Passively Mobile Communicating Machines that Use Restricted Space

TL;DR

This paper introduces a new theoretical model for passively mobile wireless sensor networks using Turing Machines, characterizes their computational power under space constraints, and establishes a space hierarchy with thresholds at log n and log log n.

Contribution

It defines the PMSPACE complexity classes for passively mobile machines with bounded memory, providing exact characterizations and space hierarchies, including a threshold at o(log log n).

Findings

Unique IDs can be generated with O(log n) memory.

PMSPACE(f(n)) equals symmetric predicates in NSPACE(nf(n)) for f(n)={}log n.

SEMILINEAR is a subset of PMSPACE(f(n)) when f(n)=o(loglog n).

Abstract

We propose a new theoretical model for passively mobile Wireless Sensor Networks, called PM, standing for Passively mobile Machines. The main modification w.r.t. the Population Protocol model is that agents now, instead of being automata, are Turing Machines. We provide general definitions for unbounded memories, but we are mainly interested in computations upper-bounded by plausible space limitations. However, we prove that our results hold for more general cases. We focus on complete communication graphs and define the complexity classes PMSPACE(f(n)) parametrically, consisting of all predicates that are stably computable by some PM protocol that uses O(f(n)) memory on each agent. We provide a protocol that generates unique ids from scratch only by using O(log n) memory, and use it to provide an exact characterization for the classes PMSPACE(f(n)) when f(n)={\Omega}(log n): they are precisely the classes of all symmetric predicates in NSPACE(nf(n)). In this way, we provide a space hierarchy for the PM model when the memory bounds are {\Omega}(log n). Finally, we explore the computability of the PM model when the protocols use o(loglog n) space per machine and prove that SEMILINEAR=PMSPACE(f(n)) when f(n)=o(loglog n), where SEMILINEAR denotes the class of the semilinear predicates. In fact, we prove that this bound acts as a threshold, so that SEMILINEAR is a proper subset of PMSPACE(f(n)) when f(n)=O(loglog n).