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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

Logic and Cognition Workshop ICLA 2019, Delhi March 2, 2019

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

• 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 different 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 different agents (beliefs can be about other beliefs etc) using DCEC.

Arrow’s Theorem

Need

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

Deontic Cognitive Event Calculus

Our Tool

CC

CEC

DCEC

µC

DCEC∗

DCEC∗

e

The deontic cognitive event calculus is one member in the cognitive calculi family.

Cognitive Caluli: briefly

• Are quantified multi-sorted modal logic.

Reasoning

Extensional Reasoning Intensional (Modal) Reasoning

Math Physics Chemistry Theory of mind reasoning Groceries Driving a car

… …

Crudely Split

Why quantified multi-sorted modal logic?

• 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
• f 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
• n 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)

A Few Applications of Cognitive Calculi

The Doctrine of Double Effect

IJCAI 2017; Autonomy Track

Toward the Engineering of Virtuous Machines

Naveen Sundar Govindarajulu, Selmer Bringsjord and Rikhiya Ghosh Rensselaer AI & Reasoning Lab Rensselaer Polytechnic Institute (RPI) Troy, New York, 12180, USA www.rpi.edu

AAAI/ACAM Conference on AI, Ethics and Society 2019

Vasanth Sarathy Human Robot Interaction Laboratory Tufts University Medford, MA, 02155, USA www.tufts.edu

The Formalization (Overview)

(Q1) Virtuous Person Vn(s) ↔ ∃≥na : Exemplar(s, a)

(Q2) Virtue Gn(τ) ↔ ∃≥na : Trait(τ, a)

¬

(R1) Admiration in DCEC holds(admires(a, b, α), t) $ B B B B B @

Θ(a, t0) ^

B B B @a, t, 2 6 6 4 (a 6= b) ^ (t0 < t) ^ happens(action(b, α), t0)^ ν(action(b, α), t) > 0 3 7 7 5 1 C C A 1 C C C C C A

(R2) Inference Schema for Trait ( σi, happens(action(αi, a), ti) g

• σi(t)
• = σ, g(αi) = α

)n

i=1

Trait(τ, a) [ITrait]

Exemplar Defninition Exemplar(e, l) $ 9!nt.9α.holds(admires(l, e, α), t)

(R3) Learning a Trait LearnTrait(l, τ, t) $ 9e 2 4 Exemplar(e, l)^ B ⇣ l, t, Trait

• τ, e

⌘ 3 5 LearnTrait(l, hσ, αi, t) !

• σ ! happens(action(l, α), t)

calculus.

Syntax S ::= Agent | ActionType | Action v Event | Moment | Fluent f ::=                                action : Agent ⇥ ActionType ! Action initially : Fluent ! Formula holds : Fluent ⇥ Moment ! Formula happens : Event ⇥ Moment ! Formula 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(t1, . . . , tn) φ ::=                q : Formula | ¬φ | φ ^ ψ | φ _ ψ | 8x : 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) D(a, t, φ) | I(a, t, φ) O(a, t, φ, (¬)happens(action(a⇤, α), t0))

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 ti, 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.

DCEC briefly

• Jones intends to convince Smith to believe that Jones

believes that were the cat, lying in the foyer now, to be let

• ut, 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

• ut, 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)]

• ut(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

• ut, it would settle, dozing, on the mat.

intensional operators

I(j, C(s, B(s, B(j,ι[c : in(c, ι(f : Foyer(f)), m : mat(m)]

• ut(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

• ut, it would settle, dozing, on the mat.

intensional operators scoped term

I(j, C(s, B(s, B(j,ι[c : in(c, ι(f : Foyer(f)), m : mat(m)]

• ut(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

• ut, it would settle, dozing, on the mat.

subjunctive conditional intensional operators scoped term

I(j, C(s, B(s, B(j,ι[c : in(c, ι(f : Foyer(f)), m : mat(m)]

