Sequence Semantics for Norms and Obligations

Sequence Semantics for
Norms and Obligations
Guido Governatori (joint work with Olivieri, Calardo, Rotolo)
DEON 2016, Bayreuth, 19 July 2016
www.data61.csiro.au

Motivation
2 | Sequence Semantics for Norms and Obligations | Guido Governatori (joint work with Olivieri, Calardo, Rotolo)

Normative Systems
A normative system is a set of norms, where the norms deﬁne:
• what are the obligations, prohibitions, permissions, . . . in the
system
• the conditions under which obligations, prohibitions,
permissions . . . are in force
• norms can be violated
• violated norms can be compensated for

Normative Systems
A normative system is a set of norms, where the norms deﬁne:
• what are the obligations, prohibitions, permissions, . . . in the
system
• the conditions under which obligations, prohibitions,
permissions . . . are in force
• norms can be violated
• violated norms can be compensated for
Most deontic logics are not able to handle properly compensatory
obligations

A Privacy Act (Governatori 2015)
Section 1: (Prohibition to collect personal medical information)
Offence: It is an offence to collect personal medical
information.
Defence: It is a defence to the prohibition of collecting
personal medical information, if an entity immediately
destroys the illegally collected personal medical
information before making any use of the personal
medical information
Section 2: An entity is permitted to collect personal medical
information if the entity acts under a Court Order
authorising the collection of personal medical information.
Section 3: (Prohibition to collect personal information) It is forbidden
to collect personal information unless an entity is
permitted to collect personal medical information.
Offence: an entity collected personal information
Defence: an entity being permitted to collect personal medical
information.

Making Sense of the Act
• Collection of medical information is forbidden.
• Destruction of the illegally collected medical information
excuses the illegal collection.
• Collection of medical information is permitted if there is an
authorising court order.
• Collection of personal information is forbidden.
• Collection of personal information is permitted if the
collection of medical information is permitted

Dilemma Structure
• b (“collection of medical information”) is forbidden
c (“destruction of medical information”) compensates the
illegal collection
• b is permitted if a (“acting under a court order”)
• d (“collection of personal information”) is forbidden
• d is permitted if b is permitted

Modelling Compensation
Contrary-to-duty obligation
Oα ¬α → Oβ
Violation triggered obligation
Oα ∧ ¬α → Oβ

Oα ¬α → Oβ
Compensation operator
a1 ⊗ a2 ⊗ · · · ⊗ an

Oα ¬α → Oβ
Compensation operator
a1 ⊗ a2 ⊗ · · · ⊗ an
ai is obligatory, but if the obligation is violated, then the fulﬁlment
of the obligation of ai+1 compensate the violation of the obligation
of ai .
TS, σ |= φ ⊗ ψ iﬀ ∀i ≥ 0, TS, σi |= φ; or
∃j, k : 0 ≤ j ≤ k, TS, σj |= ¬φ and TS, σk |= ψ.

Formalising the Dilemma
• ¬b ⊗ c
• a → Pb
• O¬d
• Pb → Pd

Formalising the Dilemma (take 2)
• ¬a → ¬b ⊗ c
• a → Pb
• O¬b → O¬d
• Pb → Pd

Formalising the Dilemma (take 2)
• ¬a → ¬b ⊗ c
• a → Pb
• O¬b → O¬d
• Pb → Pd
t0
¬a
t1
¬a, b
t3
¬a, c, d

Logic for ⊗

Language
• Any propositional letter p ∈ Prop and ⊥ are wffs;
• If a and b are wffs, then a → b is a wff;
• If a is a wff and no operator ⊗m, ⊕m, O and P occurs in a,
then Oa and Pa are a wff;
• If a1, . . . , an are wffs and no operator ⊗m, ⊕m, O and P
occurs in any of them, then a1 ⊗n · · · ⊗n an and
a1 ⊕n · · · ⊕n an are a wff, where n ∈ N+
• Nothing else is a wff.

Intuition
• ⊗-chains for prescriptive norms
• ⊕-chains for permissive norms
• O for obligations
• P for permissions

Intuition
a ⊗ b

Intuition
a ⊗ b
a ⊕ b

Intuition
a ⊗ b
a ⊕ b
⊗1
a Oa

Basic Axioms and Rules
a ≡ b
Oa ≡ Ob
O-RE
a ≡ b
Pa ≡ Pb
P-RE
n
i=1 ai ≡ bi
n
i=1 ai ≡ n
i=1 bi
⊗-RE
n
i=1 ai ≡ bi
n
i=1 ai ≡ n
i=1 bi
n
i=1
ai ≡
k−1
i=1
ai
n
i=k+1
ai ( -contraction)
aj ≡ ak, j < k

Basic Axioms and Rules
a ≡ b
Oa ≡ Ob
O-RE
a ≡ b
Pa ≡ Pb
P-RE
n
i=1 ai ≡ bi
n
i=1 ai ≡ n
i=1 bi
⊗-RE
n
i=1 ai ≡ bi
n
i=1 ai ≡ n
i=1 bi
n
i=1
ai ≡
k−1
i=1
ai
n
i=k+1
ai ( -contraction)
aj ≡ ak, j < k
a ⊗ b ⊗ a ⊗ c ≡ a ⊗ b ⊗ c

Deontic Axioms
Pa ≡ ¬O¬a (OP-duality)
Oa → Pa (O-P)
Oa → ¬O¬a (D-O)
¬O⊥ (P-O)
Oa → ¬P¬a (O¬P)
a1 ⊗ · · · ⊗ an → a1 ⊗ · · · ⊗ an−1, n ≥ 2 (⊗-shortening)
a1 ⊕ · · · ⊕ an → a1 ⊕ · · · ⊕ an−1, n ≥ 2 (⊕-shortening)

