Computer Search for Counterexamples to Wilkie’s Identity

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Abstract

Tarski raised the High School Problem, i.e., whether a set of 11 identities (denoted by HSI) serves as a basis for all the identities which hold for the natural numbers. It was answered by Wilkie in the negative, who gave an identity which holds for the natural numbers but cannot be derived from HSI. This paper describes some computer searching efforts which try to find a small model of HSI rejecting Wilkie’s identity. The experimental results show that such a model has at least 11 elements. Some experiences are reported, and some issues are discussed.