Modeling Ceteris Paribus Preferences in Formal
Concept Analysis
Sergei Obiedkov
Higher School of Economics, Moscow, Russia
Preference context
adapted from (Brafman and Domshlak 2009)
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dark interior
bright interior
Preferences
white exterior
red exterior
SUV
minivan
Cars
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c4
Preference context
adapted from (Brafman and Domshlak 2009)
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dark interior
bright interior
Preferences
white exterior
red exterior
SUV
minivan
Cars
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c4
You are buying a red car with bright interior. Will it be a
minivan or an SUV?
From preferences over objects to preferences over
descriptions
From data, derive statements like
I prefer a white car to a red car.
From preferences over objects to preferences over
descriptions
From data, derive statements like
I prefer a white car to a red car.
...and back
Use derived statements to predict preferences over new objects.
From preferences over objects to preferences over
descriptions
From data, derive statements like
I prefer a white car to a red car.
What exactly does this mean?
every white car to every red car?
most white cars to most red cars?
...and back
Use derived statements to predict preferences over new objects.
From preferences over objects to preferences over
descriptions
From data, derive statements like
I prefer a white car to a red car.
What exactly does this mean?
every white car to every red car?
most white cars to most red cars?
Ceteris paribus semantics
every white car to every red car that is otherwise similar
...and back
Use derived statements to predict preferences over new objects.
Lifting preferences to propositions
in modal preference logics (van Benthem et al. 2009)
Based on a preference relation over possible worlds:
One approach to ceteris paribus semantics
ψ is preferred to φ, Γ being equal
m
every world satisfying ψ is preferred to every world satisfying φ
that satisfies the same formulas from Γ
Lifting preferences to propositions
in modal preference logics (van Benthem et al. 2009)
Based on a preference relation over possible worlds:
One approach to ceteris paribus semantics
ψ is preferred to φ, Γ being equal
m
every world satisfying ψ is preferred to every world satisfying φ
that satisfies the same formulas from Γ
Our approach is similar, but:
φ and ψ are atomic conjunctions
Γ is a set of atomic formulas
We use formal concept analysis as a formal framework.
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dark interior
bright interior
white exterior
red exterior
SUV
minivan
Preference context
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Preference context P = (G , M, I , ≤)
(G , M, I ) is a formal context.
Preference relation ≤ is a preorder on G .
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Ceteris paribus preferences
Ceteris paribus preferences in preference logics
ψ is preferred to φ ceteris paribus with respect to a set Γ of
propositions if, for every two possible worlds w1 and w2 such
that
w1 |= φ,
w2 |= ψ,
∀γ ∈ Γ(w1 |= γ ⇐⇒ w2 |= γ),
we have
w1 ≤ w2 .
Ceteris paribus preferences
Ceteris paribus preferences in preference logics
ψ is preferred to φ ceteris paribus with respect to a set Γ of
propositions if, for every two possible worlds w1 and w2 such
that
w1 |= φ,
w2 |= ψ,
∀γ ∈ Γ(w1 |= γ ⇐⇒ w2 |= γ),
we have
w1 ≤ w2 .
P |= A 4C B
Ceteris paribus preferences in FCA
B ⊆ M is preferred to A ⊆ M ceteris paribus with respect to
C ⊆ M in P = (G , M, I , ≤) if
∀g ∈ A0 ∀h ∈ B 0 ({g }0 ∩ C = {h}0 ∩ C
⇒
g ≤ h).
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c5
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dark interior
bright interior
white exterior
red exterior
SUV
minivan
Example
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I prefer minivans to SUVs
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SUV 4∅ minivan
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×
dark interior
bright interior
white exterior
red exterior
SUV
minivan
Example
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I prefer minivans to SUVs
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×
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SUV 64∅ minivan
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c5
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dark interior
bright interior
white exterior
red exterior
SUV
minivan
Example
c5
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×
c1
c2
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×
I prefer minivans to SUVs
. . . with the same interior color.
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SUV 64∅ minivan
SUV 4{bright,dark} minivan
Semantics based on preference contexts
P |= Π
(Π is a set of preferences)
Π is sound for P
⇐⇒
∀π ∈ Π(P |= π)
Π |= A 4C B
A 4C B is a semantic consequence of Π if, for all P,
P |= Π
=⇒
P |= A 4C B.
