Abstract Information present in any training set of vectors for machine learning can be interpreted in two different ways, either as whole states or as individual atomic units. In this paper, we show that these alternative information distributions are often inherently incongruent within the training set. When learning with a Boltzmann machine, modifications in the network architecture can select one type of distributional information over the other; favouring the activation of either state exemplar or atomic characteristics. This choice of distributional information is of relevance when considering the representation of knowledge in logic. Traditional logic only utilises preference that is the correlate of whole state exemplar frequency. We propose that knowledge representation derived from atomic characteristic activation frequencies is the correlate of compositional typicality, which currently has limited formal definition or application in logic. Further, we argue by counter-example, that any representation of typicality by ‘most preferred model semantics’ is inadequate. We provide a definition of typicality derived from the probability of characteristic features; based on neural network modelling.

Abstract The field of belief revision in logic is still in evolution and holds a variety of disparate approaches; a consequence of theoretical conjecture. As a probabilistic model of supra-classical, non-monotonic (SCNM) logic, the Boltzmann machine, offers an experimental gateway into the field. How does the Boltzmann network adapt to new information? Catastrophic forgetting is the default response to retraining in any neural network. We have moderated this irrational non-monotonicity by alterations in the Boltzmann learning algorithm. The spectrum of experimental belief change is limited by the availability of ‘new’ information, a pragmatic realization co-related to the property of Rational Monotonicity in the domain of SCNM logic. Recognizing this upper boundary of defeasible belief simplifies the task of experimentally exploring machine adaptation. A minority of belief revisions involve new, but unsurprising information, that is at least partially consistent with the previous learned beliefs. In these circumstances, the Boltzmann network incrementally adjusts the priority of model state exemplars in accordance with preference; the traditional approach in SCNM logic. However, in the majority of situations the new information will be surprisingly inconsistent with the previous beliefs. In these circumstances, the pre-order on model states stratified by preference, will not have sufficient granularity to represent the conflicting requirements of ranking based on compositional atomic typicality. This novel experimental finding has not previously been considered in the logical conjecture on Belief Revision.

Abstract This paper looks at the representation of supra-classical, non-monotonic (SCNM) logic by an artificial neural network. It identifies the features of defeasiblity in this logic related to inference in the context of common-sense reasoning. It considers the machine characteristics that make a representation possible, with reference to previous literature. We describe a theoretical environment for investigating the representation and provide experimental evidence confirming that a Boltzmann machine is a suitable network representation. A Boltzmann machine can learn an input distribution corresponding to a preference relation and explicitly retrieve appropriate model states, constituting one-to-many mappings, entailed by the uncertain information contained in a premiss. The place of the Boltzmann machine in knowledge representation is discussed. In future papers, this neural network model of SCNM logic will serve as an experimental gateway for exploration of typicality and belief revision.