• ut(c) →subj doze(c, m))))

assume Premise 2 C(t , P(robert, t , happens(display(wealth, host), t )))

from {Premise 2}

assume Premise 1 C(t , ∀ a,t happens(display(wealth, a), t) ⇒ holds(wealthy(a), t))

from {Premise 1}

CC ⊢ G1 B(robert, t , holds(wealthy(host), t ))

from {Premise 2,Premise 1}

1

CC ⊢ G2 B(host, t , B(robert, t , holds(wealthy(host), t )))

from {Premise 2,Premise 1}

2 1

CC ⊢ G3 B(robert, t , B(host, t , B(robert, t , holds(wealthy(host), t ))))

from {Premise 2,Premise 1}

3 2 1

Example automated inference

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)

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)

• Not cognitively realistic

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)

• Not cognitively realistic
• Modal operators like knowledge require an argument
• r justification that can be hard to model using

possible worlds

Proof-theoretic Semantics

• Meanings for connectives are specified in terms of

proofs

• Computational Proof-Theoretic Semantics: Meanings for

connectives are specified in terms of computations over proofs

Cognitive Reducibility Argument for Proof-Theoretic Semantics

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

ν | = φ ! ψ iff if ν | = φ then ν | = ψ

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

ν | = φ ! ψ iff if ν | = φ then ν | = ψ

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

ν | = φ ! ψ iff if ν | = φ then ν | = ψ

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

ν | = φ ! ψ iff if ν | = φ then ν | = ψ

Meta logical machinery

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

ν | = φ ! ψ iff if ν | = φ then ν | = ψ

M

Meta logical machinery

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

ν | = φ ! ψ iff if ν | = φ then ν | = ψ

M

Meta logical machinery

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

ν | = φ ! ψ iff if ν | = φ then ν | = ψ

M

Meta logical machinery

Cognitive Reducibility Argument for Proof-Theoretic Semantics

β := (pi ^ pj) $ (pj ^ pi)

ν | = β

ν | = φ ! ψ iff if ν | = φ then ν | = ψ

Not defined truth functionally or model theoretically

M

Meta logical machinery

Assume we have a set of agents.

Assume we have a set of agents.

a1 a2 an

Assume we have a set of agents.

a1 a2 an

We have a set of constant symbols

Assume we have a set of agents.

a1 a2 an

We have a set of constant symbols

agents A = {a1, . . . , an}.

System C0

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

System C0

φ ::= ( p | ¬φ | φ ∨ ψ B(a, φ)

Syntax

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

System C0

φ ::= ( p | ¬φ | φ ∨ ψ B(a, φ)

Syntax

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

System C0

φ ::= ( p | ¬φ | φ ∨ ψ B(a, φ)

Syntax

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

System C0

φ ::= ( p | ¬φ | φ ∨ ψ B(a, φ)

Syntax

• ✏

_

• ✏

✏

System C0

φ ::= ( p | ¬φ | φ ∨ ψ B(a, φ)

Syntax

A structure models an agent

A structure models an agent

. . . βS

{. . .}

ΓS a1

{. . .} βS

ΓS

a2

{. . .} βS

ΓS

an

{. . .} βS

ΓS A function from agents to structures A set of pure propositional formulae

A structure has two parts

Semantics

A structure

S

Semantics Given a structure S, we define S ✏ φ below: S ✏ φ0 iff ΓS ` φ0; φ0 is purely propositional S ✏ B(a, φ) iff βS(a) ✏ φ S ✏ ¬φ iff

• S 6✏ φ
• S ✏ (φ _ ψ) iff
• S ✏ φ or S ✏ ψ
• Semantics
• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

Semantics Given a structure S, we define S ✏ φ below: S ✏ φ0 iff ΓS ` φ0; φ0 is purely propositional S ✏ B(a, φ) iff βS(a) ✏ φ S ✏ ¬φ iff

• S 6✏ φ
• S ✏ (φ _ ψ) iff
• S ✏ φ or S ✏ ψ
• Semantics
• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

Semantics Given a structure S, we define S ✏ φ below: S ✏ φ0 iff ΓS ` φ0; φ0 is purely propositional S ✏ B(a, φ) iff βS(a) ✏ φ S ✏ ¬φ iff

• S 6✏ φ
• S ✏ (φ _ ψ) iff
• S ✏ φ or S ✏ ψ
• Semantics
• ✏

_

• ✏

✏

• ✏

_

• ✏

✏

Semantics Given a structure S, we define S ✏ φ below: S ✏ φ0 iff ΓS ` φ0; φ0 is purely propositional S ✏ B(a, φ) iff βS(a) ✏ φ S ✏ ¬φ iff