Completeness
Π is complete for P if, for all A 4C B,
P |= A 4C B
=⇒
Π |= A 4C B.
Ceteris paribus preferences as implications
Ceteris paribus translation of P
KP∼ = (G × G , (M × {1, 2, 3}) ∪ {≤}, I∼ )
⇐⇒
⇐⇒
⇐⇒
⇐⇒
(g1 , g2 )I∼ (m, 1)
(g1 , g2 )I∼ (m, 2)
(g1 , g2 )I∼ (m, 3)
(g1 , g2 )I∼ ≤
g1 Im,
g2 Im,
{g1 }0 ∩ {m} = {g2 }0 ∩ {m},
g1 ≤ g2 .
Ceteris paribus translation for cars
m1
...
c1 , c4
c1 , c5
...
×
×
s1
r1
...
m2
×
s2
r2
×
×
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...
m3
s3
×
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r3
...
≤
×
c1
c2
c3
c4
c5
×
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×
dark interior
bright interior
white exterior
red exterior
SUV
minivan
Ceteris paribus preferences as implications
c5
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×
c1
×
×
×
c2
c3
×
×
c4
Ceteris paribus translation for cars
m1
...
c1 , c4
c1 , c5
...
×
×
s1
r1
...
m2
×
s2
r2
×
×
×
...
m3
s3
×
×
r3
...
≤
×
Ceteris paribus preferences as implications
Translation of ceteris paribus preferences
A ceteris paribus preference A 4C B is valid in a preference
context P = (G , M, I , ≤) if and only if the implication
(A × {1}) ∪ (B × {2}) ∪ (C × {3}) → {≤}
is valid in KP∼ .
Example
SUV 4{bright,dark} minivan
m
{SUV1 , minivan2 , bright3 , dark3 } → {≤}
(1)
Ceteris paribus preferences as implications
Translation of ceteris paribus preferences
A ceteris paribus preference A 4C B is valid in a preference
context P = (G , M, I , ≤) if and only if the implication
(A × {1}) ∪ (B × {2}) ∪ (C × {3}) → {≤}
(1)
is valid in KP∼ .
Proposition
The set
{A 4C B | (A × {1}) ∪ (B × {2}) ∪ (C × {3}) is minimal
w.r.t. KP∼ |= (1)}
is sound and complete for the preference context P.
Yet another application for hypergraph transversals.
Semantic consequence
Given Π:
A1 4C1 B1 ,
A2 4C2 B2 ,
...
An 4Cn Bn ,
do we necessarily have
A 4C B?
If yes, then Π |= A 4C B, where Π = {Ai 4Ci Bi | 1 ≤ i ≤ n}.
Semantic consequence
Given Π:
If P satisfies
A1 4C1 B1 ,
A2 4C2 B2 ,
...
An 4Cn Bn ,
do we necessarily have
A 4C B?
A1 4C1 B1 ,
A2 4C2 B2 ,
...
An 4Cn Bn ,
does it necessarily satisfy
A 4C B?
If yes, then Π |= A 4C B, where Π = {Ai 4Ci Bi | 1 ≤ i ≤ n}.
Inference
Problem: Check if Π |= A 4C B.
Inference
Problem: Check if Π |= A 4C B.
Solution: Translate preferences into implications and check
semantic consequence there.
Inference
Problem: Check if Π |= A 4C B.
Solution: Translate preferences into implications and check
semantic consequence there.
Pro: Fast (linear time).
Inference
Problem: Check if Π |= A 4C B.
Solution: Translate preferences into implications and check
semantic consequence there.
Pro: Fast (linear time).
Contra: Incorrect.
Example
{a 4d b,
a 4c d,
d 4c b}
|=
a 4c b,
although
{a1 b2 d3 →≤,
a1 d2 c3 →≤,
d1 b2 c3 →≤}
6|=
a1 b2 c3 →≤ .
Dependencies in the translated context
Ceteris paribus translation for cars
m1
...
c1 , c4
c1 , c5
...
