Logical Characterizations of Recurrent Graph Neural Networks with Reals and Floats
CoRR(2024)
摘要
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.
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