Complexity of Scott sentences

We give effective versions of some results on Scott sentences. Effectivizing a result of Montalban (2015), we show that if A has a computable Pi(alpha) Scott sentence, then the automorphism orbits of all tuples are defined by formulas that are computable Sigma(beta) for some beta < alpha. Effectivizing a result of A. Miller (1983), we show that if a countable structure A has a computable Sigma(alpha) Scott sentence and one that is computable Pi(alpha), then it has one that is computable d-Sigma(<)(alpha). We also give an effective version of a result of D. Miller (1978) on which the result of A. Miller was based. Using the non-effective results of Montalban 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 a computable finitely generated group has a computable d-Sigma(2) Scott sentence iff the orbit of some (or every) generating tuple is defined by a computable Pi(1) formula.