An Undecidable Property of Recurrent Double Sequences

Abstract

For an arbitrary finite algebra $\left(A,f\left(\cdot ,\cdot \right),0,1\right)$ one defines a double sequence $a\left(i,j\right)$ by $a\left(i,0\right)=a\left(0,j\right)=1$ and $a\left(i,j\right)=f\left(a\left(i,j-1\right),a\left(i-1,j\right)\right)$.

The problem if such recurrent double sequences are ultimately zero is undecidable, even if we restrict it to the class of commutative finite algebras.