Generic computation and its complexity

Abstract

There is a fundamental mismatch between the hardness of database queries and their Turing complexity. For example, the even query on a set has low Turing complexity but is by all accounts a hard query. The mismatch is due to the abstract, genen'c nature of database computation: data items which are indistinguishable by logical properties are treated uniformly. The issues specific to generic computation are obscured in the Turing model. Two models of generic computation are pr~ posed. They are extensions of Turing Machines with a relational store. The machines differ in the interaction between the Turing component and the relational store. The first, simpler machine, allows limited communication with the relational store. However, it is not complete. Nonetheless, it subsumes many query languages which have emerged as central, including the jixpoint and while queries. We prove a normal form for the machine, which provides a key technical tool. The normal form specialized to the while queries allows resolving the open problem of the relation of jizpoint and while: They are equivalent iff PTIME = PSPACE. The second machine allows extended communication between the relational store and the Turing component, and is complete. The machine involves parallelism. Based on it, we define complexity measures for generic mappings. We focus on notions of polynomial time and space complexit y for generic mappings, and show their robustness. The results point to a trade-off between complexity and computing with an abstract interface.