=Paper=
{{Paper
|id=Vol-2211/paper-18
|storemode=property
|title=Concepts as Modal Operators in Description Logics
|pdfUrl=https://ceur-ws.org/Vol-2211/paper-18.pdf
|volume=Vol-2211
|authors=Lukas Grätz
|dblpUrl=https://dblp.org/rec/conf/dlog/Gratz18
}}
==Concepts as Modal Operators in Description Logics==
https://ceur-ws.org/Vol-2211/paper-18.pdf
Concepts as Modal Operators in Description
Logics?
Lukas Grätz
Institut für Informatik & Institut für Philosophie, Universität Leipzig, Germany
lukas graetz studserv uni-leipzig de
Abstract. Motivated by the colloquial language term ”glass gummy
bear”, an additional type of concept composition for description log-
ics is suggested. This composition type is then axiomatically formalized
and called concept generalization. Consistency of the formalization is
checked. By proving axiom K and Gödel rule, it is shown that this logic
is in fact a multi-modal logic. Concepts could be both modal operators
and predicate symbols. A Kripke semantics is presented (the adequacy
is future work). In this semantics, the TBox axioms hold for any view,
assertions in the ABox hold for the natural view (a selected world in the
Kripke structure) only. The relationship to other formalisms is outlined.
Further examples are discussed at the end.
1 Motivation
An approach to extend description logics (DL for short) is to find contextual
knowledge representations in colloquial language, which have not yet been rep-
resented in DL: When colloquial language structures cannot be formalized, they
have to be integrated into DL. This paper is about such an integration.
Consider a knowledge base which defines Glass and GummyBear as concepts.
In the TBox, we then have to formalize our knowledge about these terms that
gummy bears are sweets that sweets are edible and that glass is not edible:
GummyBear v Edible (1)
Glass v ¬Edible (2)
My example deals with a glass gummy bear. As we know, a lot of objects
can be made out of glass including a piece of glass in the shape of a gummy
bear. I know that because I own such an object myself. However, it can by no
means be a gummy bear, because a gummy bear is known to be an edible candy.
And it is not only a general consensus that gummy bears have to be a candy.
This is a defining / essential / necessary characteristic of gummy bears! If we
omit this characteristic we will not be able to give a definition of gummy bears
to an unknowing person. Nevertheless, a glass gummy bear is by no means a
?
Partly presented as poster in RuleML+RR 2018, Esch-sur-Alzette, Luxembourg.
contradiction: Such an object is, for example, based on a real gummy bear in
form, colour and surface.
So the next task is to represent a special glass gummy bear x in DL: x
should be added to the ABox. But how? The introduction of a new concept
GlassGummyBear with GlassGummyBear v Glass in the TBox is not optimal. If
we use an intersection Glass u GummyBear(x) then the knowledge base becomes
inconsistent. If we use a union Glass t GummyBear(x) then we are inaccurate,
because Glass t GummyBear only means that something must be either glass or
gummy bear (or both, but this is excluded by the TBox).
There is one formalization approach remaining: We could treat glass not as
a concept but a role isGlassModelOfObject. So
isGlassModelOfObject(x, y) (3)
GummyBear(y) (4)
would be fine in the ABox, where y is the real gummy bear, which is modeled
by x. All glass gummy bears are then returned by the following query:
∃isGlassModelOfObject. GummyBear (5)
However, what if there is no real gummy bear which was modeled by the glass
gummy bear. Or if the real gummy bear is unknown?
The general problem is, that the “glass gummy bear” has no connection (role,
relation) to a specific gummy bear, only a relation to the concept of the gummy
bear. Hence you would need a meta-relation between an individual glass gummy
bear and the concept of a glass gummy bear.
My proposal is to extend DL by a further mechanism for the composition
of concepts: The following definition describes a glass gummy bear x without
modification of the original TBox (2), by
Glass ( GummyBear(x) (6)
in the ABox. This knowledge base is consistent. As we will see later, we can
derive Glass(x) and by (2) ¬GummyBear(x). For simplicity, this paper extends
only a subset of the DL ALC without quantifiers and roles. The notation ( is
adopted from [9].
