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projection and de-projections which serve as a basis
for inference. 3) We define procedures of constraint
propagation and inference as well as describe how
they are supported by the query language.
The paper has the following layout. Section 2
describes the formal setting by defining the notion of
nested partially ordered set and how it is used to
represent hierarchical multidimensional spaces.
Section 3 defines main operations on nested posets
and how they are used for inference. Section 4
makes concluding remarks.
2 CONCEPT-ORIENTED MODEL
The approach to inference described in this paper
relies on a novel unified model, called the concept-
oriented model (COM) (Savinov, 2011a); (Savinov,
2011b); (Savinov, 2012). One of the main principles
of COM is that an element consists of two tuples:
one identity tuple and one entity tuple. These
identity-entity couples are modeled by a novel data
modeling construct, called concept (hence the name
of the model), which generalizes classes. Concept
fields are referred to as dimensions. Yet, in this
paper we will not distinguish between identities and
entities by assuming that an element is one tuple.
Figure 1: Database is a partially ordered set.
For this paper, another COM major principle is
important which postulates that a set of data
elements is a partially ordered set (poset). Another
approach where posets are used for data modeling is
described in (Raymond, 1996). In COM, posets are
represented by tuples themselves, that is, tuple
membership relation induces partial order relation
‘<’ (less than):
ee
1
,, <〉〈 KK . Here <
1
means
‘immediately less than’ relation (‘less than’ of
rank 1). If
ba < then a is referred to as a lesser
element and b is referred to as a greater element.
Thus tuple members are supposed to be immediately
greater than the tuples they are included in. And
conversely, a tuple is immediately less than any of
its member tuples it is composed of. Since tuple
membership is implemented via references (which
are identity tuples), this principle essentially means
that an element references its greater elements.
Fig. 1 is an example of a poset graphically
represented using a Hasse diagram where an element
is drawn under its immediate greater elements and is
connected with them by edges.
At the level of concepts, tuple order principle
means that dimension types specify greater concepts.
Then a set of concepts is a poset where each concept
has a number of greater concepts represented by its
dimension types and a number of lesser concepts
which use this concept in its dimensions. For
example, assume that each book has one publisher:
CONCEPTBooks//Books<Publishers
IDENTITY
CHAR(10)isbn
ENTITY
CHAR(256)title
Publisherspublisher//Greaterconcept
According to this principle, Publishers is a
greater concept because it is specified as a type
(underlined) of the dimension publisher.
The main benefit of using partial order is that it
has many semantic interpretations: attribute-value
(greater elements are values characterizing lesser
elements), containment (greater elements are sets
consisting of lesser elements), specific-general
(greater elements are more general than lesser
elements), entity-relationship (lesser elements are
relationships for greater elements), and
multidimensional (greater elements are coordinates
for lesser elements). These interpretations allow us
to use COM as a unified model.
In the context of this paper, the most important
property of partial order is that it can be used for
representing multidimensional hierarchical spaces.
The basic idea is that greater elements are
interpreted as coordinates with respect to their lesser
elements which are interpreted as points. Thus an
element is a point for its greater elements and a
coordinate for its lesser elements. In Fig. 1, e is a
point with coordinates d and f (its greater elements)
and at the same time it is a coordinate for two points
a and c (its lesser elements).
In multidimensional space, any coordinate
belongs to some axis and any point belongs to some
set. The question is how the notions of axes and sets
can be formally represented within the order-
theoretic setting. To solve this problem we assume
that a poset consists of a number of subsets, called
domains:
m
XXXO ∪∪∪
K
21
, 0
∩
ji
XX ,
ji
. Domains are interpreted as either sets of
coordinates (axes) or sets of points (spaces), and any
element is included in some domain:
OXe
k
⊂∈ .
InferenceinHierarchicalMultidimensionalSpace
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