Formal definition
BH is defined as follows: * BH1 is NP. * BH2''k'' is the class of languages which are theDerived classes
* DP (Difference Polynomial Time) is BH2.Equivalent definitions
Defining the conjunction and the disjunction of classes as follows allows for more compact definitions. The conjunction of two classes contains the languages that are the intersection of a language of the first class and a language of the second class. Disjunction is defined in a similar way with the union in place of the intersection. * C ∧ D = * C ∨ D = According to this definition, DP = NP ∧ coNP. The other classes of the Boolean hierarchy can be defined as follows. : : The following equalities can be used as alternative definitions of the classes of the Boolean hierarchy: : : Alternatively, for every ''k'' ≥ 3: :Hardness
Hardness for classes of the Boolean hierarchy can be proved by showing a reduction from a number of instances of an arbitrary NP-complete problem A. In particular, given a sequence of instances of A such that ''xi'' ∈ A implies ''x''''i''-1 ∈ A, a reduction is required that produces an instance ''y'' such that ''y'' ∈ B if and only if the number of ''xi'' ∈ A is odd or even: * BH2''k''-hardness is proved if and the number of ''xi'' ∈ A is odd * BH2''k''+1-hardness is proved if and the number of ''xi'' ∈ A is even Such reductions work for every fixed . If such reductions exist for arbitrary , the problem is hard for PNP 'O''(log ''n'')/sup>.References