2 Proof Theoretic Definition
Definition 1 (Concept Generalization). Let A be a atomic concept and D,
E be concepts (meta variables that can be used for any concept expression). Then
A ( D is a concept (say ”A generalized D”) and is defined by the following
axioms and the generalization rule:
A ( (D t E) v A ( D t A ( E Union (7)
A(>≡A Universality (8)
A(⊥v⊥ Inconsistent Concept (9)
>(CvC Realness (10)
A ( (∃r. C) v ∃r. (A ( C) Quantification (11)
DvE
Generalization Rule (12)
A(DvA(E
Note that the generalization symbol ( has the strongest bond after ¬.
Why these axioms? First of all, it is good that we have an axiom for each
possible operand. This makes it easier to define a recursive semantics afterwards.
But then, the axioms have to be defended: The union (7) might be most difficult
to defend, because we hardly have a union operator in English. Something which
is either a leopard or a tiger could be called a tiger-or-leopard. It is reasonable
that a stone tiger-or-leopard is a stone tiger or a stone leopard.
The next three axioms deal with the well known predefined concepts ⊥ and
>. Universality (8) simply states that a concept is equivalent when used as an
adjective to the universal concept >. There is no reason that the generalization
C ( ⊥ of the empty concept ⊥ would be more than empty, hence (9). But
how to read expressions like > ( C, where > is used as an adjective? The only
interpretation I have is that these expressions would correspond to “real C”, for
example a “real gummy bear” is at most a “gummy bear”. This concludes (10).
ALC without roles is not ALC. Considering the veggie burger example in
Section 5, Axiom (11) states that roles are not affected by the use of adjectives
and that we can transit from the concept “has ingredient meat relative to the
veggie property” Veggie ( ∃ingredient. Meat to “has ingredient veggie meat”
∃ingredient. Veggie ( Meat. Note that axiom (11) is not compatible with
the proposal of my original submission, which would also require roles to be
modified by adjectives, e.g. Veggie ( ingredient. But what is the meaning of
the ingredient relationship relative to the Veggie concept? The current version
avoids relative roles and makes it easier to provide a semantics by separating
roles and concepts.
The generalization rule (12) is the most powerful part of the formalization.
If we have a TBox axiom D v E we automatically have the axiom A ( D v
A ( E for any (atomic) concept A. Under a different viewpoint the same
axioms should hold. For example, a Glass ( GummyBear v Glass ( Edible or
Veggie ( Burger v Veggie ( ∃ingredient. Meat.
Theorem 1 (Consistency). There are (infinitly) many knowledge bases (ABox
and TBox) in the extended logic where ⊥ ≡ > is not a consequence, because
knowledge base and definition 1 are satisfiable.
Proof. We can transform the extension back to the original logic by replacing the
new ( character with u. The transformed axioms (except (11)) can be proven
in ALC, moreover the transformed generalization rule is provable. Hence any
knowledge base in the extended logic without roles inherits the models of the
transfered knowledge base.
A u D v A ( D does not follow from definition 1, but A ( D v A t D
by (13). For example, we have (real) “gummy bear’ which imitates a Haribo
gummy bear, hence it is a “fake Haribo”. By using (13) we obtain that we have
something “fake”, something which is in the intersection between “fake” and
“gummy bear”. This should not be mistaken by a “fake gummy bear”!
Theorem 2. For all concepts D, E and atomic concepts A applies
A(DvA Inheritance (13)
A ( D t A ( ¬D ≡ A Excluded Middle (14)
A ( (D u E) v A ( D u A ( E Intersection (15)
Proof.
By union (7) we have A ( D t A ( ¬D ≡ C ( (D t ¬D),
and from D t ¬D ≡ > follows by (12) ≡A(>
and by universality (8) ≡ A.
Thus (14) and (13) are shown. (15) follows due to generalization (12), since
D u E v D and D u E v E.
3 Modal Logic Property
Let C be a concept. If we consider C ( as a modal operator ♦C in a multi-modal
logic, then the equivalent of axiom K
C (P ⊃ Q) ⊃ (C P ⊃ C Q) (16)
⇐⇒ ♦C (P ∧ ¬Q) ∨ (♦C ¬P ∨ ¬♦C ¬Q) (17)
is provable. We can transform (17) directly into the syntax I use, which has
neither classical implication nor a necessity operator:
Theorem 3 (Axiom K).
C ( (P u ¬Q) t C ( ¬P t ¬(C ( ¬Q) ≡ > (18)
Proof.
> ≡ C ( ¬Q t ¬(C ( ¬Q) (19)
v C ( ((P u ¬Q) t ¬P ) t ¬(C ( ¬Q) (20)
≡ C ( (P u ¬Q) t C ( ¬P t ¬(C ( ¬Q) (21)
(20) follows by using the generalization rule with ¬Q v (P u ¬Q) t ¬P .