×
×
s1
r1
...
m2
×
s2
r2
×
×
×
...
m3
s3
×
×
r3
In the translated context KP∼ , we have for each m
KP∼ |= {mi , mj } → {mk }
for {i, j, k} = {1, 2, 3}
...
≤
×
Dependencies in the translated context
Ceteris paribus translation for cars
m1
...
c1 , c4
c1 , c5
...
s1
r1
...
×
×
m2
×
s2
r2
×
×
×
...
m3
s3
×
×
r3
...
In the translated context KP∼ , we have for each m
KP∼ |= {mi , mj } → {mk }
KP∼
for {i, j, k} = {1, 2, 3}
|= m1 ∨ m2 ∨ m3
Theorem
Deciding whether Π |= A 4C B is a coNP-complete problem.
≤
×
Complexity of inference
Theorem
Deciding whether Π |= A 4C B is a coNP-complete problem.
Proof of membership in coNP.
To prove that Π 6|= A 4C B:
Show a certificate P = (G , M, I , ≤) such that P |= Π, but
P 6|= A 4C B.
There are g , h ∈ G violating A 4C B.
Let P2 be the restriction of P to g and h and use P2 as a
certificate.
P2 |= Π and P2 6|= A 4C B can be checked in polynomial
time.
Total indifference
Definition
A set of preferences Π over M induces total indifference if
P |= Π
⇐⇒
P = (G , M, I , G × G ) for some G .
Total indifference
Definition
A set of preferences Π over M induces total indifference if
P |= Π
⇐⇒
P = (G , M, I , G × G ) for some G .
Π indices total indifference if and only if Π |= ∅ 4∅ ∅. Thus,
Problem (Total indifference)
Given a set Π of ceteris paribus preferences over M, decide if it
induces total indifference.
is a special case of deciding whether Π |= A 4C B.
Total indifference
Proposition
Problem (Total indifference) is coNP-hard.
Proof.
We reduce the following coNP-complete problem to (Total
indifference):
Problem
Decide if a propositional formula φ in CNF is unsatisfiable.
From φ over variables M build Πφ over attributes M:
∅ 4{m} ∅ for each m ∈ M;
P 4∅ N for each clause of φ, where P is the set of all
variables that occur positively and N is the set of all
variables that occur negatively in this clause.
Example
Let φ be
(p ∨ ¬q) ∧ (¬q ∨ r ) ∧ (¬p ∨ q ∨ ¬r ).
Then Πφ over M = {p, q, r } is
p 4∅ q
r 4∅ q
q 4∅ pr
∅ 4{p} ∅
∅ 4{q} ∅
∅ 4{r } ∅
From satisfying assignment
p=1
q=r =0
we can build
P = ({g , h}, M, I , ≤) |= Π,
where
g 0 = {p}
h0 = {q, r }
g < h,
showing that Π does not induce total indifference.
Nonparameterized preferences
Preferences that hold all other things being equal:
A 4M\(A∪B) B
or A 4 B, for short.
Nonparameterized preferences
Preferences that hold all other things being equal:
A 4M\(A∪B) B
or A 4 B, for short.
Parameterized preferences are not more expressive than
nonparameterized preferences (but are exponentially more
concise):
Theorem
P |= A 4C B
if and only if
P |= A ∪ D 4 B ∪ E
for all D ⊆ M and E ⊆ M such that D ∩ C = B ∩ C ,
E ∩ C = A ∩ C , and D ∩ E ⊆ C .
Nonparameterized preferences
Example
The preference
SUV 4{bright,dark} minivan
is equivalent to
{SUV 4 minivan,
SUV, red 4 minivan,
SUV, white 4 minivan,
SUV 4 minivan, red,
SUV 4 minivan, white,
SUV, red 4 minivan, white,
SUV, white 4 minivan, red,
SUV, red, white 4 minivan,
SUV 4 minivan, red, white}.
Related approaches to preference handling
Modal preference logics
CP-nets (only nonparameterized preferences) and their
extensions
Outcomes are identified with attribute sets
⇒ This is like preference contexts with exactly 2|M| objects
⇒ Generally, PSPACE-complete
Future work
Complexity of inference under specified integrity
constraints
Methods for preference learning based on ceteris paribus
preferences
Passive and active learning