Scott sentences for equivalence structures

For a computable structure A, if there is a computable infinitary Scott sentence, then the complexity of this sentence gives an upper bound for the complexity of the index set I (A). If we can also show that I (A) is m-complete at that level, then there is a correspondence between the complexity of the index set and the complexity of a Scott sentence for the structure. There are results (Calvert et al. in Algebra Logic 45:306325, 2006; Carson et al. in Trans Am Math Soc 364:5715-5728, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print; McCoy andWallbaum in Trans Am Math Soc 364:5729-5734, 2012) that suggest that these complexities will always match. However, it was shown in Knight and McCoy (Arch Math Logic 53:519-524, 2014) that there is a structure (a particular subgroup of Q) for which the index set is m-complete d - Sigma(0)(2), though there is no computable d - Sigma(2) Scott sentence. In the present paper, we give an example of a particular equivalence structure for which the index set is m-complete Pi(3) but for which there is no computable Pi(3) Scott sentence. There is, however, a computable Pi(3) pseudo-Scott sentence for the structure, that is, a sentence that acts as a Scott sentence if we only consider computable structures.