Set Description Languages and Reasoning about Numerical Features of Sets
Hans Jürgen Ohlbach
Set description languages, for example description logics, can be used to specify sets and set-theoretic relationships between them. Mathematical programming, on the other hand, can be used to find optimal solutions for arithmetical equation and in-equation systems. In this paper a combination methodology is presented, which allows one to use numerical algorithms for reasoning about numerical features of sets specified with a set description language. The method is applied to description logics as an examples for a set description language. In this context new properties of logical languages become interesting which have not been considered so far.
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Modal Logics, Description Logics and Arithmetic Reasoning
Hans Jürgen Ohlbach and Jana Koehler
We introduce mathematical programming and atomic decomposition as the basic modal (T-Box) inference techniques for a large class of modal and description logics. The class of description logics suitable for the proposed methods is strong on the arithmetical side. In particular there may be complex arithmetical conditions on sets of accessible worlds (role fillers).
The atomic decomposition technique can deal with set constructors for modal parameters (role terms) and parameter (role) hierarchies specified in full propositional logic. Besides the standard modal operators, a number of other constructors can be added in a relatively straightforward way. Examples are graded modalities (qualified number restrictions) and also generalized quantifiers like `most', `n%', `more' and `many'.
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How to Augment a Formal System with a Boolean Algebra Component
Hans Jürgen Ohlbach
We introduce the atomic decomposition technique as a means to combine computation and reasoning in a given formal system with reasoning about Boolean Algebras and features of the elements of the Boolean Algebras which make sense in the given basic system. Propositional reasoning is invoked in a kind of compilation phase which eliminates the Boolean Algebras part of the problem completely and shifts the main reasoning problem to the basic system. The decomposition method works for combinations of formal systems with a Boolean Algebras component, where the Boolean terms are embedded in bridging functions mapping the Boolean parts to objects the given formal system can understand.
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A Multi-Dimensional Terminological Knowledge Representation
Franz Baader and Hans Jürgen Ohlbach
An extension of the concept description language ALC used in KL-ONE like terminological reasoning is presented. The extension includes multi-modal operators that can either stand for the usual role quantifications or for modalities such as belief, time etc. The modal operators can be used at all levels of the concept terms, and they can be used to modify both concepts and roles. This is an instance of a new kind of combination of modal logics where the modal operators of one logic may operate directly on the operators of the other logic. Different versions of this logic are investigated and various results about decidability and undecidability are presented. The main problem, however, decidability of the basic version of the logic, remains open.
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Last Modified: Monday, 25-Mar-2002 14:50:19 CET