Veeti Ahvonen

Veeti Ahvonen, Damian Heiman, Antti Kuusisto et al.

We give a novel logical characterization of encoder-decoder transformers, the foundational architecture for LLMs that also sees use in various settings that benefit from cross-attention. We study such transformers over text in the practical setting of floating-point numbers and soft-attention, characterizing them with a new temporal logic. This logic extends propositional logic with a counting global modality over the encoder input and a past modality over the decoder input. We also give an additional characterization of such transformers via a type of distributed automata, and show that our results are not limited to the specific choices in the architecture and can account for changes in, e.g., masking. Finally, we discuss encoder-decoder transformers in the autoregressive setting.

Veeti Ahvonen, Damian Heiman, Antti Kuusisto et al.

In pioneering work from 2019, Barceló and coauthors identified logics that precisely match the expressive power of constant iteration-depth graph neural networks (GNNs) relative to properties definable in first-order logic. In this article, we give exact logical characterizations of recurrent GNNs in two scenarios: (1) in the setting with floating-point numbers and (2) with reals. For floats, the formalism matching recurrent GNNs is a rule-based modal logic with counting, while for reals we use a suitable infinitary modal logic, also with counting. These results give exact matches between logics and GNNs in the recurrent setting without relativising to a background logic in either case, but using some natural assumptions about floating-point arithmetic. Applying our characterizations, we also prove that, relative to graph properties definable in monadic second-order logic (MSO), our infinitary and rule-based logics are equally expressive. This implies that recurrent GNNs with reals and floats have the same expressive power over MSO-definable properties and shows that, for such properties, also recurrent GNNs with reals are characterized by a (finitary!) rule-based modal logic. In the general case, in contrast, the expressive power with floats is weaker than with reals. In addition to logic-oriented results, we also characterize recurrent GNNs, with both reals and floats, via distributed automata, drawing links to distributed computing models.

Veeti Ahvonen, Maurice Funk, Damian Heiman et al.

Transformers are the basis of modern large language models, but relatively little is known about their precise expressive power on graphs. We study the expressive power of graph transformers (GTs) by Dwivedi and Bresson (2020) and GPS-networks by Rampásek et al. (2022), both under soft-attention and average hard-attention. Our study covers two scenarios: the theoretical setting with real numbers and the more practical case with floats. With reals, we show that in restriction to vertex properties definable in first-order logic (FO), GPS-networks have the same expressive power as graded modal logic (GML) with the global modality. With floats, GPS-networks turn out to be equally expressive as GML with the counting global modality. The latter result is absolute, not restricting to properties definable in a background logic. We also obtain similar characterizations for GTs in terms of propositional logic with the global modality (for reals) and the counting global modality (for floats).

Veeti Ahvonen, Damian Heiman, Antti Kuusisto

We give an alternative proof for the existing result that recurrent graph neural networks working with reals have the same expressive power in restriction to monadic second-order logic MSO as the graded modal substitution calculus. The proof is based on constructing distributed automata that capture all MSO-definable node properties over trees. We also consider some variants of the acceptance conditions.