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In pioneering work from 2019, Barcel\'o 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.

We examine the relationship of graded (multi)modal logic to counting (multichannel) message passing automata with applications to the Weisfeiler-Leman algorithm. We introduce the notion of graded multimodal types, which are formulae of graded multimodal logic that encode the local information of a pointed Kripke-model. We also introduce message passing automata that carry out a generalization of the Weisfeiler-Leman algorithm for distinguishing non-isomorphic graph nodes. We show that the classes of pointed Kripke-models recognizable by these automata are definable by a countable (possibly infinite) disjunction of graded multimodal formulae and vice versa. In particular, this equivalence also holds between recursively enumerable disjunctions and recursively enumerable automata. We also show a way of carrying out the Weisfeiler-Leman algorithm with a formula of first order logic that has been augmented with H\"artig's quantifier and greatest fixed points.

We investigate the descriptive complexity of a class of neural networks with unrestricted topologies and piecewise polynomial activation functions. We consider the general scenario where the running time is unlimited and floating-point numbers are used for simulating reals. We characterize these neural networks with a rule-based logic for Boolean networks. In particular, we show that the sizes of the neural networks and the corresponding Boolean rule formulae are polynomially related. In fact, in the direction from Boolean rules to neural networks, the blow-up is only linear. We also analyze the delays in running times due to the translations. In the translation from neural networks to Boolean rules, the time delay is polylogarithmic in the neural network size and linear in time. In the converse translation, the time delay is linear in both factors. We also obtain translations between the rule-based logic for Boolean networks, the diamond-free fragment of modal substitution calculus and a class of recursive Boolean circuits where the number of input and output gates match.

We consider distributed algorithms in the realistic scenario where distributed message passing is operated via circuits. We show that within this setting, modal substitution calculus MSC captures the expressive power of circuits. The translations between circuits and MSC-programs are linear in both directions. Furthermore, we show that the colouring algorithm based on Cole-Vishkin can be specified via logarithmic size programs.