The 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... 

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... 

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. 

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". 

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), ... 

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. 

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 ... 

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 ... 

Theorem 7.9 shows that In a gTS all loop, p-a and end failures can be eliminated and the proof provides a procedure for constructing the reduced gTS. From a practical point of view, a recognition schema which Is "reduced" seems to be desired since the program will always terminate and it will never loop or halt before coming up with the final answer. 

These recurrence relations are then plugged by the so-called Basic Synthesis Theorem into a LISP program schema that reflects a divide-and-conquer design strategy. This yields the following LISP program ... 

The Synthesis Method is currently too closely entangled with the divide-and-conquer schema, with the synthesis mechanism described in the two previous chapters, and with the objective of Step 6 of that mechanism. In order to make it a full-fledged stand-alone method that can be added to a more general tool-box, some future research is needed to make it schema-independent. It might still rely on the MSG Method for the inference of an example set. But the inference of a property set could probably be made predicate-variable-independent and schema-independent by some theorem-proving task that exploits the integrity constraints of a schema. 

SafeCap offers a fairly compact core DSL. The basic element of a SafeCap schema is the definition of railway topology. The main concepts of the DSL are tracks, nodes, ambits (train detection units), routes, lines and rules. The SafeCap DSL is a formal language a schema is interpreted as a hybrid transition model - a model mixing continuous and discrete behaviours. The discrete part is employed to derive static verification conditions (theorems) and, as a supplementary technique, to help discover transition traces leading to the violation of safety conditions. The continuous part refines the discrete part with the notions of train acceleration/decelerafion, point switching and driver reaction times, and so on. 

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