D. Marker, M. Messmer and A. Pillay. Model Theory of Fields, page 3. Springer-Verlag, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the length of y, by f ij ( a) c iff c is the unique element such that fl i j (c; a) holds. Furthermore, this function is semialgebraic and the following property holds: if is safe for D, then adom( D] is contained in S i;j S a f ij ( a) where a ranges over adom(D) It follows from [23] that each f ij ( y) is algebraic, that is, there exists a polynomial p ij (x; y) such that p ij (x; y) 0 iff x = f ij ( y) It is easy to see that for P = fp ij j i M; fl j 2 Gammag, Gamma(D) P (D) and P (D) is always finite. To complete the proof, we must show effectiveness. We can ....

.... formulae saying that each fl(x; y) has fewer than M satisfiers for each y, and checking if it is true by applying QE; since it is true for some M , the process terminates) Hence, we can effectively construct fl i j s, and the procedure for finding p ij s is effective (although not stated in [23], it follows from the analysis of the proof there) 2 4.4 Extensions Suppose we are given two elementary equivalent structures M and M 0 , for example, hR; i and hQ ; i. Assuming M is o minimal based on a dense order, so is M 0 , and thus the characterization of a safe queries applies ....

D. Marker, M. Messmer and A. Pillay. Model Theory of Fields. Springer Verlag, 1996.

....set, but variables x and y are not independent: this is because the only definable proper subsets of N are f1; 2g and N Gamma f1; 2g, and no Boolean combination of those gives us E. As another example, consider the field of complex numbers, whose theory is decidable and has quantifier elimination [18]. Let (x; y) x 2 1 = 0) y 2 1 = 0) x y = 0) It defines the finite set f(i; Gammai) Gammai; i)g but nevertheless x and y are not independent (since i is not definable) To avoid similar situations, we impose an extra condition on a structure, again, well known in model theory [4, ....

....has definable Skolem functions if for every formula ( x; y) there exists a definable function f ( x) with the property that M j= 8 x (9 y ( x; y) x; f ( x) In other words, f ( a) is an element of ( a; M) assuming ( a; M) is not empty. We say that a Skolem function f is invariant [18], if ( a 1 ; M) a 2 ; M) implies f ( a 1 ) f ( a 2 ) If the existence of such a Skolem function can be guaranteed for every , we say that M has definable invariant Skolem functions. Theorem 1. Assume that M has the following properties: a) its theory is decidable; b) M has ....

[Article contains additional citation context not shown here]

D. Marker, M. Messmer and A. Pillay. Model Theory of Fields. Springer Verlag, 1996.

....form. Moreover, for every FO Poly query , a collection of polynomials P can be effectively found such that and (P; are equivalent on databases on which is safe. Proof is based on Theorem 1, unform bounds for ominimal structures [26] and a characterization of semialgebraic functions in [19]. See appendix for details. 2 5 Deciding safety of conjunctive queries and relatives Safety of arbitrary calculus queries is undecidable even in the pure case [39] and of course it remains undecidable when interpreted functions are present. The main goal of this section is to show that safety is ....

D. Marker, M. Messmer and A. Pillay. Model Theory of Fields. Springer Verlag, 1996.

No context found.

D. Marker, M. Messmer and A. Pillay, Model Theory of Fields, Lecture Notes in Logic 5, Springer Verlag, 1996.

No context found.

D. Marker, M. Messmer and A. Pillay. Model Theory of Fields, page 3. Springer-Verlag, 1996.

No context found.

D. Marker, M. Messmer and A. Pillay. Model Theory of Fields, page 3. Springer-Verlag, 1996.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC