Toward proof-theoretic semantics for the deontic cognitive event - PowerPoint PPT Presentation

Logic and Cognition Workshop ICLA 2019, Delhi March 2, 2019 Toward proof-theoretic semantics for the deontic cognitive event calculus Naveen Sundar Govindarajulu, Selmer Bringsjord Rensselaer AI & Reasoning Lab Rensselaer Polytechnic Institute (RPI) Troy, New York, 12180, USA www.rpi.edu Sponsored by

Overview • Goal: Handle automation of Arrow’s theorem and similar results when applied to aggregation over cognitive states. • Our logic/tool: deontic cognitive event calculus (DCEC) • This talk: proof-theoretic semantics for a fragment of DCEC

Arrow’s Theorem (very briefly) • Without a dictator in sway, • it is impossible for a group of agents to have their individual preferences aggregated to yield preferences for the group as a whole • (with certain other desirable conditions). • First applied to voting over discrete finite choices

Arrow’s Theorem

Arrow’s Theorem • Also applies to judgements of propositions.

Arrow’s Theorem • Also applies to judgements of propositions. • Agents speculating on the value of propositions

Arrow’s Theorem • Also applies to judgements of propositions. • Agents speculating on the value of propositions • General case, we have a set of agents supplying propositions that are quite complex and conflicting.

Arrow’s Theorem • Also applies to judgements of propositions. • Agents speculating on the value of propositions • General case, we have a set of agents supplying propositions that are quite complex and conflicting.

Arrow’s Theorem

Arrow’s Theorem • Goal : Build a benevolent AI dictator that can merge complex beliefs from di ff erent agents (beliefs can be about other beliefs etc) using DCEC.

Arrow’s Theorem • Goal : Build a benevolent AI dictator that can merge complex beliefs from di ff erent agents (beliefs can be about other beliefs etc) using DCEC.

Need • A logic that can handle beliefs, knowledge, intentions, obligations, desires and other modalities

Our Tool Deontic Cognitive Event Calculus

DCEC ∗ e DCEC ∗ DCEC CEC µ C CC The deontic cognitive event calculus is one member in the cognitive calculi family.

Cognitive Caluli: briefly • Are quantified multi-sorted modal logic.

Why quantified multi-sorted modal logic? Reasoning Crudely Split Intensional (Modal) Reasoning Extensional Reasoning Theory of mind reasoning Math Physics Chemistry … Groceries Driving a car …

A Few Applications of Cognitive Calculi • False belief task • Arkoudas, Konstantine, and Selmer Bringsjord. "Toward Formalizing Common-sense Psychology: An Analysis of the False-belief Task." PRICAI 2008 : Trends in Artificial Intelligence (2008): 17-29. Expanded: “Propositional Attitudes and Causation” Int. J. Software & Informatics , 3.1 : 47–65, 2009. • Self-awareness/consciousness • Mirror task • Bringsjord, Selmer, and Naveen Sundar Govindarajulu. "Toward a Modern Geography of Minds, Machines, and Math." In Philosophy and Theory of Artificial Intelligence , pp. 151-165. Springer Berlin Heidelberg, 2013. • Floridi’s KG4 (earlier: Wise Man Puzzle, including infinitized WMP) • Bringsjord, Selmer, John Licato, Naveen Sundar Govindarajulu, Rikhiya Ghosh, and Atriya Sen. "Real Robots that Pass Human Tests of Self-consciousness." In Robot and Human Interactive Communication (RO-MAN), 2015 24th IEEE International Symposium on, pp. 498-504. IEEE, 2015. • Moral Cognition • Akrasia • Bringsjord, Selmer, G. Naveen Sundar, Dan Thero, and Mei Si. “Akratic Robots and the Computational Logic Thereof." In Proceedings of the IEEE 2014 International Symposium on Ethics in Engineering, Science, and Technology , p. 7. IEEE Press, 2014. • Doctrine of Double Effect • Govindarajulu, Naveen Sundar, and Selmer Bringsjord. "On Automating the Doctrine of Double Effect." International Joint Conference on AI (IJCAI 2017) • Govindarajulu, Naveen Sundar, and Selmer Bringsjord. “Beyond the Doctrine of Double Effect: A Formal Model of True Self-Sacrifice” International Conference on Robot Ethics and Safety Systems (ICRESS 2017) • Virtue Ethics • Govindarajulu, Naveen Sundar, Selmer Bringsjord, Rikhiya Ghosh and Vasanth Sarathy. “Towards Virtuous Machines” AAAI/ACM Conference on AI Ethics and Society (AIES 2019)

The Doctrine of Double Effect

IJCAI 2017; Autonomy Track

AAAI/ACAM Conference on AI, Ethics and Society 2019 Toward the Engineering of Virtuous Machines Naveen Sundar Govindarajulu, Vasanth Sarathy Selmer Bringsjord and Rikhiya Ghosh Human Robot Interaction Laboratory Rensselaer AI & Reasoning Lab Tufts University Rensselaer Polytechnic Institute (RPI) Medford, MA, 02155, USA Troy, New York, 12180, USA www.tufts.edu www.rpi.edu

