=Paper=
{{Paper
|id=Vol-1624/paper14
|storemode=property
|title=Generalized Metrics and Their Relevance for FCA and Closure Operators
|pdfUrl=https://ceur-ws.org/Vol-1624/paper14.pdf
|volume=Vol-1624
|authors=Tobias Gäbel-Hökenschnieder,Thorsten Pfeiffer,Mosadak Al Salamat,Stefan E. Schmidt
|dblpUrl=https://dblp.org/rec/conf/cla/Gabel-Hokenschnieder16
}}
==Generalized Metrics and Their Relevance for FCA and Closure Operators==
https://ceur-ws.org/Vol-1624/paper14.pdf
Generalized Metrics and Their Relevance for
FCA and Closure Operators
Tobias Gäbel-Hökenschnieder1 , Thorsten Pfeiffer2 , Mosadak Al Salamat1 , and
Stefan E. Schmidt1
1
Institut für Algebra, Technische Universität Dresden,
2
FINMA, Bern, Schweiz
tgh@mail.de, thorsten pfeiffer@yahoo.com, mosadak@gmx.de, midt1@msn.com
Abstract. We provide an approach to generalized metrics that covers
various concepts of distance. In particular, we consider functorial maps
which are weakly positive. Here, we focus on the supermodular case
which generalizes dimension functions. We give a lattice-theoretically
based construction for supermodular functorial maps, which generalize
those arising from Dempster-Shafer-Theory. Within this framework, gen-
eralized metrics relevant for FCA and closure operators are discussed.
Keywords: Generalized metric, supermodular, formal concept analysis,
Dempster-Shafer-Theory, closure operators
1 Introduction
Generalized metrics recently have become of increased interest for modelling a
concept of directed distances with values in a qualitative measurement space. In
particular, they allow to distinguish between deletion and error within the con-
text of transferred information. We propose a general modeling including lattices
and ordered monoids. Here, our goal is to construct generalized metrics relevant
for FCA and closure operators [3, 8, 7]. For our approach it turns out that su-
permodularity plays an important role, which goes beyond ideas of measurement
accociated with Dempster-Shafer-Theory [8].
Our modeling of generalized metrics can be very helpful to improve and better
understand the mapping of ratings, i. e. compare the rating methodologies of
different rating agencies with different result scales.
2 Motivation
Before we present generalized metrics in an abstract setting, we want to discuss a
motivating special situation where we collect properties relevant for our general
approach.
We start with a lattice L = (L, ≤L ). Then, we consider the ordered monoid
M = (M, ∪, ∅, ⊆) with M = 2L . Furthermore, we set
↓ x := {t ∈ L | t ≤L x}
and define the maps
λ : L −→ M : x 7→↓ x
and
Dλ : ≤L −→ M : (x, y) 7→↓ y − ↓ x. (1)
We can easily see that Dλ (x, y) is equal to the set {t ∈ L | t ≤L y and t L x},
where x, y ∈ L and x ≤L y.
Observation Obviously, Dλ fulfils the following properties:
– Dλ (x, x) = ∅ holds for all x ∈ L, since Dλ (x, x) =↓ x − ↓ x = ∅.
– Dλ (x, y) ∪ Dλ (y, z) = Dλ (x, z) holds for all x, y, z ∈ L with x ≤L y ≤L z,
since
Dλ (x, y) ∪ Dλ (y, z) = (↓ y − ↓ x) ∪ (↓ y − ↓ z)
= ↓z−↓x
= Dλ (x, z).
Satisfying these conditions, Dλ will be called functorial (see definition 2).
Now we want to put the previous considerations in a slightly more general setting.
As above, the starting point is a lattice L = (L, ≤L ). For a given subset A of L
we consider the ordered monoid M = (M, ∪, ∅, ⊆) with M = 2A . Then, for all
x ∈ L we define
Ax := {a ∈ A | a ≤L x}.
Based on this setup, we consider the following maps:
λ : L −→ M : x 7→ Ax,
Dλ : ≤L −→ M : (x, y) 7→ Ay − Ax. (2)
Claim 1 Dλ is functorial w. r. t. (L, M).
