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Lecture 16: coNP and PSPACE completeness
David Dill
Department of Computer Science
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•
coNP
•
PSPACE
2
Other complexity classes
The basic ideas that we have seen in NPcompleteness (reductions, complete
problems, etc.) have been applied to
many
other complexity classes (see
http://qwiki.stanford.edu/wiki/Complexity_Zoo
to get
an impression of just how many there are!)
We’ll talk about two of the most important classes that are also most closely
related to the NPcomplete problems: CoNP and PSPACE (called ps in the
textbook).
CoNP = “Complement of a problem in NP”
PSPACE = “Can be computed using polynomial space”
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View Full Document Closure properties
Closure under complementation is a very important topic in these classes.
A complexity class C is closed under complementation if,
L
∈ C
implies
L
is also
in
C
.
Theorem
P is closed under complementation.
This is easy to see. If
L
∈
P
, then there is a DTM that can compute it (and
always halts, of course) in polynomial time. Modify this DTM to swap the
accept
and
reject
results.
The book points out that this can be done at a cost of one additional step in the
computation, so the complemented DTM also runs in polynomial time.
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For a concrete example: Let
L
be the triples
a
G,v
0
,v
1
A
of a directed graph and
two vertices in the graph where there is a path from
v
0
to
v
1
in
G
.
L
= Σ
*
−
L
It includes the set of problems
a
G,v
0
,v
1
A
where there is
not
a path from
v
0
to
v
1
in
G
– plus any illformed inputs (e.g., the graph encoding is bogus, or there is
a missing separator between
G
and
v
0
or
v
0
and
v
1
).
Checking whether an input is wellformed is generally much easier than doing the
actual computation, and we only care about the hardest problems in a complexity
class. We will ignore the illformed members of
L
.
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This note was uploaded on 01/12/2010 for the course CS 154 at Stanford.
 '08
 Motwani,R
 Computer Science

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