Finite and Algorithmic Model Theory

106 Bart Kuijpers and Jan Van den BusscheFor the last part of the theorem, we remark that the query Q int on 1-dimensional datasets S that expresses “Is S a singleton that contains a naturalnumber?” is expressible in GCF-transitive-closure logic but not in first-orderlogic.2.12 Some concluding remarks on transitive-closure logicsOne of the motivations to study these different transitive-closure logics, is tocompare their expressive power and to establish which languages are computationallycomplete on linear or arbitrary (semi-algebraic) datasets.It is not clear whether transitive-closure logic with stop conditions is moreexpressive than transitive-closure logic without stop conditions. In particular,it is not clear whether transitive-closure logic without stop conditions is alsocomputationally complete on linear spatial datasets.We also remark that for CF-transitive-closure logic without stop condition,termination is decidable and for CF-transitive-closure logic with stop conditiontermination is not decidable. This does not separate these languages, however(because equivalence is undecidable).We also remark that many results on semi-algebraic functions also hold forarbitrary real closed fields. But termination of continuous semi-algebraic functionsf : R → R for arbitrary real closed fields R is not first-order expressible(for R the proof relies on Bolzano-Weierstrass).