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In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. [1] Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines.
The Turing jump of X can be thought of as an oracle to the halting problem for oracle machines with an oracle for X. [1]Formally, given a set X and a Gödel numbering φ i X of the X-computable functions, the Turing jump X′ of X is defined as
The nested set model is a technique for representing nested set collections (also known as trees or hierarchies) in relational databases.. It is based on Nested Intervals, that "are immune to hierarchy reorganization problem, and allow answering ancestor path hierarchical queries algorithmically — without accessing the stored hierarchy relation".
Using links, records link to other records, and to other records, forming a tree. An example is a "customer" record that has links to that customer's "orders", which in turn link to "line_items". The hierarchical database model mandates that each child record has only one parent, whereas each parent record can have zero or more child records.
In the context of Oracle Databases, a schema object is a logical data storage structure. [4] An Oracle database associates a separate schema with each database user. [5] A schema comprises a collection of schema objects. Examples of schema objects include: tables; views; sequences; synonyms; indexes; clusters; database links; snapshots ...
An oracle machine can be conceived as a Turing machine connected to an oracle. The oracle, in this context, is an entity capable of solving some problem, which for example may be a decision problem or a function problem. The problem does not have to be computable; the oracle is not assumed to be a Turing machine or computer program.
By induction, every set that is recursively enumerable by an oracle machine with an oracle for (), is in +. The other direction can be proven by induction as well: Suppose every formula in Σ p + 1 0 {\displaystyle \Sigma _{p+1}^{0}} can be enumerated by an oracle machine with an oracle for ∅ ( p ) {\displaystyle \emptyset ^{(p)}} .
Also, this construction is effective in that given an arbitrary oracle A we can arrange the oracle B to have P A ≤ P B and EXP NP A = EXP NP B = BPP B. Also, for a ZPP=EXP oracle (and hence ZPP=BPP=EXP<NEXP), one would fix the answers in the relativized E computation to a special nonanswer, thus ensuring that no fake answers are given.