A SAT Attack on Tarski's High School Algebra Problem

Abstract

Tarski's high school algebra problem asks whether every true identity concerning addition, multiplication, and exponentiation of positive integers follows from a list of 11 elementary identities. Surprisingly, Wilkie showed that the following identity is valid over the positive integers and yet does not follow from Tarski's axioms: \begin{align*} &\left((1+x)^y + (1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x + (1+x^2+x^4)^x\right)^y = \\ &\left((1+x)^x + (1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y + (1+x^2+x^4)^y\right)^x. \end{align*} Gurevi\v{c} gave an algebra on 59 elements that satisfies Tarski's axioms but not Wilkie's identity, and over the years several authors whittled down the size of such a countermodel, culminating in a countermodel of size 12 due to Burris and Yeats. On the other hand, Zhang proved that there is no countermodel with fewer than 11 elements. Using SAT, we prove that the smallest countermodels are of size 12, as conjectured by Burris and Yeats. Moreover, we show that there are exactly 8,957,952 countermodels on 12 elements up to isomorphism and provide a simple classification of them. Our SAT approach outperforms dedicated tools for finding countermodels in equational theories, namely Mace4 and SEM. Furthermore, using autoformalization, we prove the correctness of our main result in Lean.