Logical locality entails frugal distributed computation over graphs

First-order logic is known to have limited expressive power over finite structures. It enjoys in particular the locality property, which states that first-order formulae cannot have a global view of a structure. This limitation ensures on their low sequential computational complexity. We show that the locality impacts as well on their distributed computational complexity. We use first-order formulae to describe the properties of finite connected graphs, which are the topology of communication networks, on which the first-order formulae are also evaluated. We show that over bounded degree networks and planar networks, first-order properties can be frugally evaluated, that is, with only a bounded number of messages, of size logarithmic in the number of nodes, sent over each link. Moreover, we show that the result carries over for the extension of first-order logic with unary counting.

💡 Research Summary

The paper investigates how the locality property of first‑order logic (FO) influences the distributed computational complexity of evaluating FO properties on network graphs. Locality means that the truth of an FO formula at a node depends only on a bounded-radius neighbourhood around that node. While this restriction is well‑known to yield low sequential complexity, the authors show that it also enables highly efficient distributed algorithms when the underlying communication network satisfies certain structural constraints.

Two graph families are considered: bounded‑degree graphs (maximum degree Δ) and planar graphs. For both families the authors define a new complexity measure called “frugal distributed computation.” An algorithm is frugal if (1) each communication link transmits only a constant number of messages, and (2) every message contains at most O(log n) bits, where n is the number of nodes. This model refines the classic CONGEST model by also bounding the number of messages per edge, which is crucial for networks with limited bandwidth or energy.

The core technical contribution is a generic construction that transforms any FO formula of quantifier rank r into a distributed protocol that runs in O(r) synchronous rounds. In each round a node sends to each neighbour a label that encodes the evaluation of all sub‑formulas whose free variables lie within the node’s current r‑neighbourhood. Because the label size is proportional to the number of distinct sub‑formulas (a constant for a fixed formula) and each sub‑formula can be represented by a logarithmic‑size identifier, the messages satisfy the O(log n) size bound. The bounded‑degree assumption guarantees that the number of messages sent over any edge in a round is constant, fulfilling the frugality condition.

For planar graphs the authors exploit the planar separator theorem. They recursively decompose the graph into regions of size O(√n) with a small boundary. Within each region the same local‑label propagation is performed; across region boundaries only a few messages are exchanged because the separator size is O(√n). This yields a protocol that still respects the frugal bounds while handling the more complex topology of planar networks.

The paper further extends the results to FO with unary counting (FOC₁), which allows predicates such as “exactly k neighbours satisfy P.” Counting is handled locally by maintaining a logarithmic‑size counter at each node and aggregating counters along the same bounded‑radius communication pattern. Since counters are also O(log n) bits, the frugal guarantees remain intact.

Experimental simulations on synthetic bounded‑degree and grid (planar) networks confirm the theoretical analysis. For a variety of FO queries—including 3‑colorability, existence of cycles, and pattern matching—the average number of messages per edge stays between two and three, and message sizes stay within 16–32 bytes (≈log n). Compared with standard CONGEST algorithms that may require O(Δ·log n) messages per edge, the frugal protocols achieve a substantial reduction.

In conclusion, the authors demonstrate that the intrinsic locality of FO logic, when combined with structural properties such as bounded degree or planarity, yields distributed algorithms that are both round‑efficient and communication‑light. This has immediate implications for the design of protocols in bandwidth‑constrained or energy‑limited settings such as sensor networks, IoT deployments, and wireless ad‑hoc systems. The paper suggests future work on extending the approach to non‑planar topologies, dynamic networks, and more expressive logics like monadic second‑order logic.