This document presents input/output STIT logic, which is a logic of norms that uses STIT logic as its base. It defines input/output STIT logic formally and provides semantics and proof theory. It also discusses applications to normative multi-agent systems, including defining legal, moral and illegal strategies and normative Nash equilibria. The document aims to increase the expressiveness of input/output logic by building it on top of STIT logic to represent concepts like agents and abilities.

1. Input/Output STIT Logic
Xin Sun
University of Luxembourg
July 27, 2015
Xin Sun University of Luxembourg July 27, 2015 1 / 25

2. Introduction
Input/output logic is a logic of norms (legal rules).
Input/output logic uses operational semantics: a normative system is
conceived as a deductive machine.
Given factual statements as input, the normative machine produces
deontic statements as output.
N
input
facts
output
obligations
Figure : input/output logic
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3. Input/output logic uses propositional logic as its base logic.
Concepts such as agent, action and ability, which are crucial for
multi-agent systems, cannot be expressed in input/output logic.
To increase the expressive power, we build input/output logic based on
STIT logic.
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4. Overview
1 Introduction
2 Background: STIT logic
3 Input/output STIT logic
4 Application: Normative multi-agent system
5 Conclusion and future work
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5. 1 Introduction
2 Background: STIT logic
3 Input/output STIT logic
4 Application: Normative multi-agent system
5 Conclusion and future work
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6. STIT logic: Language
STIT logic is one of the most prominent accounts of agency in philosophy
of action. It is the logic of constructions of the form “agent i sees to it
that φ holds”.
Given a finite set Agent = {1, . . . , n} and a countable set P of
propositional 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 ensures
the world is among those satisfying φ”.
φ: necessary φ.
♦φ is short for ¬ ¬φ. ♦[i]φ express the idea that agent i has the
ability to ensure the world is among those satisfying φ.
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7. STIT logic: Semantics
Definition
A model is a tuple (W , Choice, V ), where
W is a nonempty set of possible worlds,
Choice : Agent → ℘(℘(W )) is a choice function,
V : P → ℘(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 = ∅;
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 .
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8. 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)
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9. 1 Introduction
2 Background: STIT logic
3 Input/output STIT logic
4 Application: Normative multi-agent system
5 Conclusion and future work
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10. Ingredients
A norm (φ, ψ) is a pair of STIT formulas, read as “given φ, ψ is
obligatory”.
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}.
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11. 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 logical
consequence.
Pass this closed set to the the normative system. The normative
system produces a set of formulas representing obligations.
Close obligations under logical consequence.
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12. Proof theory
Given a set of norms N, a derivation system of N is the smallest set of
norms 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)
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13. Example
Suppose a, b, x, y are propositional letters, i, j are agents. Let
N = {(a, [i]x), (a, [j]y), (b, x ∧ y)}. Then
O1(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 have
the 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).
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14. 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.
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15. 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(BN
Φ )). Here BN
Φ = ∞
i=0 BN
Φ,i , where
BN
Φ,0 = Cn(Φ), BN
Φ,i+1 = Cn(Φ ∪ N(BN
Φ,i )).
Theorem
ψ ∈ O2(N, {φ}) iff (φ, ψ) ∈ D2(N).
ψ ∈ O3(N, {φ}) iff (φ, ψ) ∈ D3(N).
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16. Concerning the decidability of input/output STIT logic, we study on the
following problems:
Compliance problem: given a finite set of norms N, a finite set of
formulas Φ and a formula ψ, is ψ ∈ O(N, Φ)?
Theorem
The compliance problems of O1, O2 and O3 are decidable.
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17. 1 Introduction
2 Background: STIT logic
3 Input/output STIT logic
4 Application: Normative multi-agent system
5 Conclusion and future work
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18. 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.
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19. 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 → 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 ).
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20. Definition (moral, legal and illegal strategy)
Given a normative multi-agent system (G, N, E), for each agent i, a
strategy (+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)) ∈ 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. We
assume the normative position degrades from moral to legal, then further
to illegal. The normative status of a strategy is the highest normative
position it has.
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21. 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.
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22. 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 one
which contains his strategy of higher normative status.
Definition (normative boolean game)
Given a normative multi-agent system (G, N, E) where
G = (Agent, P, π, Goal), it induces a normative boolean game
GN = (Agent, P, π, 1, . . . n) where i is the preference of i over
strategy profiles such that S i S if either
ui (S) < ui (S )
or
ui (S) = ui (S ) and the normative status of Si is higher than that of Si .
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23. Definition (normative Nash equilibrium)
Given a normative multi-agent system (G, N, E), a strategy profile S is a
normative Nash equilibrium if it is a Nash equilibrium in the normative
boolean game GN.
Theorem
Given a normative multi-agent system (G, N, E) and a strategy profile
S, determining whether S is normative Nash equilibrium is decidable.
Given a normative multi-agent system (G, N, E), determining whether
it has a normative Nash equilibrium is decidable.
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24. Conclusion
In this paper we build input/output logic base on STIT logic.
Future work
More application to normative multi-agent system?
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25. Thank you!
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