<s>
The	O
boolean	B-General_Concept
hierarchy	I-General_Concept
is	O
the	O
hierarchy	O
of	O
boolean	O
combinations	O
(	O
intersection	O
,	O
union	O
and	O
complementation	O
)	O
of	O
NP	O
sets	O
.	O
</s>
<s>
Equivalently	O
,	O
the	O
boolean	B-General_Concept
hierarchy	I-General_Concept
can	O
be	O
described	O
as	O
the	O
class	O
of	O
boolean	O
circuits	O
over	O
NP	O
predicates	O
.	O
</s>
<s>
A	O
collapse	O
of	O
the	O
boolean	B-General_Concept
hierarchy	I-General_Concept
would	O
imply	O
a	O
collapse	O
of	O
the	O
polynomial	O
hierarchy	O
.	O
</s>
<s>
The	O
other	O
classes	O
of	O
the	O
Boolean	B-General_Concept
hierarchy	I-General_Concept
can	O
be	O
defined	O
as	O
follows	O
.	O
</s>
<s>
The	O
following	O
equalities	O
can	O
be	O
used	O
as	O
alternative	O
definitions	O
of	O
the	O
classes	O
of	O
the	O
Boolean	B-General_Concept
hierarchy	I-General_Concept
:	O
</s>
<s>
Hardness	O
for	O
classes	O
of	O
the	O
Boolean	B-General_Concept
hierarchy	I-General_Concept
can	O
be	O
proved	O
by	O
showing	O
a	O
reduction	O
from	O
a	O
number	O
of	O
instances	O
of	O
an	O
arbitrary	O
NP-complete	O
problem	O
A	O
.	O
</s>