Proof. – Firstly, Dλ (x, x) = ∅ obviously holds for all x ∈ L.
– Secondly, we have to show that Dλ (x, y) ∪ Dλ (y, z) = Dλ (x, z) holds for all
x, y, z ∈ L with x ≤L y ≤L z:
Let x, y, z be elements in L such that x ≤L y ≤L z. For all a ∈ A, a ∈ Dλ (x, z)
is equivalent to
a L x and a ≤L z.
Let a ∈ Dλ (x, z). We distinguish two situations:
Case 1: a ≤L y Then, a L x and a ≤L y, that is a ∈ Dλ (x, y).
Case 2: a L y Then, a L y and a ≤L z, that is a ∈ Dλ (y, z).
Hence, a ∈ Dλ (x, y) ∪ Dλ (y, z).
On the other hand, assume a ∈ Dλ (x, y) ∪ Dλ (y, z). Hence, a L x and
a ≤L y, or a L y and a ≤L z. Then, a L x (since x ≤L y) and a ≤L z
(since y ≤L z) which yields a ∈ Dλ (x, z). 2
In (2), we introduced Dλ as a function with domain ≤L . Next, we want to look
for an extension of Dλ onto L × L. We achieve this by the following map:
dλ : L × L −→ M : (x, y) 7→ Dλ (x ∧ y, y) (3)
Claim 2 The map dλ is a generalized quasi metric (GQM) w. r. t. (L, M), that
is, the subsequent conditions are satisfied:
(A0) for all x, y ∈ L : ∅ ⊆ dλ (x, y),
(A1) for all x ∈ L : dλ (x, x) = ∅,
(A2) for all x, y, z ∈ L : dλ (x, z) ⊆ dλ (x, y) ∪ dλ (y, z).
We remind the reader that A is called join-dense in L if for all x, y ∈ L with
x L y there exists a ∈ A such that a L y and a ≤L x.
Claim 3 Let A be join-dense in L. Then dλ is a generalized metric (GM) w. r.
t. (L, M), that is, dλ is a GQM which additionally satisfies:
(A3) For all x, y ∈ L : dλ (x, y) = ∅ = dλ (y, x) =⇒ x = y.
A more general definition for the underlying concepts will be given in definition
3.
Proof. (A0) Obviously, for all x, y ∈ L, the condition ∅ ⊆ dλ (x, y) is
satisfied.
(A1) Clear, since for all x ∈ L : dλ (x, x) = ∅.
(A2) We have to show that dλ (x, z) ⊆ dλ (x, y) ∪ dλ (y, z) holds for all
x, y, z ∈ L. This is equivalent to
Dλ (x ∧ z, z) ⊆ Dλ (x ∧ y, y) ∪ Dλ (y ∧ z, z).
To do so, let a ∈ Dλ (x ∧ z, z). Hence, a L x ∧ z and a ≤L z,
which implies
a L x and a ≤L z.
We have to examine two cases:
Case 1: a ≤L y Hence, a L x ∧ y and a ≤L y. It follows
a ∈ Dλ (x ∧ y, y).
Case 2: a L y Hence, a L y ∧ z and a ≤L z. It follows
a ∈ Dλ (y ∧ z, z).
All in all, also (A2) is satisfied. Consequently, dλ is a GQM.
(A3) Let x, y ∈ L. We suppose dλ (x, y) = ∅. This is equivalent to
Ay − A(x ∧ y) = ∅
⇐⇒ Ay = A(x ∧ y).
Taking advantage of the precondition dλ (x, y) = ∅ = dλ (y, x),
we follow that
Ay = A(x ∧ y) = A(y ∧ x) = Ax.
Hence, y = x, as A is join-dense.
All in all, dλ is a GM w. r. t. (L, M). 2
Claim 4 The map Dλ is supermodular w. r. t. (L, M) [10], that is, for all
x, y ∈ L, the following condition holds:
(A4) Dλ (x ∧ y, y) ⊆ Dλ (x, x ∨ y)
Proof. Let a ∈ Dλ (x ∧ y, y). Since Dλ (x ∧ y, y) equals Ay − A(x ∧ y), we know
that
a ∈ Ay and a ∈ / A(x ∧ y). (4)
According to the definition of A, we obtain a ≤ y and a x∧y. Hence, a ≤ x∨y.
