Input/Output STIT Logic

Xin SunUniversity of Luxembourg

July 27, 2015

Xin Sun University of Luxembourg July 27, 2015 1 / 25

Introduction

Input/output logic is a logic of norms (legal rules).

Input/output logic uses operational semantics: a normative system isconceived as a deductive machine.

Given factual statements as input, the normative machine producesdeontic statements as output.

Ninputfacts

outputobligations

Figure : input/output logic

Xin Sun University of Luxembourg July 27, 2015 2 / 25

Input/output logic uses propositional logic as its base logic.

Concepts such as agent, action and ability, which are crucial formulti-agent systems, cannot be expressed in input/output logic.

To increase the expressive power, we build input/output logic based onSTIT logic.

Xin Sun University of Luxembourg July 27, 2015 3 / 25

Overview

1 Introduction

2 Background: STIT logic

3 Input/output STIT logic

4 Application: Normative multi-agent system

5 Conclusion and future work

Xin Sun University of Luxembourg July 27, 2015 4 / 25

1 Introduction

2 Background: STIT logic

3 Input/output STIT logic

4 Application: Normative multi-agent system

5 Conclusion and future work

Xin Sun University of Luxembourg July 27, 2015 5 / 25

STIT logic: Language

STIT logic is one of the most prominent accounts of agency in philosophyof action. It is the logic of constructions of the form “agent i sees to itthat φ holds”.

Given a finite set Agent = {1, . . . , n} and a countable set P ofpropositional letters.Let i ∈ Agent be an agent. The language of STIT logic L:

φ, ψ ::= p | ¬φ | φ ∧ ψ | [i ]φ | �φ

[i ]φ: i sees to it that φ. It can be viewed as action “agent i ensuresthe world is among those satisfying φ”.

�φ: necessary φ.

♦φ is short for ¬�¬φ. ♦[i ]φ express the idea that agent i has theability to ensure the world is among those satisfying φ.

Xin Sun University of Luxembourg July 27, 2015 6 / 25

STIT logic: Semantics

Definition

A model is a tuple (W ,Choice,V ), where

W is a nonempty set of possible worlds,

Choice : Agent 7→ ℘(℘(W )) is a choice function,

V : P 7→ ℘(W ) is the truth valuation for propositional letters.

Choice is required to satisfy some conditions:

(1) for each i ∈ Agent, Choice(i) is a partition of W ;

(2) for every x1 ∈ Choice(1), . . . , xn ∈ Choice(n), x1 ∩ . . . ∩ xn 6= ∅;

Let (w ,w ′) ∈ Ri iff there is K ∈ Choice(i) such that {w ,w ′} ⊆ K .

M,w |= �ϕ iff M,w ′ |= ϕ for all w ′ ∈W .

M,w |= [i ]ϕ iff M,w ′ |= ϕ for all w ′ such that (w ,w ′) ∈ Ri .

Xin Sun University of Luxembourg July 27, 2015 7 / 25

STIT logic: example

Example

Agent={1, 2}, M = (W ,Choice,V ),W = {w1, . . . ,w7}, Choice(1) = {{w1,w2,w3}, {w4,w5,w6,w7}},Choice(2) = {{w1,w4}, {w2,w5,w6}, {w3,w7}}, V (p) = {w1,w4},V (q) = {w2,w3,w7}.

w1 : p w2 : q w3 : q

w4 : p w5,w6 w7 : q

M,w1 |= [2]p.

M,w1 |= ¬[1]p.

M,w5 |= ♦[1](p ∨ q)

Xin Sun University of Luxembourg July 27, 2015 8 / 25

1 Introduction

2 Background: STIT logic

3 Input/output STIT logic

4 Application: Normative multi-agent system

5 Conclusion and future work

Xin Sun University of Luxembourg July 27, 2015 9 / 25

Ingredients

A norm (φ, ψ) is a pair of STIT formulas, read as “given φ, ψ isobligatory”.

A normative system N is a set of norms.

N is viewed as a function such that for a set of formulas Φ,N(Φ) = {ψ ∈ L | (φ, ψ) ∈ N for some φ ∈ Φ}.

