What reasonable first-order queries are permitted by Trakhtenbrot's theorem?
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Around 1950, B.A. Trakhtenbrot proved an important undecidability result (known, by a pure accident, as \Trakhtenbrot's theorem"): there is no algorithm to decide, given a rst-order sentence, whether the sentence is satis able in some nite model. The result is in fact true even if we restrict ourselves to languages that has only one binary relation Tra63]. It is hardly conceivable that at that time Prof. Trakhtenbrot expected his result to in uence the development of the theory of relational databases query languages, but it did. This perhaps is not surprising in view of the following two facts: 1) The theory of relational databases is strongly rooted in logic and can easily be abstracted to make it a branch of mathematical logic. 2) The main interest in the theory of relational databases is in nite relations. In the rst section we explain those two points in more detail. Then, in section 2 of the present paper, we discuss the question: \What constitutes a reasonable query to a database?" The discussion naturally leads to a certain class of queries, known as the \domain independent" queries, as an ideal query language. At this point Trakhtenbrot's theorem appears as a major obstacle, since it easily implies that this ideal class is undecidable. Real query languages cannot be allowed to have this property. Section 3 outlines then several possible ways to circumvent this di culty. The following section explains the..
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