The Formalization (Overview) ( Q 1 ) Virtuous Person V n ( s ) ↔ ∃ ≥ n a : Exemplar ( s, a ) ( Q 2 ) Virtue G n ( τ ) ↔ ∃ ≥ n a : Trait ( τ , a ) ¬ ( R 1 ) Admiration in DCEC ( R 2 ) Inference Schema for Trait ( R 3 ) Learning a Trait holds ( admires ( a, b, α ) , t ) ) n ( σ i , happens ( action ( α i , a ) , t i ) $ 2 3 Exemplar ( e, l ) ^ Θ ( a, t 0 ) ^ 0 1 LearnTrait ( l, τ , t ) $ 9 e � � σ i ( t ) = σ , g ( α i ) = α g ⇣ �⌘ 4 5 � B l, t, Trait τ , e ( a 6 = b ) ^ ( t 0 < t ) B C i =1 0 2 3 1 [ I Trait ] B C B C Trait ( τ , a ) B B 6 7 C C � � ^ happens ( action ( b, α ) , t 0 ) ^ LearnTrait ( l, h σ , α i , t ) ! σ ! happens ( action ( l, α ) , t ) B @ a, t, B B 6 7 C C @ 4 5 A A ν ( action ( b, α ) , t ) > 0 Exemplar Defninition Exemplar ( e, l ) $ 9 ! n t. 9 α . holds ( admires ( l, e, α ) , t )

DCEC briefly calculus. Syntax S ::= Agent | ActionType | Action v Event | Moment | Fluent  action : Agent ⇥ ActionType ! Action    initially : Fluent ! Formula      holds : Fluent ⇥ Moment ! Formula      happens : Event ⇥ Moment ! Formula   f ::= clipped : Moment ⇥ Fluent ⇥ Moment ! Formula     initiates : Event ⇥ Fluent ⇥ Moment ! Formula      terminates : Event ⇥ Fluent ⇥ Moment ! Formula      prior : Moment ⇥ Moment ! Formula  t ::= x : S | c : S | f ( t 1 , . . . , t n ) q : Formula | ¬ φ | φ ^ ψ | φ _ ψ | 8 x : φ ( x ) |    P ( a, t, φ ) | K ( a, t, φ ) |      C ( t, φ ) | S ( a, b, t, φ ) | S ( a, t, φ ) | B ( a, t, φ ) φ ::=  D ( a, t, φ ) | I ( a, t, φ )      O ( a, t, φ , ( ¬ ) happens ( action ( a ⇤ , α ) , t 0 )) 

DCEC briefly Sort Description Agent Human and non-human actors. Time The Time type stands for time in the domain. E.g. simple, such as t i , or complex, such as birthday ( son ( jack )) . Event Used for events in the domain. ActionType Action types are abstract actions. They are in- stantiated at particular times by actors. Exam- ple: eating. Action A subtype of Event for events that occur as actions by agents. Fluent Used for representing states of the world in the event calculus.

• Jones intends to convince Smith to believe that Jones believes that were the cat, lying in the foyer now, to be let out, it would settle, dozing, on the mat.

• Jones intends to convince Smith to believe that Jones believes that were the cat, lying in the foyer now, to be let out, it would settle, dozing, on the mat. I ( j, C ( s, B ( s, B ( j, ι [ c : in ( c, ι ( f : Foyer ( f )) , m : mat ( m )] out ( c ) → subj doze ( c, m ))))

• Jones intends to convince Smith to believe that Jones believes that were the cat, lying in the foyer now, to be let out, it would settle, dozing, on the mat. I ( j, C ( s, B ( s, B ( j, ι [ c : in ( c, ι ( f : Foyer ( f )) , m : mat ( m )] intensional operators out ( c ) → subj doze ( c, m ))))

• Jones intends to convince Smith to believe that Jones believes that were the cat, lying in the foyer now, to be let out, it would settle, dozing, on the mat. I ( j, C ( s, B ( s, B ( j, ι [ c : in ( c, ι ( f : Foyer ( f )) , m : mat ( m )] intensional operators out ( c ) → subj doze ( c, m )))) scoped term

• Jones intends to convince Smith to believe that Jones believes that were the cat, lying in the foyer now, to be let out, it would settle, dozing, on the mat. I ( j, C ( s, B ( s, B ( j, ι [ c : in ( c, ι ( f : Foyer ( f )) , m : mat ( m )] intensional operators out ( c ) → subj doze ( c, m )))) subjunctive conditional scoped term

Example automated inference assume Premise 1 C (t , ∀ a,t happens(display(wealth, a), t) ⇒ holds(wealthy(a), t)) 0 from {Premise 1} assume Premise 2 C (t , P (robert, t , happens(display(wealth, host), t ))) 0 0 0 from {Premise 2} CC ⊢ G1 B (robert, t , holds(wealthy(host), t )) 1 0 CC ⊢ from {Premise 2,Premise 1} G2 B (host, t , B (robert, t , holds(wealthy(host), t ))) 2 1 0 CC ⊢ from {Premise 2,Premise 1} G3 B (robert, t , B (host, t , B (robert, t , holds(wealthy(host), t )))) 3 2 1 0 from {Premise 2,Premise 1}

Semantics?

Semantics? • Possible-worlds semantics is not attractive for us for a number of reasons (we are okay with possible worlds being used for necessity/possibility)