Abstract
Concept refinement operators have been introduced to describe and compute generalisations and specialisations of concepts, with, amongst others, applications in concept learning and ontology repair through axiom weakening. We here provide a probabilistic proof of almost-certain termination for iterated refinements, thus for an axiom weakening procedure for the fine-grained repair of \(\mathcal {ALC}\) ontologies. We determine the computational complexity of refinement membership, and discuss performance aspects of a prototypical implementation, verifying that almost-certain termination means actual termination in practice.
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Notes
- 1.
One way to verify this is to observe that the series \(\sum _{i=0}^\infty (\log (i + \ell -\epsilon ) - \log (i + \ell ))\) diverges to minus infinity. This in turn may be verified by noting that \(\sum _{i=0}^\infty (\log (i + \ell -\epsilon ) - \log (i + \ell )) \le \sum _{i=0}^\infty (\log (i + \lceil \ell \rceil -\epsilon ) - \log (i + \lceil \ell \rceil )) = \sum _{i=\lceil \ell \rceil }^\infty (\log (i -\epsilon ) - \log (i))\), because \(\log (i + \ell - \epsilon ) - \log (i + \ell ) \le \log (i+\lceil \ell \rceil - \epsilon ) - \log (i + \lceil \ell \rceil )\), and then showing that \(-\sum _{i=\lceil \ell \rceil }^\infty (\log (i -\epsilon ) - \log (i)) = \sum _{i=\lceil \ell \rceil }^\infty \log (i) - \log (i-\epsilon )\) diverges to plus infinity by means of the integral method: the terms of the series are all positive, and \(\int _{\lceil \ell \rceil }^U \log (x) - \log (x-\epsilon ) dx\) goes to infinity when U goes to infinity. Since the integral diverges, so does the series, which gives us our conclusion.
- 2.
For example, suppose that the algorithm terminates in exactly n steps with probability \(6/\pi ^2 \cdot n^{-2}\). Using the fact that \(\sum _{n=1}^\infty n^{-2} = \pi ^2/6\), we have at once that the algorithm terminates in finite time with probability 1. However, the expectation of its runtime would be \(6/\pi ^2 \sum _{n=1}^\infty n^{-1}\), which diverges to infinity.
- 3.
The implementation is available at https://bitbucket.org/troquard/ontologyutils.
- 4.
- 5.
These ontologies can be found in the directory ontologyutils/src/master/resources/Random/ of the implementation.
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Confalonieri, R., Galliani, P., Kutz, O., Porello, D., Righetti, G., Troquard, N. (2022). Almost Certain Termination for \(\mathcal {ALC}\) Weakening. In: Marreiros, G., Martins, B., Paiva, A., Ribeiro, B., Sardinha, A. (eds) Progress in Artificial Intelligence. EPIA 2022. Lecture Notes in Computer Science(), vol 13566. Springer, Cham. https://doi.org/10.1007/978-3-031-16474-3_54
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