Fine hierarchies of generic computation
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Suppose that you are a user of a commercial relational database, accessible over the Internet, whose owner has decided to copy the price lists of the US telephone companies — first order queries are for free just like local calls, because they are local by the theorem of Gaifman
These are the rules. Well, what is your strategy, to compute all you want to know about the database, paying as little as possible? And how much will the total price be?
We answer this question, showing that the question whether you can get your answer without any costs at all, depends on whether or not the theories of databases in the provided query language are finitely axiomatizable.
Thus, assuming there is a limit on the number of variables allowed in queries, if the database query language is the fixpoint logic, you can get everything for free. When it is Datalog however, even with inequality and negation of the edb's, you have to pay. We present a method, which for graphs of n vertices costs about $ log2 log2n. Thus querying a graph with 1 Terabyte vertices costs $7.00. We demonstrate that this price cannot be substantially reduced without causing a large computational overhead.
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