# Function-variable Elimination and Its Limitations

### Kiyoshi Akama, Ekawit Nantajeewarawat

#### Abstract

The famous proof method by the conventional Skolemization and resolution has a serious limitation. It does not guarantee the correctness of proving theorems in the presence of built-in constraints. In order to understand this difficulty, we use meaning-preserving Skolemization (MPS) and equivalent transformation (ET), which together provide a general framework for solving query-answering (QA) problems on first-order logic. We introduce a rule for function variable elimination (FVE), by which we regard the conventional Skolemization as a kind of the composition of MPS and FVE. We prove that the FVE rule preserves the answers to a class of QA problems consisting of only user-defined atoms, while we cannot prove it in the presence of built-in constraints. By avoiding the application of the FVE rule in MPS & ET computation, we obtain a more general solution for proof problems, which guarantees the correctness of computation even in the presence of built-in constraints.

#### References

- Akama, K. and Nantajeewarawat, E. (2011a). MeaningPreserving Skolemization. In Proceedings of the 3rd International Conference on Knowledge Engineering and Ontology Development, pages 322-327, Paris, France.
- Akama, K. and Nantajeewarawat, E. (2011b). MeaningPreserving Skolemization. Technical report, Hokkaido University, Sapporo, Japan.
- Akama, K. and Nantajeewarawat, E. (2012). Proving Theorems Based on Equivalent Transformation Using Resolution and Factoring. In Proceedings of the Second World Congress on Information and Communication Technologies, WICT 2012, pages 7-12, Trivandrum, India.
- Akama, K. and Nantajeewarawat, E. (2013). Embedding Proof Problems into Query-Answering Problems and Problem Solving by Equivalent Transformation. In Proceedings of the 5th International Conference on Knowledge Engineering and Ontology Development, pages 253-260, Vilamoura, Portugal.
- Akama, K. and Nantajeewarawat, E. (2014). Equivalent Transformation in an Extended Space for Solving Query-Answering Problems. In Proceedings of the 6th Asian Conference on Intelligent Information and Database Systems, LNAI 8397, pages 232-241, Bangkok, Thailand.
- Akama, K. and Nantajeewarawat, E. (2015). FunctionVariable Elimination and Its Limitations. Technical report, Hokkaido University, Sapporo, Japan.
- Chang, C.-L. and Lee, R. C.-T. (1973). Symbolic Logic and Mechanical Theorem Proving. Academic Press.
- Motik, B., Sattler, U., and Studer, R. (2005). Query Answering for OWL-DL with Rules. Journal of Web Semantics, 3(1):41-60.
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#### Paper Citation

#### in Harvard Style

Akama K. and Nantajeewarawat E. (2015). **Function-variable Elimination and Its Limitations** . In *Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management - Volume 2: KEOD, (IC3K 2015)* ISBN 978-989-758-158-8, pages 212-222. DOI: 10.5220/0005597202120222

#### in Bibtex Style

@conference{keod15,

author={Kiyoshi Akama and Ekawit Nantajeewarawat},

title={Function-variable Elimination and Its Limitations},

booktitle={Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management - Volume 2: KEOD, (IC3K 2015)},

year={2015},

pages={212-222},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005597202120222},

isbn={978-989-758-158-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 7th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management - Volume 2: KEOD, (IC3K 2015)

TI - Function-variable Elimination and Its Limitations

SN - 978-989-758-158-8

AU - Akama K.

AU - Nantajeewarawat E.

PY - 2015

SP - 212

EP - 222

DO - 10.5220/0005597202120222