Suppose a ≤ x. As a ≤ y, it follows a ≤ x ∧ y which is a contradiction to (4).
Therefore, a ∈ A(x ∨ y) − Ax = Dλ (x, x ∨ y). 2
3 Abstract Approach
We want to put our recent examinations from the special case into a more general
setting. For that, we start with some necessary definitions [1, 3, 10].
Definition 1 M = (M, ∗, ε, ≤) is an ordered monoid if M := (M, ∗, ε) is a
monoid and (M, ≤) is a poset such that a ≤ b implies c∗a ≤ c∗b and a∗c ≤ b∗c,
for all a, b, c ∈ M .
Definition 2 Let P = (P, ≤P ) be a poset and M = (M, ∗, ε, ≤) be an ordered
monoid. A function
∆ : ≤P −→ M
is called functorial w. r. t. (P, M), if
– for all p ∈ P : ∆(p, p) = ε,
– for all p, t, q ∈ P with p ≤P t ≤P q : ∆(p, t) ∗ ∆(t, q) = ∆(p, q).
Furthermore, ∆ is called weakly positive, if ε ≤ ∆(p, q) for all (p, q) ∈ ≤P .
∆ is called supermodular w. r. t. (P, M), if ∆(p ∧ q, q) ≤ ∆(p, p ∨ q) holds for
all (p, q) ∈ ≤P .
Furthermore, ∆ is called submodular w. r. t. (P, M), if ∆(p∧q, q) ≥ ∆(p, p∨q)
holds for all (p, q) ∈ ≤P .
Definition 3 Let P be a set, and M = (M, ∗, ε, ≤) be an ordered monoid. A
function d : P × P −→ M is called generalized quasi-metric (GQM) w. r.
t. (P, M), if
(A0) for all (p, q) ∈ ≤P : ε ≤ d(p, q)
(A1) for all p ∈ P : d(p, p) = ε
(A2) for all p, t, q ∈ P : d(p, t) ∗ d(t, q) ≤ d(p, q)
If in addition, (A3) holds, d is a generalized metric (GM) w. r. t. (P, M):
(A3) for all (p, q) ∈ P × P : d(p, q) = ε = d(q, p) =⇒ p=q
For a given ∆ : ≤P −→ M , does there exist a generalized quasi-metric
d : P × P −→ M w. r. t. (P, M) which extends ∆ such that d|≤P = ∆?
Theorem 1 Let P = (P, ≤P ) be a lattice. If a map ∆ : ≤P −→ M is weakly
positive, supermodular and functorial w. r. t. (P, M), then
d : P × P −→ M, (p, q) 7→ ∆(p ∧ q, q)
is a GQM w. r. t. (P, M).
Proof. Obviously, conditions (A0) and (A1) from definition 3 hold for d.
For (A2), we have to show that d(p, q) ≤P d(p, t) ∗ d(t, q) holds for all p, t, q ∈ P .
According to the definition of d this means
∆(p ∧ q, q) ≤P ∆(p ∧ t, t) ∗ ∆(t ∧ q, q).
We will prove this inequality immediately in Claim 3 below. However, first of
all, we need to show two properties in preparation for that.
Claim 1 (Interval Property)
t ≤P x ≤P y ≤P z =⇒ ∆(x, y) ≤ ∆(t, z).
z
Proof. Since ∆ is functorial, we obtain
y
∆(t, z) = ∆(t, x) ∗ ∆(x, y) ∗ ∆(y, z).
x
As ∆ is weakly positive, we get
t
∆(x, y) = ε ∗ ∆(x, y) ∗ ε ≤ ∆(t, z).
