Opening the Black-box: Deep Neural Networks as Weighted Conditional Knowledge Bases (Extended Abstract)

In this abstract we report the results of the paper “Weighted defeasible knowledge bases and a multipreference semantics for a deep neural network model” in Proc. JELIA 2021 [15], which investigates the relationships between a multipreferential semantics for defeasible reasoning in knowledge representation and a deep neural network model. Weighted knowledge bases for description logics are considered under a “concept-wise” multipreference semantics. The semantics is further extended to fuzzy interpretations and exploited to provide a preferential interpretation of Multilayer Perceptrons. Preferential approaches have been used to provide axiomatic foundations of nonmonotonic and common sense reasoning [11, 31, 33, 26, 28, 32, 3, 22]. They have been extended to description logics (DLs), to deal with inheritance with exceptions in ontologies, by allowing for non-strict forms of inclusions, called typicality or defeasible inclusions, with different preferential semantics [19, 7] and closure constructions [9, 8, 20, 5, 34, 6, 16]. The paper exploits a concept-wise multipreference semantics as a semantics for weighted knowledge bases, i.e. knowledge bases in which defeasible or typicality inclusions of the form T(C) v D (meaning “the typical C’s are D’s” or “normally C’s are D’s”) are given a positive or negative weight. For instance, A multipreference semantics, taking into account preferences with respect to different concepts, was first introduced by the authors as a semantics for ranked DL knowledge bases [13]. For weighted knowledge bases, a different semantic closure construction is developed, still in the spirit of other semantic constructions in the literature, and is further extended to the fuzzy case. A preference relation <Ci on the domain ∆ of a DL interpretation can be associated to each concept Ci to represent the relative typicality of domain individuals with respect to Ci. Preference relations with respect to different concepts do not need to agree, as a domain element x may be more typical than y as a horse but less typical as a zebra. The plausibility/implausibility of properties for a concept is represented by their (positive or negative) weight. For instance, a weighted TBox (called TEmployee) associated to concept Employee might contain the following weighted defeasible inclusions: (d1) T(Employee) v Young , 50 (d3) T(Employee) v ∃has classes.>, -70 (d2) T(Employee) v ∃has boss.Employee , 100; meaning is that, while an employee normally has a boss, he is not likely to be young or have classes. Furthermore, between the two defeasible inclusions (d1) and (d3), the second one is considered to be less plausible than the first one.