Chase-equivalent

The main observation behind this reduction algorithm is that its output SO tgd (e.g., cto) is chase-equivalent to the input SO tgd (e.g., a). 

Definition 7. Let M i and M2 be two schema mappings from S to T that are specified by SO tgds (or in particular by GLAV mappings). We say that M and M2 are chase-equivalent if, for every source instance 7, we have that chaseMi(I) ... 

Theorem 5 (Yu and Popa 2005). Every SO tgd a is chase-equivalent to its reduced form cto. 

We also note that op represents a relaxation of the composition M o M (since op is chase-equivalent but not logically equivalent to ct, which expresses M o M ). Such relaxation of composition appears early in the work of Madhavan and Halevy [Madhavan and Halevy 2003],4 The concept used there is based, implicitly, on CQ-equivalence however, their results are limited to GLAV mappings, which, in general, are not powerful enough to express composition (even under the relaxed form) [Fagin et al. 2005b],... 

The callose-like material labelled rapidly, to reach a peak at about 14 hours of chasing and its radioactivity then slowly diminished. Label in the residual fraction was still increasing at 40 hours of chasing, while the precursor pool reached its maximal labelling at about 6 hours of chasing. Equivalent results were obtained when the radioisotope was supplied as CO2. The soluble precursors and callose were at their peak of labelling after one day of chasing, while the label in the residual fraction came to a steady value at five days. 

The schema mapping Mt used in Sect. 3.2 is an exact chase-inverse in the sense that it can recover the original source instance I exactly. In general, however, equality with I is too strong of a requirement, and all we need is a more relaxed form of equivalence of instances, where intuitively the equivalence is modulo nulls. In this section, we start with a concrete example to show the need for such relaxation. We... 

Thus, we recovered the two original facts of I but also the additional fact P(ni, n2) (via joining 2( i,2) and 2(2> 2))- Therefore, M is not an exact chase-inverse of M. Nevertheless, since n and n2 are nulls, the extra fact P(n, n2) does not add any new information that is not subsumed by the other two facts. Intuitively, the last instance is equivalent (although not equal) to the original source instance I. 

The existence of a chase-inverse for M implies that M has no information loss, since we can recover an instance that is the same modulo homomorphic equivalence as the original source instance. At the same time, a chase-inverse is a relaxation of the notion of an exact chase-inverse hence, it may exist even when an exact chase-inverse does not exist. 

As it can be seen, the recovered source instance U is not homomorphically equivalent to the original source instance there is a homomorphism from U to 7, but no homomorphism can map the constant CS in 7 to the null X in U. Intuitively, there is information loss in the evolution mapping M", which does not export the major field. Later on, in Sect. 5.2, we will show that in fact M" has no chase-inverse thus, we cannot recover a homomorphically equivalent source instance. 

Data exchange equivalence. First, we observe that the source instance U that is recovered by contains all the information that has been present in the original source instance I and has been exported by M". Indeed, if we now apply the mapping M" on U, we obtain via the chase an instance that is the same as J modulo null renaming (i.e., the chase may generate a different null C2 instead of ci). Thus, the following holds ... 

Like U, the instance U is data exchange equivalent to I with respect to M". (The only difference from U is in the major field, which is not used by the chase with M".) Furthermore, such instance U would be obtained if we use the following inverse instead of... 

Putting it all together, we now formally capture the two desiderata discussed above (data exchange equivalence and homomorphic containment) into the following definition of a relaxed chase-inverse. 

The notion of relaxed chase-inverse originated in Fagin et al. [2009b], under the name of universal-faithful inverse. The definition given in Fagin et al. [2009b] had, however, a third condition called universality, which turned out to be redundant (and equivalent to homomorphic containment). Thus, the formulation given here for a relaxed chase-inverse is simpler. 

Theorem 4. Let M be a GLAV schema mapping from a schema St to a schema S2 that has a chase-inverse. Then the following statements are equivalent for every GLAV schema mapping M from S2 to Sj ... 

The next step is to apply the chase and backchase algorithm to And rewritings of q that are equivalent given the union of the constraints in Ali and Al i. The following query over schema Si is such an equivalent rewriting and will be returned by the... 

We also remark that the language needed to express quasi-inverses requires disjunction. As a result, PRISM uses an extension of the chase and backchase algorithm that is able to handle disjunctive dependencies this extension was developed as part of MARS [Deutsch and Tannen 2003], Finally, we note that we may not always succeed in finding equivalent reformulations, depending on the input query, the evolution mappings and also on the quasi-inverses that are chosen. Hence, PRISM must still rely on a human DBA to solve exceptions. 

The CT reaction usually limits the rate of PC biosynthesis. The first evidence in support of this conclusion was drawn from the relative pool sizes of the aqueous precursors (in rat liver, choline = 0.23 mM, phosphocholine =1.3 mM, CDP-choline = 0.03 mM). Calculation of these values assumes that 1 g wet tissue is equivalent to 1 ml and that there is no compartmentation of the pools. The second assumption may not be valid as there is evidence for compartmentation of PC precursors (M.W. Spence, 1989). The concentration of phosphocholine is 40-fold higher than that of CDP-choline, consistent with a bottleneck in the pathway at the reaction catalyzed by CT. Pulse-chase experiments illustrate this bottleneck more vividly. After a 0.5 h pulse of hepatocytes with [methyl- H]choline, more than 95% of radioactivity in the precursors of PC was in phosphocholine, with the remainder in choline and CDP-choline. When the radioactivity was chased with unlabeled choline, labeled phosphocholine was quantitatively converted to PC (Fig. 5). The radioactivity in CDP-choline remained low during the chase and CDP-choline was rapidly converted to PC. There was minimal radioactivity in choline which suggests that choline is immediately phosphorylated after it enters the cell. It is important to note that if a cell or tissue is in a steady state, pool sizes and reaction rates do not change. Thus, although the rate of PC synthesis is determined by the CT reaction, the rates of the reactions catalyzed by choline kinase and cholinephosphotransferase are the same as that of the reaction catalyzed by CT, otherwise, the pool sizes of precursors would change. For example, if the choline kinase reaction were faster than the CT reaction, the amount of phosphocholine would increase. Thus, CT sets the pace of the pathway. 

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