The boolean hierarchy is the

hierarchy A hierarchy (from Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important ...

of boolean combinations (

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...

and complementation) of NP sets. Equivalently, the boolean hierarchy can be described as the class of boolean circuits over NP predicates. A collapse of the boolean hierarchy would imply a collapse of the polynomial hierarchy.

Formal definition

BH is defined as follows: * BH₁ is NP. * BH_2''k'' is the class of languages which are the

of a language in BH_2''k''-1 and a language in coNP. * BH_2''k''+1 is the class of languages which are the

of a language in BH_2''k'' and a language in NP. * BH is the union of the BH_i

Derived 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. :

\mathsf_ = \mathsf \wedge \mathsf_

\mathsf_ = \mathsf \vee \mathsf_

The following equalities can be used as alternative definitions of the classes of the Boolean hierarchy: :

\mathsf_ = \bigvee_^k \mathsf

\mathsf_ = \mathsf \vee \bigvee_^k \mathsf

Alternatively, for every ''k'' ≥ 3: :

\mathsf_k = \mathsf \vee \mathsf_

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 ''x_i'' ∈ 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 ''x_i'' ∈ A is odd or even: * BH_2''k''-hardness is proved if and the number of ''x_i'' ∈ A is odd * BH_2''k''+1-hardness is proved if and the number of ''x_i'' ∈ A is even Such reductions work for every fixed . If such reductions exist for arbitrary , the problem is hard for P^{NP 'O''(log ''n'')/sup>.

References

Hierarchy

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