The expressiveness of looping terms in the semantic programming

We consider the language of $\Delta_0$-formulas with list terms interpreted over hereditarily finite list superstructures. We study the complexity of reasoning in extensions of the language of $\Delta_0$-formulas with non-standard list terms, which represent bounded list search, bounded iteration, and bounded recursion. We prove a number of results on the complexity of model checking and satisfiability for these formulas. In particular, we show that the set of $\Delta_0$-formulas with bounded recursive terms true in a given list superstructure $HW(\mathcal{M})$ is non-elementary (it contains the class $\mathrm{kExpTime}$, for all $k\geqslant 1$). For $\Delta_0$-formulas with restrictions on the usage of iterative and recursive terms, we show lower complexity.