inclusive product updates from the PeekBasket ¬ t ! subjective view. t Bolander: Multi-agent Abduction, 6 Dec 2018 – p. 13/22