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My understanding about Codd's concept of "safe queries" was created to ensure that a query would always terminate. One key ability of a Turing machine is that it can work on infinite calculations (and thus isn't guaranteed to terminate). If the safe query restriction were removed, would relational calculus be Turing-complete since that means it doesn't have to terminate?

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    Another thing: just because something allows writing non-terminating programs, that doesn't necessarily mean it's Turing-complete. Example: a programming language that has an infinite loop as its only construct. – 2010-03-27

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I don't know if this is correct, but I have a hypothesis. If we allowed free variables in the formula, then we could think of queries as functions. For example, consider the following query:

{  |  \in R and A = x}

If the above query has a result M, we can consider that to be a function of the form lambda x.M. If we allow relations to be free variables, we can define a function of the form lambda R.R. Now, if we allow "higher-order queries", ie queries that can query queries, we can represent the Church numerals (with q being a query):

0 = lambda qR.R 1 = lambda qR.q R 2 = lambda qR.q (q R) 3 = lambda qR.q (q (q R))

Therefore, I think that relational calculus can also be a lambda calculus, and therefore be Turing-complete. Can anyone tell me if I'm on the right path?