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This is primarily a reference request. Has any book or text studied definability in structures with formulas of a certain type? Like, sets defined by universal formulas, or existential formulas, or positive primitive formulas. Is there in fact a large research area associated with this topic?

user107952
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1 Answers1

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To push this off the unanswered queue:

Yes, quite a lot of work has been done on this general topic. Here are some relevant observations:

  • Quantifier elimination (or model completeness, or similar) is generally applied in conjunction with a theorem to the effect that we have a good understanding of the behavior of "simple" formulas. E.g. to prove that a theory $T$ is decidable, we might first show that the $T$-provability relation restricted to quantifier-free sentences is decidable, and then prove quantifier elimination. (Keep in mind that simplicity properties like this can always be "brute forced" in in a precise sense.)

  • There are various results about the classes of structures axiomatized by sentences of a certain form (e.g. that a $\forall$-axiomatizable class of structures is closed under taking substructures) and these have corresponding results about definable sets (e.g. that if $X=\varphi^\mathcal{M}$ with $\varphi$ a $\forall$-formula then whenever $\mathcal{M}\subseteq\mathcal{N}$ we have $\varphi^\mathcal{M}=\varphi^\mathcal{N}\cap\mathcal{M}$).

  • In a much narrower context, in appropriate structures the optimal quantifier complexity (modulo "bounded quantification" in the relevant sense) of a formula defining a given relation is tightly tied to the computational complexity of that relation; we see this in classical computability theory via Post's theorem, as well as in most (all?) forms of generalized computability theory.

There are many other examples, and if memory serves a lot of relevant information can be found in Hodges' big model theory book.

Noah Schweber
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