Theorem 4 (Gödel rule).
D≡>
(22)
¬(C ( ¬D) ≡ >
Proof. The Gödel rule is equivalent to
¬D ≡ ⊥
(23)
C ( ¬D ≡ ⊥
and since ⊥ is a subconcept of every concept:
¬D v ⊥
(24)
C ( ¬D v ⊥
Axiom (9) ⊥ ≡ C ( ⊥ results in
¬D v ⊥
. (25)
C ( ¬D v C ( ⊥
This is a special case of the generalization rule.
4 Kripke Structure
As we learned in the previous section, the DL extension is in fact a modal logic,
where concepts can be both modal operators and predicates. Now we are curious,
if there is some sort of Kripke structure for a semantic representation: In this
section, I will present a tentative semantics and then go on to a more formal
FOL embedding.
It is possible to prove the generalization rule and some axioms in definition
1 by using axiom K and the Gödel rule. However, the universality C ( > ≡ C
does not seem to be provable in standard modal logics, and so the most important
property of inheritance is still missing. On the other hand, this is not surprising,
because an adequate semantics should provide models for the complete KB,
including the ABox.
For example, we want to find a model of Glass ( GummyBear(x): Should
the concept Glass be a modal operator and the composite concept Glass (
GummyBear a predicate that is applied to x? Or is GummyBear the predicate and
must be bracketed correctly as Glass ( (GummyBear(x))? As we will see, the
bracketing makes no difference.
I assume that the ABox always makes statements about the natural view, i.e.
a selected current world w∗ ∈ W in the Kripke structure. Thus a glass gummy
bear would be formalized in the ABox as Glass ( GummyBear(a) in the natural
view and not in a transferred view as GummyBear(a). The same should apply
to queries like Glass ( GummyBear. A query GummyBear would return no glass
gummy bears.
Otherwise, the views behave like the classical semantics of DL: Each view
w ∈ W contains a subset of individuals ∆.
wI ⊆ ∆ (26)
The TBox should hold for every view. Inheritance is created by additional lim-
itation of the model space, so that the subset of individuals in each view is at
most reduced and may never be supplemented by new individuals
∀v, w ∈ W. RC (v, w) =⇒ wI ⊆ v I (27)
where RC is the relation corresponding to the ”modal operator” C (. This
is also known as shrinking domain assumption in modal logics. It takes getting
used to the fact that the natural view contains all individuals; otherwise it seems
reasonable that e.g. from Glass ( GummyBear(a) also Glass(a) follows.
The last trick is that the modal operator > is defined as the identity relation
R> = {(w, w)|w ∈ W }, (28)
thus universality and realness is guaranteed. The relation
R⊥ = {(w, w⊥ )|w ∈ W } (29)
I
points to the empty view w⊥ = ∅, in which there are no individuals.
The complete Kripke model can be given by the following tuple:
hW, w∗ , RC1 , ..., RCn , Ii (30)
Please note that adequacy of the calculus in definition 1 still needs to be proved
relative to the semantics presented here. Currently, both calculus and semantics
are still controversial.
4.1 FOL Embedding
ALC can be embedded into first order logic (F OL) using a translation function
πx : LALC → LF OL , which converts ALC formulas into F OL with one free
variable x (see [11,1])1 . Slightly modifying π gives us the desired properties (for
C, D concepts, atomic concepts A and viewpoints v1 )
v1
πx : L(
ALC → LF OL
C t D 7→ v1 πx (C) ∨ v1 πx (D) (31a)
v1
¬C 7→ ¬ πx (C) (31b)
v1 v1
C u D 7→ πx (C) ∧ πx (D) (31c)
v1
∀r. C 7→ ∀y pr (x, y) → πy (C) (31d)
v1
∃r. C 7→ ∃y pr (x, y) ∧ πy (C) (31e)
A 7→ ∃v2 qA (v1 , v2 ) ∧ e(v2 , x) (31f)
v2
A ( D 7→ ∃v2 qA (v1 , v2 ) ∧ e(v2 , x) ∧ πx (D) (31g)
1
For each transformation rule, two variants are needed for the permutations of the
two variables x and y. In the modified embedding presented here, the transformation
rules for the permutations v2 πx , v1 πy and v2 πy can be defined likewise.