• S 6✏ φ
• S ✏ (φ _ ψ) iff
• S ✏ φ or S ✏ ψ
• Semantics
• ✏

_

• ✏

✏

Semantics Given a structure S, we define S ✏ φ below: S ✏ φ0 iff ΓS ` φ0; φ0 is purely propositional S ✏ B(a, φ) iff βS(a) ✏ φ S ✏ ¬φ iff

• S 6✏ φ
• S ✏ (φ _ ψ) iff
• S ✏ φ or S ✏ ψ
• Semantics

(a) ShadowProver Input (b) ShadowProver Output

https://github.com/naveensundarg/prover Prover

Lemma 1 (Preservation of ✏ for IRES) Given Γ ✏ φ1 _ . . . χ . . . _ φn Γ ✏ ψ1 _ . . . ¬χ . . . _ ψm we have Γ ✏ φ1 _ . . . _ . . . φn _ ψ1 _ . . . _ ψm

Proof for Lemma 1

Let ρ1 ⌘ φ1 _ . . . χ . . . _ φn ρ2 ⌘ ψ1 _ . . . ¬χ . . . _ ψm ρ ⌘ φ1 _ . . . _ . . . φn _ ψ1 _ . . . _ ψm

⌘ _ _ _ _ _ Consider some S such that: S ✏ ρ1 S ✏ ρ2 We need to prove S ✏ ρ.

Two possible cases

Case 1:

Case 2:

Exactly one of S ✏ χ or S ✏ ¬χ holds.

Neither S ✏ χ nor S ✏ ¬χ hold.

Output of the inference Inputs to the inference

Case 1: Exactly one of S ✏ χ or S ✏ ¬χ holds.

Assume that S ✏ χ but S 6✏ ¬χ

φ1 ∨ . . . χ . . . ∨ φn ψ1 ∨ . . . ¬χ . . . ∨ ψm then there is some other literal u in ρ2 such that S ✏ u

By IRes , this literal is also present in ρ at the top level, thus giving us S ✏ ρ.

Case 2:

Neither S ✏ χ nor S ✏ ¬χ hold.

Then there are literals u1 and u2 such that

ρ1 ≡ (. . . u1 . . . ∨ χ . . .) ρ2 ≡ (. . . u2 . . . ∨ ¬χ . . .) such that S ✏ u1 and S ✏ u2 Since ρ ≡ (. . . ∨ u1 . . . ∨ u2 . . .) we have S ✏ ρ.

Lemma 2 (Preservation of ✏ for IB) Given Γ ✏ B(a, φ1 ∨ . . . χ . . . ∨ φn) Γ ✏ B(a, ψ1 ∨ . . . ¬χ . . . ∨ ψm) we have Γ ✏ B(a, φ1 ∨ . . . ∨ . . . φn ∨ ψ1 ∨ . . . ∨ ψm)

Proof for Lemma 2

Omitted from the slides as the proof is similar to the proof for Lemma 1

What about Intentions?

• A bit harder, but we can assume intentions have a structure

similar to beliefs but independent of beliefs.

{. . .}

A set of pure propositional formulae Structure

. . . a1 a2 an

{. . .}

βS

{. . .}

ΓI

S

ΓB

S

ιS

What about Knowledge?

• Knowledge is justified true belief
• Expand the structure just like we did for intentions but with the

condition that the known propositions should be derivable in the top level system.

What about Obligations?

• Option 1: Expand the structure just like what we did for intentions

and knowledge?

• Problem: Handling dyadic obligations O(a, b)
• Solution (sort of): Define obligation in terms of knowledge,

belief and intentions

O(a, φ, ψ) ≡ h B(a, φ) → K(a, I(a, ψ) i

Wrapping it all up

Given a set of beliefs, use the proof theoretic semantics shown here to find out maximal sets of beliefs with satisfiable models Can be implemented given a theorem prover Building an Arrow dictator for complex modalities.

Prover Discard

a1

Future

• Soundness theorems and completeness theorems for

more modal operators

• Use the deontic cognitive event calculus to prove some

versions of Arrow’s theorem.

Questions