Perceptron Connectives in Knowledge Representation

Abstract

We discuss the role of perceptron (or threshold) connectives in the context of Description Logic, and in particular their possible use as a bridge between statistical learning of models from data and logical reasoning over knowledge bases. We prove that such connectives can be added to the language of most forms of Description Logic without increasing the complexity of the corresponding inference problem. We show, with a practical example over the Gene Ontology, how even simple instances of perceptron connectives are expressive enough to represent learned, complex concepts derived from real use cases. This opens up the possibility to import concepts learnt from data into existing ontologies.

Succinctness and Complexity of ALC with Counting Perceptrons

“…Proposition 15 generalises the complexity results of Galliani et al (2020) for ALC with the standard tooth.…”

“…In contrast, we recall that, adding the regular tooth of Galliani et al (2020) to ALC results in logics of the same expressivity regardless of whether the weights are possibly negative or not.…”

“…Finally, using the transformations of Galliani et al (2020), eliminating the regular teeth altogether is efficient, and we obtain a problem of deciding the satisfiability of an ALCQ concept wrt an ALCQ TBox, which is EXPTIMEcomplete (Tobies 2000).…”

“…They have been used, e.g., in learning scenarios (Galliani et al 2020) and in modelling concept combination problems (Righetti et al 2019). Therefore, designing description logic languages that support the tooth has the potential to help general practitioners to write ontologies in a more accessible and agile way.…”

“…Adding the perceptron operator to ALC does not increase the expressivity since every instance of the operator can be replaced with an equivalent Boolean combination of ALC concepts that suitably unfolds the tooth into a DNF, as shown by Porello et al (2019). Furthermore, adding the perceptron operator to ALC does not affect the complexity of reasoning (Galliani et al 2020), namely, it is still EXPTIME-complete to do reasoning wrt a general TBox: there exists a linear transformation such that any instance of an entailment in ALC using the ∇ ∇ can be transformed into an equivalent instance of entailment in pure ALC (although obfuscating the original readability of the concepts defined with ∇ ∇).…”