Structure Theorem and Strict Alternation Hierarchy for FO^2 on Words

It is well-known that every first-order property on words is expressible using at most three variables. The subclass of properties expressible with only two variables is also quite interesting and well-studied. We prove precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words. Our results apply to both the case with and without a successor relation. For both languages, our structure theorems show exactly what is expressible using a given quantifier depth, n, and using m blocks of alternating quantifiers, for any m \leq n. Using these characterizations, we prove, among other results, that there is a strict hierarchy of alternating quantifiers for both languages. The question whether there was such a hierarchy had been completely open. As another consequence of our structural results, we show that satisfiability for first-order logic with two variables without successor, which is NEXP-complete in general, becomes NP-complete once we only consider alphabets of a bounded size

Extending Two-Variable Logic on Trees

The finite satisfiability problem for the two-variable fragment of first-order logic interpreted over trees was recently shown to be ExpSpace-complete. We consider two extensions of this logic. We show that adding either additional binary symbols or counting quantifiers to the logic does not affect the complexity of the finite satisfiability problem. However, combining the two extensions and adding both binary symbols and counting quantifiers leads to an explosion of this complexity. We also compare the expressive power of the two-variable fragment over trees with its extension with counting quantifiers. It turns out that the two logics are equally expressive, although counting quantifiers do add expressive power in the restricted case of unordered trees

One-Dimensional Fragment Over Words and Trees

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential (universal) quantifiers that leave at most one variable free. We investigate this fragment over words and trees, presenting a complete classification of the complexity of its satisfiability problem for various navigational signatures and comparing its expressive power with other important formalisms. These include the two-variable fragment with counting and the unary negation fragment.Peer reviewe

The Descriptive Complexity of Graph Neural Networks

We analyse the power of graph neural networks (GNNs) in terms of Boolean circuit complexity and descriptive complexity. We prove that the graph queries that can be computed by a polynomial-size bounded-depth family of GNNs are exactly those definable in the guarded fragment GFO+C of first-order logic with counting and with built-in relations. This puts GNNs in the circuit complexity class TC^0. Remarkably, the GNN families may use arbitrary real weights and a wide class of activation functions that includes the standard ReLU, logistic "sigmod", and hyperbolic tangent functions. If the GNNs are allowed to use random initialisation and global readout (both standard features of GNNs widely used in practice), they can compute exactly the same queries as bounded depth Boolean circuits with threshold gates, that is, exactly the queries in TC^0. Moreover, we show that queries computable by a single GNN with piecewise linear activations and rational weights are definable in GFO+C without built-in relations. Therefore, they are contained in uniform TC^0