On Interpretability in the Theory of Concatenation

Abstract

We prove that a variant of Robinson arithmetic $Q$ with nontotal operations is interpretable in the theory of concatenation $\mathrm{TC}$ introduced by A. Grzegorczyk. Since $Q$ is known to be interpretable in that nontotal variant, our result gives a positive answer to the problem whether $Q$ is interpretable in $\mathrm{TC}$. An immediate consequence is essential undecidability of $\mathrm{TC}$.