Motivated by the colloquial language term "glass gummy bear", an additional type of concept composition for description logics is suggested. This composition type is then axiomatically formalized and called concept generalization. Consistency of the formalization is checked. By proving axiom K and Godel 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.
An approach to extend description logics (DL for short) is to nd contextual knowledge representations in colloquial language, which have not yet been represented 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 de nes 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:
(
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 de ning / essential / necessary characteristic of gummy bears! If we omit this characteristic we will not be able to give a de nition 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
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)
would be ne 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:
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 speci c 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 de nition describes a glass gummy bear x without
modi cation of the original TBox (
De nition 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 de ned by the following axioms and the generalization rule:
A ( (D t E) v A ( D t A ( E
A ( > A ( ? v ? > ( C v C
A A ( (9r: C) v 9r: (A ( C)
D v E A ( D v A ( E
Union Universality Inconsistent Concept Realness Quanti cation Generalization Rule (7) (8) (9) (10) (11) (12) 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 de ne a recursive semantics afterwards. But then, the axioms have to be defended: The union (7) might be most di cult 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 prede ned 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 a ected by the use of adjectives and that we can transit from the concept \has ingredient meat relative to the veggie property" Veggie ( 9ingredient: Meat to \has ingredient veggie meat" 9ingredient: Veggie ( Meat. Note that axiom (11) is not compatible with the proposal of my original submission, which would also require roles to be modi ed 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 di erent viewpoint the same axioms should hold. For example, a Glass ( GummyBear v Glass ( Edible or
Theorem 1 (Consistency). There are (in nitly) many knowledge bases (ABox and TBox) in the extended logic where ? > is not a consequence, because knowledge base and de nition 1 are satis able.
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 de nition 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 ( D t A ( :D
A ( D v A
A A ( (D u E) v A ( D u A ( E Inheritance Excluded Middle Intersection (13) (14) (15) Proof.
By union (7) we have and from D t :D and by universality (8) > follows by (12)
A ( D t A ( :D
C ( (D t :D); A ( > 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
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) ()
C (P ^ :Q) _ ( C :P _ : C :Q) 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).
Proof.
C ( (P u :Q) t C ( :P t :(C ( :Q) > >
C ( :Q t :(C ( :Q) v C ( ((P u :Q) t :P ) t :(C ( :Q)
C ( (P u :Q) t C ( :P t :(C ( :Q) (20) follows by using the generalization rule with :Q v (P u :Q) t :P .
(16) (17) (18) (19) (20) (21) Theorem 4 (Godel rule).
Proof. The Godel rule is equivalent to 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 de nition 1 by using axiom K and the Godel 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 nd 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 di erence.
I assume that the ABox always makes statements about the natural view, i.e. a selected current world w 2 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 2 W contains a subset of individuals . (26) The TBox should hold for every view. Inheritance is created by additional limitation 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 8v; w 2 W: RC (v; w) =) wI vI 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 de ned as the identity relation thus universality and realness is guaranteed. The relation points to the empty view w?I = ;, in which there are no individuals.
The complete Kripke model can be given by the following tuple: R> = f(w; w)jw 2 W g; R? = f(w; w?)jw 2 W g hW; w ; RC1 ; :::; RCn ; Ii Please note that adequacy of the calculus in de nition 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 rst order logic (F OL) using a translation function x : LALC ! LF OL, which converts ALC formulas into F OL with one free variable x (see [11,1])1. Slightly modifying gives us the desired properties (for C, D concepts, atomic concepts A and viewpoints v1) (27) (28) (29) (30) (31a) (31b) (31c) (31d) (31e) (31f) (31g) v1 x : LALC ! LF OL
( C t D 7! v1 x(C) _ v1 x(D)
:C 7! : v1 x(C) C u D 7! v1 x(C) ^ v1 x(D) 8r: C 7! 8y pr(x; y) ! 9r: C 7! 9y pr(x; y) ^ v1 y(C)
v1 y(C)
A 7! 9v2 qA(v1; v2) ^ e(v2; x)
A ( D 7! 9v2 qA(v1; v2) ^ e(v2; x) ^ v2 x(D) 1 For each transformation rule, two variants are needed for the permutations of the two variables x and y. In the modi ed embedding presented here, the transformation rules for the permutations v2 x, v1 y and v2 y can be de ned likewise. where qA( ; ), pr( ; ) and e( ; ) are F OL-predicates. There are two main di erences 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, concepts 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) ing:
The relation predicates for special concepts q> and q? are de ned as followq>(v1; v2) :$ v1 = v2 q?(v1; v2) :$ False 5
Further Examples (32) (33) (34) (36) (37) (38) Now that everything is de ned 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 de ned as:
(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 y, whereas birds can y. This could be solved by saying that normal birds (like pigeons) y { penguins are birds but not normal.
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: 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:
?
What kind of meat does a veggie burger contain? To answer this, we use the FOL embedding: (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) v a(Veggie ( Burger) =def 9v1 qVeggie(v; w) ^ e(w; x) ^ w a(Burger) Axiom (42) yields: =) 9w qVeggie(v; w) ^ e(w; a) ^ w a(9ingr: Meat) =def 9w qVeggie(v; w) ^ e(w; a) ^ 9y [pingr(a; y) ^ w y(Meat)] =def 9w; y; u qVeggie(v; w) ^ e(w; a) ^ pingr(a; y) ^ qMeat(w; u) ^ e(u; y) Shrinking domain assumption (32): qMeat(w; u) ^ e(u; y) ! e(w; y) =) 9w; y; u pingr(a; y) ^ qVeggie(v; w) ^ e(w; y) ^ qMeat(w; u) ^ e(u; y) =def v a(9ingr: Veggie ( Meat) 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
Language permits many paradox concepts built as a combination of incompatible terms. There are di erent features and mechanisms in natural language that make these paradox combinations possible: Analogies, metaphors, ambiguity, shifting de nitions 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 connected, 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 satis es the axioms in De nition 1 (except (11)) by having a di erent 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) (54) quite uncommon for modal logics. This means we cannot have
(Glass ( Edible u Glass ( :Edible) (x): 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 blending 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 combines incompatible concepts too, the idea of a conceptual blending di ers from the viewpoint semantics presented here. On a closer look there are many di erences between the use cases of conceptual blending and adjectivized concepts: the former modi es 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 dened in a vague cognitive way"[5] whereas inconsistencies are avoided (but not excluded) when using adjectivized concepts. Also note that the examples di er: \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 y, 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 representations. If you have a large knowledge base, and a xed 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 xed and exact de nition. 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 generalization concepts for vagueness or falsehood is conceivable. Consider a false replace: This is not a real replace, but it can be called a replace 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 nitely 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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Springer, Berlin, Heidelberg (2010) 8. Klarman, S., Gutierrez-Basulto, V.: Description logics of context. J. Log. Comput.
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