Construction of Minkowski Sums by Cellular Automata

TL;DR

The paper addresses the problem of making the Minkowski sum $A+B$ of two CA-constructible sets $A$ and $B$ constructible as a CA column. It introduces a signal-based CA framework that encodes $A$ along a diagonal and uses slope-encoded signals to bring in $B$, with adjustments to slopes ($1$, $-1$, $\tfrac{1}{2}$, $-\tfrac{1}{2}$) to ensure all sums $a+b$ are captured in the left column. The construction generalizes prior CA-based representations and yields a new CA-based method for obtaining $S+(S+S)$ from $S$ (the squares), thereby providing a constructive approach to integers expressible as sums of three squares and addressing related questions. This advances the understanding of how arithmetic operations on constructible sets can be realized within cellular automata, with implications for automata theory and combinatorics on words. The results demonstrate that Minkowski sums of CA-constructible sets are themselves CA-constructible, via explicit signal-based schemes.

Abstract

We give a construction in a column of a one-dimensional cellular automaton of the Minkowski sum of two sets which can themselves occur in columns of cellular automata. It enables us to obtain another construction of the set of integers that are sums of three squares, answering a question by the same author.

Figures (3)

• Figure 1: some instances of signals
• Figure 2: CA for $S$ (left) and $S+S$ (right)
• Figure 3: Construction of $S+(S+S)$ by a cellular automaton

Theorems & Definitions (3)

• Definition 1
• Theorem 2
• proof