Fig. 1
3
Claim 2 (Meet Property) y z
x ≤P y =⇒ ∆(x ∧ z, y ∧ z) ≤ ∆(x, y). ∆(x, y)
x y∧z
Proof. To show this implication, we rewrite the
right hand side: ∆(x ∧ z, y ∧ z)
x∧z
∆(x ∧ z, y ∧ z) = ∆(x ∧ (y ∧ z), y ∧ z)
Fig. 2
We continue denoting y ∧ z by y 0 and derive
∆(x ∧ (y ∧ z), y ∧ z) = ∆(x ∧ y 0 , y 0 )
≤ ∆(x, x ∨ y 0 ),
y
due to supermodularity of ∆. We know that x ∨ y0 z
x ∨ y 0 ≤P y, since x ≤P y and y 0 ≤P y. Hence,
∆(x, y)
with Claim 1, we get
x y0
∆(x ∧ z, y ∧ z) ≤ ∆(x, y).
∆(x ∧ z, y 0 )
3 x∧z
Fig. 3
Claim 3
∆(p ∧ q, q) ≤ ∆(p ∧ t, t) ∗ ∆(t ∧ q, q).
p t q
p∧t p∧q t∧q
p∧t∧q
Fig. 4
Proof. Taking advantage of Claim 1, we know that
∆(p ∧ q, q) ≤ ∆(p ∧ t ∧ q, q).
Since ∆ is functorial, it follows
∆(p ∧ t ∧ q, q) = ∆(p ∧ t ∧ q, t ∧ q) ∗ ∆(t ∧ q, q).
With Claim 2, we finally receive
∆(p ∧ t ∧ q, t ∧ q) ∗ ∆(t ∧ q, q) ≤ ∆(p ∧ t, t) ∗ ∆(t ∧ q, q).
Therefore, ∆(p ∧ q, q) ≤ ∆(p ∧ t, t) ∗ ∆(t ∧ q, q). 2
The latter theorem can be applied to various concepts of distance between ob-
jects of a given lattice. In the following, we will study some interesting applica-
tions in different context, starting with formal concept analysis.
4 Application to FCA
Let K = (G, M, I) be a finite formal context. Then the set of formal concepts of
K is given by
BK := {(X, Y ) ∈ 2G × 2M | X 0 = Y and , Y 0 = X}
and the formal concept lattice of K is defined as
BK := (BK, ≤BK )
with c1 ≤BK c2 iff A1 ⊆ A2 holds for all c1 = (A1 , B1 ), c2 = (A2 , B2 ) ∈ BK.
Remarkably, the map
dext : BK × BK −→ N such that (c1 , c2 ) 7→ #(A2 − A1 )
is a GM w. r. t. (BK, M) with M := (N, +, 0, ≤). The reason for this is based
in Theorem 1, as we will outline below.
For all c1 = (A1 , B1 ), c2 = (A2 , B2 ) ∈ BK it follows
dext (c1 , c2 ) = #A2 − #(A1 ∩ A2 ),
since A2 − A1 = A2 − (A1 ∩ A2 ) and # is the counting measure.
To verify that dext is a GM, we define
Dext : ≤BK −→ N such that(c1 , c2 ) 7→ #A2 − #A1 .
Claim 4 Dext is functorial w. r. t. (BK, M), weakly positive and supermodular.
Proof. The properties of being weakly positive and functorial are clear due to
the definition of Dext via the counting measure. Let us have a closer look at the
supermodularity:
Let c1 , c2 ∈ BK. We have to show that
!
Dext (c1 ∧ c2 , c2 ) ≤ Dext (c1 , c1 ∨ c2 ).
Transforming the left hand side, we obtain
Dext (c1 ∧ c2 , c2 ) = Dext A1 ∩ A2 , (A1 ∩ A2 )0 , A2 , B2
= #A2 − #(A1 ∩ A2 )
= dext (c1 , c2 ).
On the right hand side, we get
A1 , A2 , (B1 ∩ B2 )0 , B1 ∩ B2
Dext (c1 , c1 ∨ c2 ) = Dext
| {z }
=(A1 ∪A2 )00
00
= # (A1 ∪ A2 ) − #A1
≥ #(A1 ∪ A2 ) − #A1
= #A2 − #(A1 ∩ A2 )
= dext (c1 , c2 ).
Hence,
Dext (c1 ∧ c2 , c2 ) ≤ Dext (c1 , c1 ∨ c2 )
and the supermodularity is shown. 2
Obviously, by theorem 1 together with claim 4 it immediately follows that dext
is a GM w. r. t. (BK, M).
