Simulation of the Nested Relational Algebra by the at Relational Algebra, with an Application to the Complexity of Evaluating Powerset Algebra Expressions

Paredaens and Van Gucht proved that the at relational algebra has the same expressive power as the nested relational algebra, as far as queries over at relations and with at results are concerned. We provide a new, very direct proof of this fact using a simulation technique. Our technique is also applied to partially answer a question posed by Suciu and Paredaens regarding the complexity of evaluating powerset algebra expressions. Speciically, we show that when only unary at relations are into play, any powerset algebra expression is either equivalent to a nested algebra expression, or its evaluation will produce intermediate results of exponential size.