We have pointed out that, during the last fifteen years, DLs have been well-investigated with respect to their expressive power and the complexity of the standard inferences. Sound and complete inference algorithms are now available for a great variety of DLs. However, in some applications (e.g., the ones for the Classic DL-system at AT&T) it has turned out that additional non-standard inferences, such as lcs, msc, and matching, are needed to support the construction and maintenance of DL knowledge bases. To this end, first ad hoc implementations of such non-standard inferences have been integrated into Classic. However, the inference algorithms underlying these implementations were mostly incomplete and formal properties of the nonstandard inferences, such as their computational complexity, had not been studied. This situation corresponds to the level of development for standard inferences in the first phase, where DL-systems have been developed without having complete algorithms at hand and without an exact understanding of the complexity of the underlying reasoning problems (see Section 2.1).
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