Schema theorem

This basic difference equation - known as the Schema Theorem [holl92] - expresses the fact that the sample representation of schemas whose average fitness remains above average relative to the whole population increases exponentially over time. As it stands, however, this equation addresses only the reproduction operator, and ignores effects of both crossover and mutation. 

Finally, in order to also take into account the mutation operator, we note that the probability that a schema S survives under mutation is given by pu S) = (1 — Pm) where pm is the single-bit mutation probability and 0( S) is the number of fixed-bits (i.e. the order) or S. With this we can now express the Schema Theorem that (partially) respects the operations of reproduction, crossover and mutation ... 

The schema theorem provides a very powerful statement about the behavior of schemata in a chromosome. Mathematically, it states... 

We conclude from this basic theorem that the sample representation of low-order schemas with above average fitness relative to the fitness of the population increases exponentially over time. ... 

THEOREM 4.17 Every partially computable function can be computed using the set of program schemas obtained by applying one WHILE construction to a STEP scheme and using interpretations of function letters limited to S(x), Z(x), ... 

Techniques de construction et theoremes d existence en geometrie algebrique IV Les schemas de Hilbert, Seminaire Bourbaki expose 221 (1961), IHP, Paris. 

Plusieurs chapitres sont ensuite consacres A la discussion du theoreme enonce ci-dessus. L hypothese de normalite, faite sur S, semble essentielle nous donnons en effet, des exemples de schemas abeiiens sur une base locale, non normale, qui ne sont pas projectifs (chap. XII). L hypothese portant sur la con-nexite des fibres de G ne peut pas, elle non plus, Stre supprimee, mime si l on suppose X separe sur S. Toutefois, nous montrons qu un schema en... 

Theorems VII 2.1. Soient S un schema normal, G un S-schAma en groupes, lisse sur S, A fibres connexes, et soit X un S-espace homogene sous... 

Theoreme X 10. Soient S un schema, G un S-schema en groupes lisse, de presentation finie, separ sur S, k fibres maximales affines. Alors les conditions suivantes sont quivalentes ... 

Theorems 5 3 1 Soient S un schema affine, G un S-groupe affine et plat sur S. Les foncteurs Hn(G, ) sont les foncteurs derives d ... 

Theorems 5.3 3- Soient S un schema affine et G un S-groupe diago-nalisable. Pour tout G-Og-module quasi-coherent P, on a Hn(G, P) 0, n > 0. ... 

Theorem 2. Let Q be an SPJ query and let pM be a schema p-mapping. The problem of finding the probability for a by-tuple answer to Q with respect to pM is P-complete with respect to data complexity and is in PTIME with respect to mapping complexity. ... 

Theorem 4. Let pGM be a general p-mapping between a source schema S and a target schema f. Let Ds be an instance of S. Let Q be an SPJ query with only equality conditions over f. The problem of computing Qtable(Ds) with respect to pGM is in PTIME in the size of the data and the mapping. ... 

Expressive power A natural question to ask at this point is whether probabilistic mediated schemas provide any added expressive power compared to deterministic ones. Theorem 8 shows that if we consider one-to-many schema mappings, where one source attribute can be mapped to multiple mediated attributes, then any combination of a p-med-schema and p-mappings can be equivalently represented using a deterministic mediated schema with p-mappings, but may not be represented using a p-med-schema with deterministic schema mappings. Note that we can easily extend the definition of query answers to one-to-many mappings, as one mediated attribute can correspond to no more than one source attribute. 

In contrast, Theorem 9 shows that if we restrict our attention to one-to-one mappings, then a probabilistic mediated schema does add expressive power. 

Theorem 9. There exists a source schema S, a p-med-schema M, a set of one-to-one p-mappings pM between S and possible mediated schemas in M, and an instance D of S, such that for any deterministic mediated schema T and any one-to-one p-mapping pM between S and T, there exists a query Q such that 2m,pm(-0) Qt,pm(D). ... 

Note that the second part of Step 1 can map one source attribute to multiple mediated attributes thus, the mappings in the result pM are one-to-many mappings and so typically different from the p-mapping generated directly on the consolidated schema. The following theorem shows that the consolidated mediated schema... 

Theorem 10 (Merge Equivalence). For all queries Q, the answers obtained by posing Q over a p-med-schema M = M Mi) with p-mappings pM, ..., pMi is equal to the answers obtained by posing Q over the consolidated mediated schema T with consolidated p-mapping pM. ... 

Theorem 1 (Fagin et al. 2005b). Let Mi and M2 be two consecutive schema mappings. The following hold ... 

Additionally, the above theorem also applies in the context of source schema evolution, provided that the source evolution mapping M" has a chase-inverse. We summarize the applicability of Theorem 3 to the context of schema evolution as follows. 

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 ... 

As an immediate application of the preceding theorem, we conclude that the schema mapping M" in Sect. 5.1 has no chase-inverse, because A41 is a relaxed chase-inverse of M" but not a chase-inverse of M". 

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