Book ChapterDOI

# A Comparison of Different Conceptual Structures Projection Algorithms

22 Jul 2007-pp 165-178
TL;DR: 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.
Abstract: Knowledge representation (KR) is used to store and retrieve 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 operations. 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

• This representation must be able to store and retrieve meaningful data so that reasoning operations can be performed.
• These networks are displayed as a discrete graphical structure of vertices and arcs [1].
• Graph diagrams that are built out of the blocks of conceptual structures are conceptual graphs (CG) [4,5].
• The referent (if present) contains the individual instantiation for the type field.

### 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].
• 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.
• 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.
• This is because all checks are being done locally.
• It should be noted, that this algorithm does not take into account any support or hierarchy knowledge information.

### 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].
• When the projection operation is performed using the query graph from Figure 1 onto the KB graph and hierarchy of Figure 2, the two projections, P1 and P2, discovered are displayed in Figure 3. 1.
• Using the type hierarchy, both object and ball are matches; note, if no hierarchywere present, then there would be only one projection.
• The figures in this section were generated by CharGer [16].

### 3.2 Croitoru Projection

• Madalina Croitoru’s projection algorithm is based on SCGs as described in her two 2004 papers [8,17].
• This algorithm begins by starting from the foundational injective algorithm given by Mugnier and Chein [11], using SCGs with support which as stated in the Mugnier and Chein 1992 paper [11] is NP-complete.
• These matching graphs indicate which relation vertices should be used as potential candidates for projection; therefore, reducing the search space.

### 3.3 Notio Projection

• The Notio project is a general conceptual graph implementation with a well defined API [18].
• It is currently being used by several projects [19,20,21] for working with basic reasoning operations with a CG KB.
• This is the author’s derived theoretical algorithm (see Algorithm 1) from the Notio implementation code [18,22] for the injective projection algorithm (note: Southey never wrote any papers or documentation on the actual implemented algorithm).
• It should be noted for Algorithm 1, all the vertices are all labeled, but the edges are directed.
• Notio (if a possible mapping was indicated from step 2) will attempt to match all the relation vertices from the KB graph (along with their neighboring concepts along their edges) to query graph vertices with the same edge relationships, also known as In step 13.

### 4 New Algorithm

• After examining the above algorithms it was discovered that even though the running times were acceptable, the actual projection algorithms were not general.
• That is, the user was confined by what parts of a valid conceptual graph could be present in the data or could only have one projection even if more than one was present.
• The desire to allow the user to use a directed, connected, bipartite conceptual graph (see Definition 3) that was cyclic for both the query and KB graphs prompted a new projection algorithm to be designed.

### 4.1 Supporting Information

• In order to produce a new algorithm, new data structures and supporting routines were needed.
• These authors are not the first researcher to think about using triples.
• Kabbaj and Moulin in 2001 [23] looked at CG operations using a bootstrapping step.
• One data structure holds the matching possibilities of the query concepts with the KB graph concepts, called the match list, and the second structure holds the matching triples from the KB graph for each concept in the query graph, called the anchor list.
• Set of projections from query onto KB 34: end function have been defined around the triple relationship of the C-R-C.

### 4.2 Actual Algorithm

• The overall algorithm (see Algorithm 2) for the projection of the query graph onto the KB is based on looking at all triples that are in the query graph and checking for a complete subgraph match of the query graph onto the KB graph during preprocessing.
• Because each triple in the query graph is unique, even if the node type is not, all projections can be found in the KB graph.
• Then after all matches of conceptual units and triples are found, the actual projection graphs are built.
• Because the temporary data structures are saved from the preprocessing, matching does not have to happen again at build time.
• Because the anchor list contains all available projections, both injective and non-injective or homomorphism projections are found.

### 4.3 Execution Time

• Now that the algorithm is split into two sections, there is a running time for answering the decision question of whether or not there is a projection, it will be called the matching algorithm, and a running time for the actual projection.
• The labeling drives the execution time of the matching algorithm when doing an injective projection toward the running time for a subgraph ’labeled’ isomorphism problem which can be solved in polynomial time as opposed to a straight subgraph isomorphism problem which is known to be NP-complete.
• The size of the graphs in the KB affects the base of the execution time, but the number of times the Projection function is executed is based on the number of triples in the query graph.
• In a typical query-answer scenario where the query graph would potentially contain normally two to four triples compared to possibly a thousand in the KB graph, this algorithm takes into account that the query graph is small.
• Since in the most common case there is only one projection, the actual projection creation algorithm becomes polynomial.

### 5 Comparison and Conclusion

• Four different, yet related, projection algorithms have been described.
• It is not clear from the Mugnier and Chein 1992 paper if they can handle two concept pairs with the same relationship between them in a projection operation.
• Notio and the new algorithm have a complete separation between the preprocessing algorithm and projection; where, Croitoru uses the preprocessing algorithm inside of the actual projection, therefore, giving the same running time for both the overall algorithm and the actual projection.
• The new algorithm splits the overall projection algorithm into two parts, matching and projection construction.
• Therefore, in a typical scenario where the query graph is small, the new algorithm is not only able to find all projections for full conceptual graphs, but can use the data structures of the KB to do it faster.

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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
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
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.
c
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
n1
,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).
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
n1
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 ofthegraphbeitaSCGorthedenedtypehierarchy
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].

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