# A Comparison of Different Conceptual Structures Projection Algorithms

## 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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##### Citations

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]....

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

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

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...Projection algorithm is focused on structural similarity between CG and the execution time is at best NP-complete [26]....

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...In order to avoid the NP-complete problem of graph matching algorithm, we introduce the use of a standard CG in the dissimilarity computation....

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##### References

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16,023 citations

### "A Comparison of Different Conceptua..." refers background in this paper

...When u 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]:...

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3,204 citations

### "A Comparison of Different Conceptua..." refers background in this paper

...There is a projection from G to G′ if and only if G′ can be derived from G by the elementary specialization rules [4,6]....

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...Conceptual Structures (CS) are a logic based representation of C.S. Peirce’s existential graphs [3] developed by John Sowa [4]....

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...Graph diagrams that are built out of the blocks of conceptual structures are conceptual graphs (CG) [4,5]....

[...]

...Peirce’s existential graphs [3] developed by John Sowa [4]....

[...]

...When u 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]:...

[...]

2,725 citations

### "A Comparison of Different Conceptua..." refers background in this paper

...Graph diagrams that are built out of the blocks of conceptual structures are conceptual graphs (CG) [4,5]....

[...]

2,319 citations

### "A Comparison of Different Conceptua..." refers methods in this paper

...As discussed in the Messmer and Bunke paper [12], a naive strategy with forward-checking for establishing a subgraph isomorphism is Ullman’s backtracking in search tree algorithm [13]....

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...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....

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "A comparison of different conceptual structures projection algorithms" ?

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