Events

It is known that these theories over classical Peano arithmetic (PA) are mutually interpretable and exceeds the strength of PA. Over intuitionistic Heyting arithmetic (HA), on the other hand, finite iterations of strictly positive fixpoints have been shown to be conservative.

After introducing the setting and presenting the earlier results, as well as the technical tools, the main section of the dissertation can be roughly divided into two parts. The first presents a novel proof of the conservativity result above. The proof interprets the theories into the logic of partial terms where a realizability interpretation is used to reduce the problem to fixpoints for almost negative operator forms. A diagonal argument using a hierarchy of almost negative formulae with corresponding partial satisfaction predicates yields the result.

The second part generalises the tri-interpretation result from the second paragraph to intuitionistic theories, by proposing a new generalisation of positivity called guarded positivity with the aim to better capture the behaviour of intuitionistic implications and their interplay with transfinite iterations of truth and fixpoint predicates. As a corollary, these transfinite theories are stronger than HA.

A discussion of the results and the methods used concludes the dissertation.

Examination committee:

• Vera Koponen, chair (Uppsala University)
• Takako Nemoto (Tohoku University)
• Benno van den Berg (University of Amsterdam)

It is therefore useful to estimate the cognitively adequate complexity of description logic so that the simplest justification(s) can be identified and selected. This work focusses on the cognitive complexity of deciding the consistency of ABoxes in the description logic $\mathcal{ALE}$. Conceptual considerations entail that the complexity of the ABox consistency problem is expressed as a linear preorder on a large enough set of ABoxes. This linear preorder can be estimated by an ACT-R model called SHARP. The model is a tableau algorithm implemented in a cognitive architecture, thereby modelling a human brain executing the tableau algorithm, which forms an approximation of the cognitive complexity ordering.

A complexity ordering that is a linear preorder can be obtained from experimental data by aggregating the different individual orders from a sample into one so-called proto-order, in which the largest transitive subrelation is the desired linear pre-order. The predicted complexity ordering by SHARP agrees well with the empirical data.

To enable applications in ontology debugging, a surrogate model is needed that accurately approximates SHARP’s output with great computational efficiency. Such a surrogate model can be obtained by the Random Forest algorithm trained on an appropriate dataset generated by SHARP. The complexity ordering thus obtained agrees well with the one predicted by SHARP but can be estimated in a fraction of the time enabling it for practical applications.