Tolerance to Asynchrony in Algorithms for Multiplication and Modulo

TL;DR

The state transitions that are formed under these algorithms satisfy lattice-linearity, where these algorithms induce a lattice among the global states, which implies that these algorithms can be implemented in asynchronous environments.

Abstract

In this article, we study some parallel processing algorithms for multiplication and modulo operations. We demonstrate that the state transitions that are formed under these algorithms satisfy lattice-linearity, where these algorithms induce a lattice among the global states. Lattice-linearity implies that these algorithms can be implemented in asynchronous environments, where the nodes are allowed to read old information from each other. It means that these algorithms are guaranteed to converge correctly without any synchronization overhead. These algorithms also exhibit snap-stabilizing properties, i.e., starting from an arbitrary state, the sequence of state transitions made by the system strictly follows its specification.

Figures (5)

• Figure 1: A finite automaton $M_3$ computing $n\mod 3$ for any $n\in \mathbb{N}$.
• Figure 2: Definition of the transition function $\delta$.
• Figure 3: Multiplication of 00011011 and 0101 in base 2.
• Figure 4: Demonstration of multiplication of 100 and 100 in base 2: (a) top down (b) bottom up.
• Figure 5: Processing $11011 \mod 11$ following \ref{['algorithm:log-modulo']}.

Theorems & Definitions (27)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Example 1
• Example 2
• Lemma 1
• proof
• Lemma 2