where qA (·, ·), pr (·, ·) and e(·, ·) are F OL-predicates. There are two main differ-
ences between this embedding and the classical embedding: First we have the
extra viewpoint parameter v1 and then the atomic concepts are not translated
into unary predicates A 7→ qA (x). In contrast to the classical embedding, con-
cepts are translated into relations between viewpoints, e.g. a “fake gun” is “fake”
in the designated, common sense viewpoint v ∗ but a gun in the faker’s viewpoint.
e(v, x) denotes that an individual x belongs to a viewpoint v. For example,
we want to ensure that Glass ( GummyBear(x) exists in a Glass and a Glass (
GummyBear viewpoint by the shrinking domain assumption (32) for viewpoints
v1 , v2 , atomic concepts A and individuals x (in correspondence to the inheritance
property (13)):
e(v2 , x) ∧ qA (v1 , v2 ) → e(v1 , x) (32)
The relation predicates for special concepts q> and q⊥ are defined as follow-
ing:
q> (v1 , v2 ) :↔ v1 = v2 (33)
q⊥ (v1 , v2 ) :↔ False (34)
5 Further Examples
Now that everything is defined and the description logic has been extended, this
is the right place to sketch possible use cases.
First we have, a closer look at the biggest city as described in [7]. ”Biggest
city“ turns out to be a relative term, since we distinguish between the biggest city
in Asia Asian City ( The Biggest, the biggest city in Europe and the biggest
city on earth. But how about an alternative formalization using conventional
DL? It turns out that all we need would be a greaterThan relation between
cities. For instance, the biggest city in Asia could be defined as:
Biggest Asian City ≡ Asian City u ¬∃greaterThan. Asian City (35)
Let us try more promising examples: Another common relative terminology
is the word ”normal“. A prominent example of an inconsistent knowledge base
is that penguins cannot fly, whereas birds can fly. This could be solved by saying
that normal birds (like pigeons) fly – penguins are birds but not normal.
Bird ( Normal v Flies (36)
Pigeon v Bird ( Normal (37)
Penguin v Bird (38)
The next example deals with melted things, like melted ice cream, melted
water ice or melted chocolate. Melted ice cream is something that you would
not call ice cream, because it is no longer creamy. Melted water ice, however, is
still some aggregate form of water and can become ice again. Melted chocolate
remains tasty but has lost its original form forever. The TBox could be formalized
as follows:
Melted ( IceCream v ¬IceCream (39)
Melted ( WaterIce v Water (40)
Melted ( Chocolate v Chocolate (41)
An alternative formalization of ”melted“ by using a time logic is conceivable,
which would require another extension of DL.
Last but not least, the TBox for a vegetarian burger Veggie ( Burger:
Burger v ∃ingredient. Meat (42)
Burger v ∃ingredient. Bread (43)
Veggie u Meat ≡ ⊥ (44)
Veggie ( Bread v Bread (45)
What kind of meat does a veggie burger contain? To answer this, we use the
FOL embedding:
v
πa (Veggie ( Burger) (46)
w
=def ∃v1 qVeggie (v, w) ∧ e(w, x) ∧ πa (Burger) (47)
Axiom (42) yields:
=⇒ ∃w qVeggie (v, w) ∧ e(w, a) ∧ w πa (∃ingr. Meat)
(48)
w
=def ∃w qVeggie (v, w) ∧ e(w, a) ∧ ∃y [pingr (a, y) ∧ πy (Meat)] (49)
=def ∃w, y, u qVeggie (v, w) ∧ e(w, a) ∧ pingr (a, y) ∧ qMeat (w, u) ∧ e(u, y) (50)
Shrinking domain assumption (32): qMeat (w, u) ∧ e(u, y) → e(w, y)
=⇒ ∃w, y, u pingr (a, y) ∧ qVeggie (v, w) ∧ e(w, y) ∧ qMeat (w, u) ∧ e(u, y) (51)
=def v πa (∃ingr. Veggie ( Meat) (52)
Hence a veggie burger contains ”veggie meat“. It is easy to show that ”veggie
meat“ cannot be ”meat“ using (44). Likewise, a veggie burger contains ”veggie
bread“ (which is real bread).
6 Discussion
Language permits many paradox concepts built as a combination of incompatible
terms. There are different features and mechanisms in natural language that
make these paradox combinations possible: Analogies, metaphors, ambiguity,
shifting definitions of terms, lexicalization and modalities are only a subset of
these features. This contrasts with formal logics which are powerful because they
do not possess these features, hence avoid misinterpretations. Then again, the
gap between logic and language is the reason for the development of logics that
reintegrate some useful language features.