Remark. In analogy to the above, the map
dint : BK × BK −→ N such that (c1 , c2 ) 7→ #(B1 − B2 )
is a GM w. r. t. (BK, M).
5 Application to Dempster-Shafer-Theory
Choosing L := 2U and A := L, where U is a finite set, allows us a link to
Dempster-Shafer-Theory.
Let m be a mass function on L, that is
X
m : L −→ R≥0 : X 7→ mX is a map such that m(∅) = 0 and mX = 1.
X∈L
We define X
Belm : L −→ R≥0 : X 7→ mT
T ⊆X
as the so-called belief map w. r. t. m and
X
Plm : L −→ R≥0 : X 7→ mT
T ∈L:T ∩X6=∅
as the so-called plausibility map w. r. t. m.
Obviously, for all X ∈ Belm , the equation Belm X + Plm (U − X) = 1 holds. That
is:
1 − Plm (U − X) = Belm X
1 − Belm X = Plm (U − X). (5)
Claim 5 Let ∆ be the function which maps every pair (X, Y ) with X ⊆ Y ⊆ U
to
∆(X, Y ) := Belm Y − Belm X. (6)
∆ is functorial, weakly positive, and supermodular w. r. t. the power set lattice
of U into the naturally ordered additive monoid of non-negative real numbers.
Applying this claim to theorem 1, we receive that
d : L × L −→ R≥0 , (X, Y ) 7→ ∆(X ∩ Y, Y ) = Belm Y − Belm (X ∩ Y )
is a GM w. r. t. the power set of U into the naturally ordered additive monoid
of non-negative real numbers.
Remark. With the plausibility map introduced above, a submodular pendant
to the supermodular map ∆ in (6) can be constructed via
∆(X,
e Y ) := Plm Y − Plm X where X ⊆ Y ⊆ U.
∆
e is indeed submodular, as the following inequation holds:
Plm (X ∪ Y ) + Plm (X ∩ Y ) ≤ Plm X + Plm Y.
This can be shown by using the equality
Plm X = 1 − Belm (U − X)
that we have already observed in (5).
e induces a GM d˜ via
Remark. Dually to theorem 1, ∆
d˜: L × L −→ R≥0 , (X, Y ) 7→ ∆(X,
e X ∪ Y ) = Plm (X ∪ Y ) − Plm X.
6 A Fundamental Construction of Generalized Quasi
Metrics
Let P = (P, ≤P ) be a poset, M = (M, ∗, ε) be a monoid, and ∗ : P × M −→ P
be a map such that the following properties are satisfied:
1 For all p ∈ P and all x, y ∈ M : p ∗ (x ∗ y) = (p ∗ x) ∗ x
2 For all p ∈ P : p∗ε=p
3 For all p, y ∈ P, x ∈ M : p ≤P q =⇒ p ∗ x ≤P q ∗ x
Then we call the triple (P, M, ∗) a poset right monoid action.
In this setup, we consider the map ∇ : P × P −→ M defined by
∇(p, q) := {x ∈ M | q ≤P p ∗ x} for all p, q ∈ P.
Claim 6 For all p, q, r ∈ P the following reverse triangle inequality holds:
∇(p, q) ∗ ∇(q, r) ⊆ ∇(p, r)
Proof. We choose z ∈ ∇(p, q) ∗ ∇(q, r). That is, there exits x ∈ ∇(p, y) and
y ∈ ∇(q, r) such that z = x ∗ y.
Since x ∈ ∇(p, q), we know that q ≤P p ∗ x. Analogously, y ∈ ∇(q, r) implies
r ≤P q ∗ y.
Using property 3 , we get
3
r ≤P q ∗ y ≤P (p ∗ x) ∗ y
1
= p ∗ (x ∗ y)
= p ∗ z.
All in all, r ≤P p ∗ z which implies z ∈ ∇(p, r). 2
Let (L, ∗, , ≤) be a residual complete lattice, that is an ordered monoid for which
∗ preserves arbitrary infima in each component. Furthermore, we consider a map
ν : M −→ L which satisfies the condition
ν(x ∗ y) ≤ νx ∗ νy for all x, y ∈ M. (7)
On this basis, we construct the following map
d : P × P −→ L via (p, q) 7→ inf ν ∇(p, q) .
