Succinct Population Protocols for Presburger Arithmetic

TL;DR

This paper demonstrates that any quantifier-free Presburger arithmetic formula can be computed by a leaderless population protocol with polynomially many states, improving previous exponential bounds and introducing new constructions for different input sizes.

Contribution

It proves that all quantifier-free Presburger formulas are computable by leaderless protocols with polynomial states, advancing the understanding of population protocol capabilities.

Findings

Leaderless protocols with polynomial states for Presburger formulas

Protocols with a small number of leaders for large inputs

Protocols with a single leader for small inputs

Abstract

Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula $\varphi$ of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with $2^{O(\text{poly}(|\varphi|))}$ states that computes $\varphi$. More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree. In this paper, we prove that every formula $\varphi$ of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with $O(\text{poly}(|\varphi|))$ states. Our proof is based on several new constructions, which may be of independent interest. Given a formula $\varphi$ of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with $O(|\varphi|^3)$ leaders) that computes $\varphi$; this completes the work initiated in [STACS'18], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes $\varphi$. Our last construction gets rid of this leader for small inputs.