A Comparison of Different
Conceptual Structures Projection Algorithms
Heather D. Pfeiffer and Roger T. Hartley
New Mexico State University
{hdp,rth}@cs.nmsu.edu
Abstract. Knowledgerepresentation(KR)isusedtostoreandretrieve
meaningful data. This data is saved using dynamic data structures that
are suitable for the style of KR being implemented. The KR allows the
system to manipulate the knowledge in the data by using reasoning oper-
ations. The data structure, together with the contents of the transformed
knowledge, is known as the knowledge base (KB). An algorithm and the
associated data structures make up the reasoning operation, and the
performance of this operation is dependent on the KB.
In this paper, the basic reasoning operation for a query-answer system,
projection, is explored using different theoretical algorithms. Within this
discussion, the associated algorithms will be using different KBs for their
Conceptual Graph (CG) knowledge representation. The basic projection
algorithm defined using the CG representation is looking for a graph
morphism of a query graph onto a graph from the KB.
The overall running time for the projection operation is known to be
a NP class problem; however, by modifying the algorithm, taking into
account the associated KB, the actual time needed for discovering and
creating the projection/s can be improved. In fact, a new projection
algorithm will be defined that, given a typical query onto a carefully
defined KB, presents a running time for the actual projection that only
grows with the number of projections present.
1 Introduction
Query-Answer systems are very important in business and industry today. How-
ever, these systems need to be able to represent knowledge in the computer in
order to use reasoning techniques when attempting to answer a query for a prob-
lem domain. In the computer, the description of the problem to be solved has
become known as knowledge representation, KR. This representation must be
able to store and retrieve meaningful data so that reasoning operations can be
performed. The most common reasoning technique used in query-answer systems
is projection of the query onto the stored knowledge. Later this work will discuss
more about this technique, but first more about storing meaningful data will be
presented.
One type of KR is semantic networks. These networks are displayed as a dis-
crete graphical structure of vertices and arcs [1]. Within the graphical structure,
U. Priss, S. Polovina, and R. Hill (Eds.): ICCS 2007, LNAI 4604, pp. 165–178, 2007.
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Springer-Verlag Berlin Heidelberg 2007
166 H.D. Pfeiffer and R.T. Hartley
the vertices are called nodes and may be displayed as circles or boxes. The arcs
are called links and are displayed as lines with arrows between the nodes. The
nodes are related to each other through their links where the links are assigned
a one-to-one correspondence to a conceptual meaning [2]. The nodes are some-
times called conceptual units and may be seen as objects within the network.
These objects are of many different types including entities, attributes, events or
even states. On top of the semantic network, abstract hierarchies are organized
according to levels of generalization for the conceptual units. The links of the
network form relational connections between the conceptual units, such that the
valence (or parity) of the connection is the number of units that are associated
with a particular unit. In a semantic network links are usually dyadic (binary)
connecting two conceptual units together.
Even though there are multiple semantic network representations available,
the representation that shows much flexibility is conceptual structures. Concep-
tual Structures (CS) are a logic based representation of C.S. Peirce’s existential
graphs [3] developed by John Sowa [4]. Conceptual structures are like a set of
logic building blocks; the definitions for some of the blocks are presented begin-
ning with the type block:
Definition 1. A type is a labeling for an abstract idea which is either a concep-
tual unit or a relationship. These types are members to a set, T, that may form
several structures including hierarchy trees, lattices, and other related structures.
When the structure is a type hierarchy lattice, the set is labeled T
C
,andthefunc-
tion ctype maps a conceptual unit to the type label in the structure. When the
structure is a relation hierarchy tree, the set is labeled T
R
,andthefunctionrtype
maps a relationship to the type label in the structure.
A referent block would have the following definition:
Definition 2. A referent is an abstract conceptual unit that has been instanti-
ated with a factual value.
Graph diagrams that are built out of the blocks of conceptual structures are
conceptual graphs (CG) [4,5]. For this work, a conceptual graph has the following
definition:
Definition 3. A conceptual graph is a bipartite, connected, directed graph G =
(V,E), such that the set of all vertices (nodes) V is partitioned into two disjoint
sets V
C
and V
R
. The vertices are labeled, and the set V
C
is called the concept
nodes and the set V
R
is called the conceptual relations nodes. e ∈ E is an ordered
pair that connects an element of V
C
to an element of V
R
using a directed arc.
The label of a concept node is a pair, c =< type, referent >. The type is an
element of the set T
C
. The referent (if present) contains the individual instanti-
ation for the type field.
The label of a conceptual relation node is a pair, cr =<type,signature>,
wheretypeisanelementofthesetT
R
, and the signature is a pair, s =<I,O>
where I is the arcs that are directed into the conceptual relation and O is
A Comparison of Different Conceptual Structures Projection Algorithms 167
the arcs that are directed out from the conceptual relation. The signature is
further defined by its subset category of either relation or actor.Therela-
tion is a tuple, r =<type,c
1
,c
2
, ..., c
n
>where type is defined above and in
the signature I ⊆ V
C
and O ∈ V
C
. The actor is a slightly different tuple,
a =<type,c
1
,c
2
, ..., {..., c
n−1
,c
n
} > where type is defined above and in the sig-
nature I ⊆ V
C
and O ⊆ V
C
.
