# Asynchronous Arbitrary Pattern Formation: the effects of a rigorous approach

## TL;DR

This paper introduces a rigorous, logic-based distributed algorithm for asynchronous arbitrary pattern formation without chirality, revealing flaws in previous approaches and establishing its equivalence with leader election.

## Contribution

It provides the first fully rigorous algorithm for APF without chirality, using logical predicates to ensure correctness and challenge prior unverified results.

## Key findings

- The new algorithm fully characterizes APF and its equivalence with leader election.
- Previous algorithms in the literature lack rigorous correctness proofs.
- Some established results in the literature are shown to be incorrect.

## Abstract

Given any multiset F of points in the Euclidean plane and a set R of robots such that |R|=|F|, the Arbitrary Pattern Formation (APF) problem asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections, uniform scalings. Initially, each robot occupies a distinct position. When active, a robot operates in standard LCM cycles. Robots are asynchronous, oblivious, anonymous, silent and execute the same distributed algorithm. So far, the problem has been mainly addressed by assuming chirality, that is robots share a common left-right orientation. We are interested in removing such a restriction. While working on the subject, we faced several issues that required close attention. We deeply investigated how such difficulties were overcome in the literature, revealing that crucial arguments for the correctness proof of the existing algorithms have been neglected. The systematic lack of rigorous arguments with respect to necessary conditions required for providing correctness proofs deeply affects the validity as well as the relevance of strategies proposed in the literature. Here we design a new deterministic distributed algorithm that fully characterizes APF showing its equivalence with the well-known Leader Election problem in the asynchronous model without chirality. Our approach is characterized by the use of logical predicates in order to formally describe our algorithm as well as its correctness. In addition to the relevance of our achievements, our techniques might help in revising previous results. In fact, it comes out that well-established results like [Fujinaga et al, SIAM J. Comp. 44(3) 2015], more recent approaches like [Bramas et al, PODC and SSS 2016] and 'unofficial' results like [Dieudonne et al, arXiv:0902.2851] revealed to be not correct.