Przemysław Andrzej Wałęga

Przemysław Andrzej Wałęga, Bernardo Cuenca Grau

Bridges between graph neural networks (GNNs) and logical formalisms have been established by fixing architectural choices, such as the types of aggregation, combination, and activation functions. These choices define restricted classes of GNNs for which tight correspondences with logical formalisms can be obtained, by showing that logical formulae can be translated into equivalent GNNs and, conversely, that GNNs can be translated into equivalent formulae. In this paper we take a semantic perspective by establishing the logical expressiveness of classes of GNN classifiers that are preserved under structural properties: embeddings (extensions), injective homomorphisms, and homomorphisms. We show that, for each such property, there exists a fragment of graded modal logic characterising the class of GNNs. In particular, preservation under embeddings, injective homomorphisms, and homomorphisms corresponds to existential graded modal logic, its existential-positive fragment, and existential-positive modal logic, respectively. These results characterise the expressiveness of broad classes of GNNs independently of specific architectural choices, but we also show that each of these classes admits a GNN architecture of the same expressiveness. Technically, our approach uses a new well-quasi-order result for trees of bounded height, yielding finite representations of unravelling-invariant classes.

Stan P Hauke, Przemysław Andrzej Wałęga

In recent years, there has been growing interest in understanding the expressive power of graph neural networks (GNNs) by relating them to logical languages. This research has been been initialised by an influential result of Barceló et al. (2020), who showed that the graded modal logic (or a guarded fragment of the logic C2), characterises the logical expressiveness of aggregate-combine GNNs. As a ``challenging open problem'' they left the question whether full C2 characterises the logical expressiveness of aggregate-combine-readout GNNs. This question has remained unresolved despite several attempts. In this paper, we solve the above open problem by proving that the logical expressiveness of aggregate-combine-readout GNNs strictly exceeds that of C2. This result holds over both undirected and directed graphs. Beyond its implications for GNNs, our work also leads to purely logical insights on the expressive power of infinitary logics.

Marco Sälzer, Przemysław Andrzej Wałęga, Martin Lange

In recent years, the expressive power of various neural architectures -- including graph neural networks (GNNs), transformers, and recurrent neural networks -- has been characterised using tools from logic and formal language theory. As the capabilities of basic architectures are becoming well understood, increasing attention is turning to models that combine multiple architectural paradigms. Among them particularly important, and challenging to analyse, are temporal extensions of GNNs, which integrate both spatial (graph-structure) and temporal (evolution over time) dimensions. In this paper, we initiate the study of logical characterisation of temporal GNNs by connecting them to two-dimensional product logics. We show that the expressive power of temporal GNNs depends on how graph and temporal components are combined. In particular, temporal GNNs that apply static GNNs recursively over time can capture all properties definable in the product logic of (past) propositional temporal logic PTL and the modal logic K. In contrast, architectures such as graph-and-time TGNNs and global TGNNs can only express restricted fragments of this logic, where the interaction between temporal and spatial operators is syntactically constrained. These provide us with the first results on the logical expressiveness of temporal GNNs.