Xin Sun University of Luxembourg July 27, 2015 10 / 25

Semantics

Let Cn(Φ) = {φ ∈ L : Φ |= φ}.

Definition

Given a set of norms N and a set of formulas Φ,O1(N,Φ) = Cn(N(Cn(Φ))).

Intuition

Take a set of formulas representing facts and close it under logicalconsequence.

Pass this closed set to the the normative system. The normativesystem produces a set of formulas representing obligations.

Close obligations under logical consequence.

Xin Sun University of Luxembourg July 27, 2015 11 / 25

Proof theory

Given a set of norms N, a derivation system of N is the smallest set ofnorms which extends N and is closed under certain derivation rules.

D1(N) is the derivation system decided by the rules SI, WO and AND.

SI (strengthening the input): from (φ1, ψ) to (φ2, ψ) whenever|= φ2 → φ1

WO (weakening the output): from (φ, ψ1) to (φ, ψ2) whenever|= ψ1 → ψ2

AND (conjunction of the output): from (φ, ψ1) and (φ, ψ2) to(φ, ψ1 ∧ ψ2)

Xin Sun University of Luxembourg July 27, 2015 12 / 25

Example

Suppose a, b, x , y are propositional letters, i , j are agents. LetN = {(a, [i ]x), (a, [j ]y), (b, x ∧ y)}. ThenO1(N, {a}) = Cn(N(Cn({a}))) = Cn({[i ]x , [j ]y}). �

Example

Let N = {(a ∨ b, [j ]x)}, then ([i ]b, [j ](x ∨ y)) ∈ D1(N) because we havethe following derivation

1 (a ∨ b, [j ]x) Assumption

2 ([i ]b, [j ]x) 1, SI

3 ([i ]b, [j ](x ∨ y)) 2, WO

Theorem

ψ ∈ O1(N, {φ}) iff (φ, ψ) ∈ D1(N).

Xin Sun University of Luxembourg July 27, 2015 13 / 25

Other input/output STIT logics

More derivation rules

OR (disjunction of the input): from (φ1, ψ) and (φ2, ψ) to(φ1 ∨ φ2, ψ)

CT (cumulative transitivity): from (φ, ψ1) and (φ ∧ ψ1, ψ2) to (φ, ψ2)

Adding OR to D1(N) gives D2(N).D2(N): SI, WO, AND, OR.

Adding CT to D1(N) gives D3(N).D3(N). SI, WO, AND, CT.

Xin Sun University of Luxembourg July 27, 2015 14 / 25

O2(N,Φ) =⋂{Cn(N(Ψ)) : Φ ⊆ Ψ = Cn(Ψ),Ψ is disjunctive}. A set

Ψ is disjunctive if for all φ ∨ ψ ∈ Ψ, either φ ∈ Ψ or ψ ∈ Ψ.

O3(N,Φ) = Cn(N(BNΦ )). Here BN

Φ =⋃∞

i=0 BNΦ,i , where

BNΦ,0 = Cn(Φ),BN

Φ,i+1 = Cn(Φ ∪ N(BNΦ,i )).

Theorem

ψ ∈ O2(N, {φ}) iff (φ, ψ) ∈ D2(N).ψ ∈ O3(N, {φ}) iff (φ, ψ) ∈ D3(N).

Xin Sun University of Luxembourg July 27, 2015 15 / 25

Concerning the decidability of input/output STIT logic, we study on thefollowing problems:

Compliance problem: given a finite set of norms N, a finite set offormulas Φ and a formula ψ, is ψ ∈ O(N,Φ)?

Theorem

The compliance problems of O1,O2 and O3 are decidable.

Xin Sun University of Luxembourg July 27, 2015 16 / 25

1 Introduction

2 Background: STIT logic

3 Input/output STIT logic

4 Application: Normative multi-agent system

5 Conclusion and future work

Xin Sun University of Luxembourg July 27, 2015 17 / 25

Normative multi-agent system

In a normative multi-agent system, agents behavior are regulated by norms.

Definition (Normative multi-agent system)

A normative multi-agent system is a triple NorMAS = (G ,N,E ) where

G is a boolean game.

