Acknowledgements. We Thank Moshe Vardi for Suggesting the Extension from Corollary20. Relational Queries in Nrc

Extending relational algebra and rela-tional calculus with set-valued attributes and aggregate functions, ACM Transac-a tuple t in I, produces the output of the query on I ? ftg. Then both proofs in 9] show how to use this assumption to produce an expression in rst-order logic plus g that computes the transitive closure of a chain. Since the construction of 9] does not assume any auxiliary data, we can apply it here to obtain that, if either query is maintainable in rst-order in the presence of auxiliary data of moderate degree, then with such auxiliary data the transitive closure of a chain is computable, which contradicts Corollary 13. 2 Using essentially the same argument, but employing Corollary 21 in place of Corollary 13, we can also prove that Corollary 24. Neither transitive closure nor same-generation can be maintained in NRC aggr in the presence of auxiliary data whose degrees are bounded by a constant. 2 8 Future Work There are many open questions we would like to address in the future. We are interested in developing techniques for proving languages local. So far, there appears to be no commonality between Gaifman's proof of locality for rst-order 16] and our proof of (restricted) locality of NRC aggr. We also believe that this restriction can be eliminated, but we have not been able to prove it. Conjecture 1 Every relational query in NRC aggr is local. The previous results do not seem to apply to ordered structures: indeed, by taking any input and returning the graph of the underlying linear order, we violate the bounded degree property. Thus, it does not hold in NRC aggr (b), which is NRC aggr augmented with a linear order on type b. However, we still believe that the bounded degree property can be partially recovered: Conjecture 2 Every relational query in NRC aggr (b) that is order-independent has the bounded degree property. Theorem 2 to Theorem 6, and Tim Griin for a careful reading of the manuscript. Part of this work was done while Wong was visiting the University of Melbourne and Bell Laboratories. Wong would like to thank these organizations and fellow coauthors for their hospitality during this work. aggr have the bounded degree property. We immediately conclude from Corollary 20 that Corollary21. (cf. 23]) NRC aggr cannot express the following queries: (deter-ministic) transitive closure of a graph, connectivity test, testing for a (binary, ternary, …