REVIEW 2 cited by
Complexity of Scott Sentences
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
Complexity of Scott Sentences
read the original abstract
We give effective versions of some results on Scott sentences. We show that if $\mathcal{A}$ has a computable $\Pi_\alpha$ Scott sentence, then the orbits of all tuples are defined by formulas that are computable $\Sigma_\beta$ for some $\beta <\alpha$. (This is an effective version of a result of Montalb\'{a}n.) We show that if a countable structure $\mathcal{A}$ has a computable $\Sigma_\alpha$ Scott sentence and one that is computable $\Pi_\alpha$, then it has one that is computable $d$-$\Sigma_\beta$ for some $\beta < \alpha$. (This is an effective version of a result of A. Miller.) We also give an effective version of a result of D. Miller. Using the non-effective results of Montalb\'{a}n and A. Miller, we show that a finitely generated group has a $d$-$\Sigma_2$ Scott sentence iff the orbit of some (or every) generating tuple is defined by a $\Pi_1$ formula. Using our effective results, we show that for a computable finitely generated group, there is a computable $d$-$\Sigma_2$ Scott sentence iff the orbit of some (every) generating tuple is defined by a computable $\Pi_1$ formula.
Forward citations
Cited by 2 Pith papers
-
Scott complexity of trees of finite rank via degrees of categoricity
Trees A_{m+1} of rank m+1 have Scott rank 2m+1 and exact Scott sentence complexity Π_in^{2m+2}, via a transfer principle from relativized categoricity hardness plus parameter reservation.
-
Scott complexity of trees of finite rank via degrees of categoricity
Computable trees of rank m+1 have Scott rank exactly 2m+1 with Scott sentence complexity in {Σ_{2m+1}, dΣ_{2m+1}, Π_{2m+2}}, and exactly Π_4 for rank 2.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.