Claim 7 The map d satisfies the triangle inequality, that is
d(p, r) ≤ d(p, q) ∗ d(q, r) holds for all p, q, r ∈ P.
Proof. Let p, q, r ∈ P . First, we transform the inequality’s right hand side:
d(p, q) ∗ d(q, r) = inf ν ∇(p, q) ∗ inf ν ∇(q, r)
= inf ν ∇(p, q) ∗ ν ∇(q, r) .
Hence, we have to show that inf ν ∇(p, q) ≤ inf ν ∇(p, q) ∗ ν ∇(q, r) .
Let t ∈ ν ∇(p, q) ∗ ν ∇(q, r) . It follows that there exists x ∈ ∇(p, q) and
y ∈ ∇(q, r) such that t = νx ∗ νy, which is greater than or equal to ν(x ∗ y) due
to property, i. e. we obtain
ν(x ∗ y) ≤ t. (8)
Consequently, with claim 6, we get
x ∗ y ∈ ∇(p, q) ∗ ∇(q, r) ⊆ ∇(p, r).
Applying ν on both sides yields
ν(x ∗ y) ∈ ν ∇(p, r) .
Hence,
(8)
inf ν ∇(p, r) ≤ ν(x ∗ y) ≤ t
and the triangle inequality of d is shown. 2
Definition 4 Let M = (M, ∗, ε) be a monoid and let L = (L, ∗, , ≤) be an
ordered monoid. Then, a map ν : M −→ L is a monoid norm w. r. t. (M, L)
if ν(ε) = and ν(x ∗ y) ≤ νx ∗ νy holds for all x, y ∈ M .
Theorem 2 Let (P, M, ∗) be a poset right monoid action with P = (P, ≤P ) and
M = (M, ∗, ε). Further, let L = (L, ∗, , ≤) be a residual complete lattice and
ν : M −→ L be a monoid norm w. r. t. (M, L).
Then
d : P × P −→ L, (p, q) 7→ inf ν ∇(p, q)
is a GQM w. r. t. (P, L).
7 Application to join geometries
Let P = (P, ≤) be a complete lattice. Then an element x ∈ P is called compact
in P if for every subset T of P with x ≤ sup T there exists a finite subset U of
T such that x ≤ sup U .
Definition 5 A join geometry is defined as a pair (P, E) consisting of a com-
plete lattice P and a join-dense subset E consisting of compact elements in P.
For the following, let (P, E) be a join geometry such that for all p, q ∈ P there
exists a compact element r ∈ P such that q ≤ p ∨ r.
Then the triple (P, M, ∗) is a poset right monoid action for M = (M, ∪, ∅) with
M := 2E
f in and
∗ : P × M −→ P, (x, D) 7→ x ∨ sup D.
Moreover, the map
ν : M −→ N ∪ {∞}, D 7→ #D
is a monoid norm w. r. t. (M, L) for
L := N ∪ {∞}, +, 0, ≤
(which forms a residual complete lattice). Obviously, by theorem 2 it follows
that
d : P × P −→ N ∪ {∞}, (p, q) 7→ inf ν ∇(p, q)
= min{#D | D ∈ 2E
f in : q ≤ p ∨ sup D}
is a GM w. r. t. (P, L).
This result has an important specialisation for closure operators on power sets
of finite sets.
8 Application to Closure Operators
Let U be a finite set and γ be := P, ⊆ with P := 2U .
a closure operator on P
Further, let M := N, +, 0, ≤ . Then the map
d : P × P −→ N, (X, Y ) 7→ min #T | T ∈ P : Y ⊆ γ(X ∪ T )
is a GQM w. r. t. (P, M), which we want to call the closure distance.
In particular, the restriction of d onto γP × γP is a GM w. r. t. (γP, M).
In context of information pooling, for a group of received elements, we can
construct the corresponding closure and with the closure distance d from above,
the distance to a given closure can be evaluated. This works for arbitrary closure
operators, which also includes closure systems of a matroid, for instance.
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