N is a finite set of norms.

E ⊆ L is a finite set of formulas representing the environment.

Xin Sun University of Luxembourg July 27, 2015 18 / 25

Each agent i has a goal, represented by a propositional formula φi .

Each agent i has a set of propositional letter Pi he can control.

A strategy of i is a valuation over Pi .

Definition (boolean game)

A boolean game is a 4-tuple (Agent,P, π,Goal), where

1 Agent = {1, . . . , n} is the of agents.

2 P is a finite set of propositional letters.

3 π : Agent 7→ 2P is a control assignment function such that{π(1), . . . , π(n)} forms a partition of P.

4 Goal = 〈φ1, . . . , φn〉 is a set of goals for each agent.

Agents’ utilities are induced by their goals. For every strategy profiles S ,ui (S) = 1 if S � φi . Otherwise ui (S) = 0.Agent’s preference over strategy profile: S ≤i S

′ iff ui (S) ≤ ui (S′).

Xin Sun University of Luxembourg July 27, 2015 19 / 25

Definition (moral, legal and illegal strategy)

Given a normative multi-agent system (G ,N,E ), for each agent i , astrategy (+p1, . . . ,+pm,−q1, . . . ,−qn) is moral if

[i ](p1 ∧ . . . ∧ pm ∧ ¬q1 ∧ . . . ∧ ¬qn) ∈ O(N,E ).

The strategy is legal if

[i ](¬(p1 ∧ . . . ∧ pm ∧ ¬q1 ∧ . . . ∧ ¬qn)) 6∈ O(N,E ).

The strategy is illegal if

[i ](¬(p1 ∧ . . . ∧ pm ∧ ¬q1 ∧ . . . ∧ ¬qn)) ∈ O(N,E ).

Moral, legal, and illegal are the three normative positions of strategies. Weassume the normative position degrades from moral to legal, then furtherto illegal. The normative status of a strategy is the highest normativeposition it has.

Xin Sun University of Luxembourg July 27, 2015 20 / 25

Example

Let (G ,N,E ) be a normative multi-agent system as following:

G = (Agent,P, π,Goal) is a boolean game with

Agent = {1, 2},P = {p, q},π(1) = {p}, π(2) = {q},Goal1 = 〈p ∧ q, p ∨ q〉.

N = {(>, [1]p)}.E = ∅.

+q −p+p (1, 1) (0, 1)

−p (0, 1) (0, 0)

Then out(N,E ) = Cn({[1]p}). Therefore normative status of+p,+q,−q,−p is respectively moral, legal, legal and illegal. a

Xin Sun University of Luxembourg July 27, 2015 21 / 25

Preference refinement

Agent’s preference is changed by the normative status of strategies:

1 an agent prefers strategy profiles with higher utility.

2 for two strategy profiles of the same utility, the agent prefers the onewhich contains his strategy of higher normative status.

Definition (normative boolean game)

Given a normative multi-agent system (G ,N,E ) whereG = (Agent,P, π,Goal), it induces a normative boolean gameGN = (Agent,P, π,≺1, . . . ≺n) where ≺i is the preference of i overstrategy profiles such that S ≺i S

′ if either

ui (S) < ui (S′)

or

ui (S) = ui (S′) and the normative status of S ′

i is higher than that of Si .

Xin Sun University of Luxembourg July 27, 2015 22 / 25

Definition (normative Nash equilibrium)

Given a normative multi-agent system (G ,N,E ), a strategy profile S is anormative Nash equilibrium if it is a Nash equilibrium in the normativeboolean game GN .

Theorem

Given a normative multi-agent system (G ,N,E ) and a strategy profileS , determining whether S is normative Nash equilibrium is decidable.

Given a normative multi-agent system (G ,N,E ), determining whetherit has a normative Nash equilibrium is decidable.

Xin Sun University of Luxembourg July 27, 2015 23 / 25

Conclusion

In this paper we build input/output logic base on STIT logic.

Future work

More application to normative multi-agent system?

Xin Sun University of Luxembourg July 27, 2015 24 / 25

Thank you!

Xin Sun University of Luxembourg July 27, 2015 25 / 25