Axioms for ⊗ and O
a1 ⊗ · · · ⊗ an → Oa1 (⊗-O)
a1 ⊗ · · · ⊗ an ∧
k<n
i=1
¬ai → Oak+1 (O-detachment)
a1 ⊗ · · · ⊗ an ∧ ¬a1 → a2 ⊗ · · · ⊗ an (⊗-detachment)
a1 ⊗ · · · ⊗ an ∧
k<n
i=1
(Oai ∧ ¬ai ) → Oak+1
(O-violation-detachment)
a1 ⊗ · · · ⊗ an ∧ Oa1 ∧ ¬a1 → a2 ⊗ · · · ⊗ an
(⊗-violation-detachment)

Axioms for ⊗, ⊕, O, P
a1 ⊕ · · · ⊕ an → Pa1 (⊕-P)
a1 ⊕ · · · ⊕ an ∧
k<n
i=1
¬Pai → Pak+1 (P-detachment)
a1 ⊕ · · · ⊕ an ∧ ¬Pa1 → a2 ⊕ · · · ⊕ an (⊕-detachment)

Summary of Logics
Basic Systems
E⊗ CPC + O-RE + ⊗-RE + ⊗-contraction
E⊕ CPC + P-RE + ⊕-RE + ⊕-contraction
E⊗⊕ E⊗ + E⊕
Basic Deontic Systems
D⊗ E⊗ + OP-duality + O-P + P-O + ⊗-shortening
D⊗⊕ E⊗⊕ + O-P + P-O + D-O + O¬P + ⊗-shortening +
⊕-shortening
DO⊗ D⊗ + ⊗-O
Basic Full Deontic System
DOP⊗⊕ D⊗⊕ + ⊗-O
Plus combinations of detachment axioms

Sequence Semantics

Idea
Generalisation of neighbourhood Semantics:
• A proposition can be represented as a set of possible worlds
• Neighbourhood structure for O (and eventually P), i.e., set of
sets of possible worlds;
• Extended neighbourhood structure for ⊗ (and evantually ⊕),
i.e., set of sequences of sets of possible worlds.

(Bi)Sequence Semantics
A bi-sequence frame is a structure F = W , CO, CP, NO, NP ,
where
• W is a non empty set of possible worlds;
• CO and CP are two functions with signature W → 2(2W )n
,
such that for every world w ∈ W , for every X ∈ CO
w and
Y ∈ CP
w , X and Y are closed under s-zipping;
• NO and NP are two functions with signature W → 22W
.

Valuations
• usual for atoms and boolean conditions,
• w |= a1 ⊗ · · · ⊗ an iff a1 V , . . . , an V ∈ CO
w ,
• w |= a1 ⊕ · · · ⊕ an iff a1 V , . . . , an V ∈ CP
w ,
• w |= Oa iff a V ∈ NO
w ,
• w |= Pa iff a V ∈ NP
w .

Completeness Results I
Theorem
E⊗, E⊕ and E⊗⊕ are sound and complete w.r.t. the class of (bi)sequence frames.
Theorem
D⊗⊕ is sound and complete w.r.t. the class of bi-sequence frames such that:
1. NP
w ⊇ NO
w (see O-P)
2. X ∈ NO
w implies −X ∈ NO
w (see D-O)
3. ∅ ∈ NO
w (see P-O)
4. X ∈ NO
w implies −X ∈ NP
w (see O¬P)
5. X1, . . . , Xn ∈ CO
w for n ≥ 2 then X1, . . . , Xn−1 ∈ CO
w (see ⊗-shortening)
6. X1, . . . , Xn ∈ CP
w for n ≥ 2 then X1, . . . , Xn−1 ∈ CP
w (see ⊕-shortening)
Theorem
DO⊗ is sound and complete w.r.t. the class of D⊗⊕ frames such that
X1, . . . , Xn ∈ CO
w then X1 ∈ NO
w

Completeness Results II
Theorem
Let S be a system such that S ⊇ DOP⊗⊕. If S contains any of the axioms listed below,
the canonical frame enjoys the corresponding property: For any world w
1. O-detachment:
If X1, . . . , Xn ∈ CO
w and w ∈ Xi for 1 ≤ i ≤ k and k < n, then Xk+1 ∈ NO
w .
2. ⊗-detachment:
If X1, . . . , Xn ∈ CO
w and w ∈ X1, then X2, . . . , Xn ∈ CO
w .
3. O-violation-detachment:
If X1, . . . , Xn ∈ CO
w and, for 1 ≤ i ≤ k and k < n, w ∈ Xi and Xi ∈ NO
w , then
Xk+1 ∈ NO
w .
4. ⊗-violation-detachment:
If X1, . . . , Xn ∈ CO
w and X1 ∈ NO
w and w ∈ X1, then X2, . . . , Xn ∈ CO
w .
5. ⊕-P:
If X1, . . . , Xn ∈ CP
w then X1 ∈ NP
w .
6. P-detachment:
If X1, . . . , Xn ∈ CP
w and Xi ∈ NP
w for 1 ≤ i ≤ k < n, then Xk+1 ∈ NP
w .
7. ⊕-detachment:
If X1, . . . , Xn ∈ CP
w and Xi ∈ NP
w for 1 ≤ i ≤ k and k < n, then
Xk+1, . . . , Xn ∈ CP
w .

Conclusions and Future Work
• novel semantics (generalising neighbourhood semantics)
• decidability and complexity results
• what are the counterparts of axioms like M, C, . . .

Questions?
guido.governatori@data61.csiro.au

Sequence Semantics for Norms and Obligations

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