In this paper, the topic is description logics only, a particular logic family
for formalizing ontologies. My work was motivated on adjective-noun phrases,
adjectives that can shift the meaning of the noun. A linguistic perspective on
privative adjectives like in “stone lion”, “fake gun” in formal semantics can be
found in [10].
Klarman [7] had a similar approach for relative terminologies. This is con-
nected, but independent from his later work on a framework for contextual
knowledge under a multi-modal description logic, which distinguishes between
object concepts and contextual concepts [8]. The formalization in [7] uses a set
theoretic semantics, which satisfies the axioms in Definition 1 (except (11)) by
having a different modal-operator inspired notation hAsian CityiThe Biggest
instead of Asian City ( The Biggest. This operator is not limited to atomic
concepts hAi – complex concept constructions hCi are also allowed. Set theoretic
semantics brings also a controversial equivalence
hDiC u hDi¬C ≡ ⊥ (53)
quite uncommon for modal logics. This means we cannot have
(Glass ( Edible u Glass ( ¬Edible) (x). (54)
I think that (54) is reasonable in colloquial language: First, observe that every
glass gummy bear is a glass model of a gummy bear, hence a glass model of
something edible. Second, observe that x is a glass model and that every glass
model is a glass model of something not edible.
Another approach extending description logics is by using conceptual blend-
ing and combinatorial creativity for concept invention [5,4,12]. The blending
process to invent words like “houseboat” involves generalization and iterative
consistency checking and evaluation. Although combinatorial creativity com-
bines incompatible concepts too, the idea of a conceptual blending differs from
the viewpoint semantics presented here. On a closer look there are many differ-
ences between the use cases of conceptual blending and adjectivized concepts:
the former modifies the relational role structure for generating a generic space,
the latter is independent from the role structure. The consistency in conceptual
blending is a result of an optimization process with “optimality principles de-
fined in a vague cognitive way”[5] whereas inconsistencies are avoided (but not
excluded) when using adjectivized concepts. Also note that the examples differ:
“houseboat” is a dvandva, a special form of a compound word where neither
house nor boat dominates the meaning. The production (or invention) of new
compound words is an infrequent morphological process. On the other side, the
use of adjective noun phrases does not require a production process: It seems
that everyone understands combinations like “stone lion”, “veggie burger” and
“fake gun” immediately.
There are other approaches for modeling that birds fly, whereas penguins
don’t and that gummy bears are edible whereas glass gummy bears are not edible:
Defeasable rules provide such a mechanism. [2] provides defeasable description
logics using defeasable subsumption and [6] solves this problem by introducing
a “normal” operator T(•). A general approach uses the KLM postulates [3].
Although ALC is equivalent to a propositional multi-modal logic, it is not
so uncommon to extend DL with modal logic in new dimensions. Besides this
work, there are several attempts, see [11] for an overview.
A new type of concept composition for description logics was presented, which
was motivated by colloquial language examples. This could be useful for semantic
web ontologies, where the focus is on natural and compact knowledge represen-
tations. If you have a large knowledge base, and a fixed TBox including the
fact that a “gummy bear” is edible, you might not want to alter the TBox for
allowing glass gummy bears, because thousands of ABox assertions depend on a
fixed and exact definition. With the description logic extension presented here,
there is no need to alter the TBox, you could use Glass ( GummyBear even if
the TBox avoids using the ( operator.
Beside the special concepts ⊥ and >, the introduction of common generaliza-
tion concepts for vagueness or falsehood is conceivable. Consider a false fireplace:
This is not a real fireplace, but it can be called a fireplace in the broader sense.
A ”possible car” ♦ ( Car could be a car-like vehicle that we only see from
distance and therefore cannot exactly identify.
By allowing only atomic concepts A ( on the left side of the ( operator, we
have a simple formalization with only finitely many modal operators. Complex
concepts C ( could be allowed by converting them into expressions using atomic
concepts only, once a normalization procedure is introduced.
In conclusion, the DL extension is more expressive then basic DL, the veggie
burger is a nice example. The scope of the approach is not yet determined.
Although inspired by language, it has a clear semantics and no vagueness or
ambiguity. This is a great advantage of logical formalism over natural language,
if it is not the main goal to formalize natural language.
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