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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.

AbstractKnowledge 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.

Summary (3 min read)

1 Introduction

  • Query-Answer systems are very important in business and industry today.
  • 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
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.
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].

Citations
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Book ChapterDOI
22 Jul 2007
TL;DR: This paper advocates a projection operator based on the classical arc consistency algorithm used in constraint satisfaction problems that has the required properties : polynomiality, local validation, parallelization, structural interpretation.
Abstract: The projection problem (conceptual graph projection, homomorphism, injective morphism, i¾?-subsumption, OI-subsumption) is crucial to the efficiency of relational learning systems. How to manage this complexity has motivated numerous studies on learning biases, restricting the size and/or the number of hypotheses explored. The approach suggested in this paper advocates a projection operator based on the classical arc consistency algorithm used in constraint satisfaction problems. This projection method has the required properties : polynomiality, local validation, parallelization, structural interpretation. Using the arc consistency projection, we found a generalization operator between labeled graphs. Such an operator gives the structure of the classification space which is a concept lattice.

11 citations


Cites background from "A Comparison of Different Conceptua..."

  • ...In conceptual graph community, different algorithms are proposed for the projection problem [4,5,6]....

    [...]


01 Jan 2007
TL;DR: It is believed that changing the reasoning operation's algorithm and providing adequate data structures for them can improve the implementation of the operation for use in intelligent systems; therefore, making them faster and more efficient.
Abstract: Knowledge representation (KR) is used to store and retrieve meaningful information. Meaning cannot be directly stored in the computer; therefore, a series of levels of representation transforms knowledge to a format that a computer can process. This transformed knowledge is saved using dynamic data structures that are suitable for the style of KR being implemented, and through the KR the system manipulates the knowledge in the data using reasoning operations. The data structure, together with the contents of the transformed knowledge, is called 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 it uses. In this work, the basic reasoning operations for knowledge management will be explored using a particular style of KR called Conceptual Graphs (CGs). These operations, projection and maximal join, are the foundation for query/answer and hypothesis generation (abduction) systems, respectively. It is believed that changing the reasoning operation's algorithm and providing adequate data structures for them can improve the implementation of the operation for use in intelligent systems; therefore, making them faster and more efficient. Different algorithms and data structures execution times are analyzed over the most general form of CGs knowledge base showing that flexible, fast and efficient operations can improve higher level systems.

9 citations


Book ChapterDOI
23 Oct 2018
TL;DR: This work proposes a search engine for searching Web data represented in UNL (Universal Networking Language) based on semantic graph matching, which includes semantic expansion for graph nodes and relation matching based on relation meaning.
Abstract: Explosive growth of the Web has made searching Web data a challenging task for information retrieval systems. Semantic search systems that go beyond the shallow keyword matching approaches and map words to their conceptual meaning representations offer better results to the users. On the other hand, a lot of representation formats have been specified to represent Web data into a semantic format. We propose a search engine for searching Web data represented in UNL (Universal Networking Language). UNL has numerous attractive features to support semantic search. One of the main features is that UNL does not depend on domain ontology. Our proposed search engine is based on semantic graph matching. It includes semantic expansion for graph nodes and relation matching based on relation meaning. The search results are ranked depending on the semantic similarity between the user query and the retrieved documents. We developed a prototype implementing the proposed semantic search engine, and our evaluations demonstrate its effectiveness across a wide-range of semantic search tasks.

7 citations


Journal ArticleDOI
01 May 2012
TL;DR: This paper focuses on a graph-based approach for text representation and presents a novel error tolerance dissimilarity algorithm for deviation detection, which has managed to identify deviating sentences and it strongly correlates with expert judgments.
Abstract: The rapid increase in the amount of textual data has brought forward a growing research interest towards mining text to detect deviations. Specialized methods for specific domains have emerged to satisfy various needs in discovering rare patterns in text. This paper focuses on a graph-based approach for text representation and presents a novel error tolerance dissimilarity algorithm for deviation detection. We resolve two non-trivial problems, i.e. semantic representation of text and the complexity of graph matching. We employ conceptual graphs interchange format CGIF --a knowledge representation formalism to capture the structure and semantics of sentences. We propose a novel error tolerance dissimilarity algorithm to detect deviations in the CGIFs. We evaluate our method in the context of analyzing real world financial statements for identifying deviating performance indicators. We show that our method performs better when compared with two related text based graph similarity measuring methods. Our proposed method has managed to identify deviating sentences and it strongly correlates with expert judgments. Furthermore, it offers error tolerance matching of CGIFs and retains a linear complexity with the increasing number of CGIFs.

6 citations


Proceedings ArticleDOI
01 Dec 2009
TL;DR: A dissimilarity algorithm to detect outliers from a collection of text represented with Conceptual Graph Interchange Format (CGIF) and results indicate that the method outperforms the existing method and correlates better to human judgements.
Abstract: The graphical text representation method such as Conceptual Graphs (CGs) attempts to capture the structure and semantics of documents. As such, they are the preferred text representation approach for a wide range of problems namely in natural language processing, information retrieval and text mining. In a number of these applications, it is necessary to measure the dissimilarity (or similarity) between knowledge represented in the CGs. In this paper, we would like to present a dissimilarity algorithm to detect outliers from a collection of text represented with Conceptual Graph Interchange Format (CGIF). In order to avoid the NP-complete problem of graph matching algorithm, we introduce the use of a standard CG in the dissimilarity computation. We evaluate our method in the context of analyzing real world financial statements for identifying outlying performance indicators. For evaluation purposes, we compare the proposed dissimilarity function with a dice-coefficient similarity function used in a related previous work. Experimental results indicate that our method outperforms the existing method and correlates better to human judgements. In Comparison to other text outlier detection method, this approach managed to capture the semantics of documents through the use of CGs and is convenient to detect outliers through a simple dissimilarity function. Furthermore, our proposed algorithm retains a linear complexity with the increasing number of CGs.

5 citations


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  • ...This avoids the NP-complete problem of graph matching algorithm....

    [...]

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Frequently Asked Questions (2)
Q1. What have the authors contributed in "A comparison of different conceptual structures projection algorithms" ?

In this paper, the basic reasoning operation for a query-answer system, projection, is explored using different theoretical algorithms. 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. 

Future work is to determine if the actual projection algorithm for all injective projections can be performed in polynomial time by experimental results [ 24 ].