Researchers M. Chein and M.-L. Mugnier [6] from the LIRMM group at the
Universite Montpellier and other researchers [7,8] have done research on a sub-
set of conceptual graphs known as simple conceptual graphs (SCGs) (see Sowa
3.1.2 [4]). As explained in Baget and Mugnier [7], these SCGs are connected,
bipartite graphs where the arcs are labeled and finite but not directed, SG =
((V
c
,V
r
),U,λ).
2 Foundational Projection
In general, the matching part of the projection algorithm is unification [9], and
there are known linear unification algorithms for acyclic (tree) graphs [10]. Also,
SCGs have been evaluated as both graph homomorphism and graph isomor-
phism. In their original paper from 1992 [11], Mugnier and Chein looked at
general projection running times and injective projection. However, CGs and
SCGs are not necessarily trees and only part of the algorithms presented next
apply to injective projection, so these linear algorithms give guidance, but do
not always directly apply.
As discussed in the Messmer and Bunke paper [12], a naive strategy with
forward-checking for establishing a subgraph isomorphism is Ullman’s back-
tracking in search tree algorithm [13]. Since Messmer and Bunke feel that it
is a common technique with a good baseline subgraph isomorphism algorithm,
the Ullman algorithm and its known complexity (from [13,12]) will be reiterated
here for defining a basis for investigating projection algorithms. The basic idea
of Ullman’s algorithm is to take one vertex of the input vertices (query graph) at
a time and map it onto a model (a graph from the KB) such that the resulting
mapping represents a subgraph isomorphism for a subgraph of the model (KB
graph) projected from the input graph (query graph) (see page 307 and 322 of
Messmer and Bunke [12]). If at some point, the mapping being built does not
represent a subgraph isomorphism then the algorithm backtracks and tries a
different mapping. This process is continued until all vertices, v
1
,...,v
M
in V
I
of the input graph are successfully mapped onto V of the model. This either
produces a subgraph isomorphism from G to G
I
or stops when a vertex in V
I
can not be mapped to at least one vertex in V . In the second case, the algo-
rithm backtracks to a new v
1
in V or v
n−1
in V and tries to remap the subgraph
isomorphism.
Even though this basic algorithm works well for small model and input graphs,
it performs poorly as the graphs become larger. This is because all checks are
being done locally. Ullman added a forward-checking procedure to know when it
is not possible for v
n
to be mapped onto an available vertex in V
I
(see page 322
168 H.D. Pfeiffer and R.T. Hartley
in Messmer and Bunke [12]), so that the algorithm can backtrack immediately
and save computational steps. In the best case Ullman’s algorithm is bounded
by: O(NIM) where N =#model graphs, I =#labeled vertices in input graph
which come from the M set of labels, M =#labeled vertices in model graph that
are unique. In the worst case the algorithm is bounded by: O(NI
M
M
2
) where
N =#model graphs, I =#vertices in the input graph and are unlabeled,
M =#vertices in the model graph and are unlabeled.
As can be seen, even with this general algorithm, labeling of vertices greatly
improves the efficiency of the algorithm. However, it should be noted, that
this algorithm does not take into account any support or hierarchy knowledge
information.
2.1 Operator
The project operator is defined through a mapping π :u → v,whereπu is a
sub-element of v.Whenu and v are defined to be conceptual graphs, for graph
u to be a subgraph of graph v then all of the nodes and arcs of u are in v [14],
and the project operator π holds to the following rules [4,15]:
– Type preserving: For each concept c in u, πc is a concept in πu where type(πc)
≤ type( c ), and ≤ is the subtype relation. If c is an individual, that is an
actual instance of an object, then referent ( c )=referent ( πc).
– Structure preserving: For each conceptual relation r in u, πr is a conceptual
relation in πu where type(πr) = type( r
). If the ith edge of r is linked to a
concept c in u, the ith edge of πr must be linked to πc in πu.
Color: bluepropObject
Fig. 1. Query Graph
2.2 Operation
A projection operation uses the project operator, which is a matching on a graph
morphism, graph data structures with either the support information for SCGs
or hierarchies when full CGs, and the actual projection algorithm. Stated in
Baget and Mugnier, ”the elementary reasoning operation, projection, is a kind of
graph homomorphism that preserves the partial order defined on labels” [7]. Not
only does projection use a project operator (see its definition in the subsection
above), but the support S ofthegraphbeitaSCGorthedefinedtypehierarchy
if a CG produces a generalization subgraph during the projection operation.
For the rest of this work, the projection operation evaluation and comparison
will be restricted to injective projection. This projection mapping is not necessar-
ily one-to-one; that is, a concept or relation in u may have more than one concept
or relation in v that πu is a valid mapping. In this respect, there is more than one
valid projection from u to v.When the projection operation is performed using
A Comparison of Different Conceptual Structures Projection Algorithms 169
CubeBetweenBalls
prop
prop
Object
Ball
Color: blue
Cube: A
Ballbetween
ontop
T
CubeBetweenBalls
prop
prop
Object
Ball
Color: blue
Cube: A
Ballbetween
ontop
Object Cube
Ball
Fig. 2. KB Graph with Type Hierarchy
P1
Color: bluepropObject
P2
Color: blueBall prop
P1
Color: bluepropObject
P2
Color: blueBall prop
Fig. 3. Projection Results
the query graph from Figure 1 onto the KB graph and hierarchy of Figure 2, the
two projections, P 1 and P 2, discovered are displayed in Figure 3.
1
Using the type
hierarchy, both object and ball are matches; note, if no hierarchy were present, then
there would be only one projection. This is a simple injective projection because
of the small graphs, however, it can become complex very quickly.
1
The figures in this section were generated by CharGer [16].