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:
* BH
1 is
NP.
* BH
2''k'' is the class of languages which are the
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 ...
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
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 ...
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 BH
2.
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 ''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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