Transcript LANDSCAPE OF COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M.

LANDSCAPE OF COMPUTATIONAL COMPLEXITY
Spring 2008
State University of New York at Buffalo
Department of Computer Science & Engineering
Mustafa M. Faramawi, MBA
A complete language for EXPSPACE:
PIM, “Polynomial Ideal
Membership”—the simplest natural
completeness level that is known
not to have polynomial-size circuits.
Dr. Kenneth W. Regan
Arithmetical Hierarchy (AH)
PIM
n
EXPSPACE
Succinct 3SAT
L ≠ NL …………………….. L ≠ PH
co-NEXP
p
K
p
P  EXP,


2
SAT
NLIN
NTIME [n2]
NP  NEXP
2
NP
TC0
PARITY
ACC0
CVP
AC0
P
P
Polynomial Hierarchy (PH)
GAP
NL
p
2
WS5
NC1
co-NP
FACT
The levels of AH and
PH are analogous,
except that we believe
NP  co-NP ≠ P and
p2  p2 ≠ PNP, which
stand in contrast to
RE  co-RE = REC and
2  2 = RECRE
REG
L
DTIME [n3]
DTIME [n2]
WS5
C
=
p
2
poly.P
=
Low-Level Classes
QBF
PNP
NP
B
Complexity “Main Sequence”
MAJ-SAT
D
Polynomial Time
co-NP

REG ≠ L
Deterministic and Nondeterministic
Time Hierarchies Within NP
PP Probabilistic
PSPACE
P
REG
WS5, the word problem for the symmetric group S5, is a regular
language that is complete for NC1 under AC0 many-one reductions.
BPP: Bounded-Error Probabilistic
Polynomial Time. Many believe BPP = P.
poly.P
= poly.P
= poly.P
AC0 ≠ ACC0
UGAP
etc
BQP: Bounded-Error Quantum
Polynomial Time. Believed larger
than P since it has FACTORING, but
not believed to contain NP.
p
TAUT
NP
= .REC
= .REC
PSPACE  EXPSPACE
p
D = {Turing machines Me:
Me does not accept e} =
the complement of K.
(“Diagonal Language”)
co-RE
REC
NC1  PSPACE, … , NL  PSPACE
PSPACE
D
RE
AC0  ACC0  PP, also TC0  PP
L
DLIN
= .REC
Best Known Separations:
QBF
For any fixed k,
there is a
problem in this
intersection that
can NOT be
solved by circuits
of size O(nk)
RECRE
NP ≠  2   2 ……… NP ≠ EXP
EXP
TOT = {Me : Me is total,
i.e. halts for all inputs}
2
= .REC
P ≠ NP  co-NP ……… P ≠ PSPACE
nxn Chess
n
TOT
2
Unknown
but Commonly Believed:
NEXP
A
K(2)
Analogy between
Arithmetical and
Polynomial Hierarchies
FACTORING is not
believed to be FACT
complete for BQP
or for NP  co-NP.
E

p
2
co-NP
NP
P
p
2
BQP
BPP
Realm of Feasibility?