We contend that reasoning about knowledge is both natural and pragmatic for verification of electronic voting protocols. We present a model in which desirable properties of elections are naturally expressed using standard knowledge operators, and show that the associated logic is decidable (under reasonable assumptions of bounded agents and nonces).
We present a general technique for constructing Graph Neural Networks (GNNs) capable of using multi-relational domain knowledge. The technique is based on mode-directed inverse entailment (MDIE) developed in Inductive Logic Programming (ILP). Given a data instance e and background knowledge B, MDIE identifies a most-specific logical formula ⊥ B (e) that contains all the relational information in B that is related to e. We represent ⊥ B (e) by a "bottom-graph" that can be converted into a form suitable for GNN implementations. This transformation allows a principled way of incorporating generic background knowledge into GNNs: we use the term 'BotGNN' for this form of graph neural networks. For several GNN variants, using real-world datasets with substantial background knowledge, we show that BotGNNs perform significantly better than both GNNs without background knowledge and a recently proposed simplified technique for including domain knowledge into GNNs. We also provide experimental evidence comparing BotGNNs favourably to multi-layer perceptrons that use features representing a "propositionalised" form of the background knowledge; and BotGNNs to a standard ILP based on the use of most-specific clauses. Taken together, these results point to BotGNNs as capable of combining the computational efficacy of GNNs with the representational versatility of ILP.
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