Hilbert’s tenth problem for term algebras with a substitution operator

The analogue of Hilbert’s 10th Problem for a first-order structure A with signature L asks whether there exists an algorithm that given an L-sentence of the form ∃ x → [ s = t ] decides whether ∃ x → [ s = t ] is true in A. In this paper, we consider term algebras over a finite signature with at least one constant symbol and one function symbol of arity at least two. We investigate the structure we obtain by extending the term algebra with a substitution operator. We prove undecidability of the analogue of Hilbert’s 10th problem without relying on the solution to the original Hilbert’s 10th Problem.