On Equivalent Characterizations of NP in Abstract Models of Computation

We investigate machine models similar to Turing machines that are augmented by the operations of a first-order structure $\mathcal{R}$, and we show that under weak conditions on $\mathcal{R}$, the complexity class $\text{NP}(\mathcal{R})$ may be characterized in three equivalent ways: (1) by polynomial-time verification algorithms implemented on $\mathcal{R}$-machines, (2) by the $\text{NP}(\mathcal{R})$-complete problem $\text{SAT}(\mathcal{R})$, and (3) by existential second-order metafinite logic over $\mathcal{R}$ via descriptive complexity. By characterizing $\text{NP}(\mathcal{R})$ in these three ways, we extend previous work and embed it in one coherent framework. Some conditions on $\mathcal{R}$ must be assumed in order to achieve the above trinity because there are infinite-vocabulary structures for which $\text{NP}(\mathcal{R})$ does not have a complete problem. Surprisingly, even in these cases, we show that $\text{NP}(\mathcal{R})$ does have a characterization in terms of existential second-order metafinite logic, suggesting that descriptive complexity theory is well suited to working with infinite-vocabulary structures, such as real vector spaces. In addition, we derive similar results for $\exists\mathcal{R}$, the constant-free Boolean part of $\text{NP}(\mathcal{R})$, by showing that $\exists\mathcal{R}$ may be characterized in three analogous ways. We then extend our results to the entire polynomial hierarchy over $\mathcal{R}$ and to its constant-free Boolean counterpart, the Boolean hierarchy over $\mathcal{R}$. Finally, we give a characterization of the polynomial and Boolean hierarchies over $\mathcal{R}$ in terms of oracle $\mathcal{R}$-machines.