Effective model theory: approach via σ-definability

Rice sequences of relations

“…Stukachev [15,21] then proved this theorem for the stronger notion of Sequivalence, as in the statement of the above-mentioned theorem, and for structures of arbitrary cardinality. Stukachev used the same structure that Soskova used, although the proof that it works for S-equivalence is completely different and more model theoretical.…”

“…This theorem is credited to Vaǐtsenavichyus [16] in Stukachev [15] and appears in some form in Beljaeve & Taȋclin [17]. There is also a natural way of going the other way around: from relations in HF A to sequences of relations on A.…”

“…For more background, see Barwise's book [14, ch. II] or Stukachev's survey paper [15]. Definition 4.1.…”

“…(2) h(R) is S-definable in HF A with parameters. [16] in Stukachev [15] and appears in some form in Beljaeve & Taȋclin [17].…”

Computable Functors and Effective Interpretability

“…Despite extensive study of effective interpretability, or Σ-definability, over the last couple of decades, the associated notion of bi-interpretability has not been considered until recently [Mon, Definition 5.2]. Let us remark that the notion of Σ-equivalence between structures, which says that two structures are Σ-definable in each other, has been studied ( [Stu13]), but the notion of bi-interpretability we are talking about is much stronger. Informally: two structures A and B are effectively bi-interpretable if they are effectively interpretable in each other, and furthermore, the compositions of the interpretations are ∆ c 1 -definable in the respective structures.…”