Embedded Finite Models Beyond Restricted Quantifier Collapse

Abstract

We revisit evaluation of logical formulas that allow both uninterpreted relations, constrained to be finite, as well as an interpreted vocabulary over an infinite domain. This formalism was denoted embedded finite model theory in the past. It is clear that the expressiveness and evaluating complexity of formulas of this type depends heavily on the infinite structure. If we embed in a wild structure like the integers with additive and multiplicative arithmetic, logic is extremely expressive and formulas are impossible to evaluate. On the other hand, for some well-known decidable structures, the expressiveness and evaluating complexity are similar to the situation without the additional infrastructure. The latter phenomenon was formalized via the notion of ``Restricted Quantifier Collapse'': adding quantification over the infinite structure does not add expressiveness. Beyond these two extremes little was known. In this work we show that the possibilities for expressiveness and complexity are much wider. We show that we can get almost any possible complexity of evaluation while staying within a decidable structure. We also show that in some decidable structures, there is a disconnect between expressiveness of the logic and complexity, in that we cannot eliminate quantification over the structure, but this is not due to an ability to embed complex relational computation in the logic. We show failure of collapse for the theory of finite fields and the related theory of pseudo-finite fields, which will involve coding computation in the logic. As a by-product of this, we establish new lower-bounds for the complexity of decision procedures for several decidable theories of fields, including the theory of finite fields. In the process of investigating this landscape, we investigate several weakenings of collapse.