Succinctness and Complexity of ALC with Counting Perceptrons

Abstract

Perceptron operators have been introduced to knowledge representation languages such as description logics in order to define concepts by listing features with associated weights and by giving a threshold. Semantically, an individual then belongs to such a concept if the weighted sum of the listed features it belongs to reaches that threshold. Such operators have been subsequently applied to cognitively-motivated modelling scenarios and to building bridges between learning and reasoning. However, they suffer from the basic limitation that they cannot consider the weight or number of role fillers. This paper introduces an extension of the basic perceptron operator language to address this shortcoming, defining the language ALCP and answering some basic questions regarding the succinctness and complexity of the new language. Namely, we show firstly that in ALCP+, when weights are positive, the language is expressively equivalent to ALCQ, whilst it is strictly more expressive in the general case allowing also negative weights. Secondly, ALCP+ is shown to be strictly more succinct than ALCQ. Thirdly, capitalising on results concerning the logic ALCSCC, we show that despite the added expressivity, reasoning in ALCP remains EXPTIME-complete.