<s>
In	O
theoretical	O
computer	O
science	O
,	O
nondeterministic	B-Application
constraint	I-Application
logic	I-Application
is	O
a	O
combinatorial	O
system	O
in	O
which	O
an	O
orientation	O
is	O
given	O
to	O
the	O
edges	O
of	O
a	O
weighted	O
undirected	O
graph	O
,	O
subject	O
to	O
certain	O
constraints	O
.	O
</s>
<s>
The	O
constraint	B-Application
logic	I-Application
problem	I-Application
and	O
its	O
variants	O
have	O
been	O
proven	O
to	O
be	O
PSPACE-complete	O
to	O
determine	O
whether	O
there	O
exists	O
a	O
sequence	O
of	O
moves	O
that	O
reverses	O
a	O
specified	O
edge	O
and	O
are	O
very	O
useful	O
to	O
show	O
various	O
games	O
and	O
puzzles	O
are	O
PSPACE-hard	O
or	O
PSPACE-complete	O
.	O
</s>
<s>
This	O
is	O
a	O
form	O
of	O
reversible	B-Application
logic	I-Application
in	O
that	O
each	O
sequence	O
of	O
edge	O
orientation	O
changes	O
can	O
be	O
undone	O
.	O
</s>
<s>
In	O
the	O
simplest	O
version	O
of	O
nondeterministic	B-Application
constraint	I-Application
logic	I-Application
,	O
each	O
edge	O
of	O
an	O
undirected	O
graph	O
has	O
weight	O
either	O
one	O
or	O
two	O
.	O
</s>
<s>
More	O
general	O
forms	O
of	O
nondeterministic	B-Application
constraint	I-Application
logic	I-Application
allow	O
a	O
greater	O
variety	O
of	O
edge	O
weights	O
,	O
more	O
edges	O
per	O
vertex	O
,	O
and	O
different	O
thresholds	O
for	O
how	O
much	O
incoming	O
weight	O
each	O
vertex	O
must	O
have	O
.	O
</s>
<s>
Typically	O
,	O
constraint	B-Application
logic	I-Application
problems	I-Application
are	O
defined	O
around	O
finding	O
valid	O
configurations	O
of	O
constraint	O
graphs	O
.	O
</s>
<s>
The	O
reduction	B-Algorithm
follows	O
from	O
QSAT	O
and	O
is	O
outlined	O
below	O
.	O
</s>
<s>
This	O
reduction	B-Algorithm
follows	O
from	O
Planar	O
QSAT	O
.	O
</s>
<s>
The	O
proof	O
that	O
these	O
problems	O
are	O
hard	O
involves	O
a	O
reduction	B-Algorithm
from	O
quantified	O
Boolean	O
formulas	O
,	O
based	O
on	O
the	O
logical	O
interpretation	O
of	O
and/or	O
constraint	O
graphs	O
.	O
</s>
<s>
It	O
requires	O
additional	O
gadgets	B-Algorithm
for	O
simulating	O
quantifiers	B-Language
and	O
for	O
converting	O
signals	O
carried	O
on	O
red	O
edges	O
into	O
signals	O
carried	O
on	O
blue	O
edges	O
(	O
or	O
vice	O
versa	O
)	O
,	O
which	O
can	O
all	O
be	O
accomplished	O
by	O
combinations	O
of	O
and-vertices	O
and	O
or-vertices	O
.	O
</s>
<s>
The	O
proof	O
of	O
this	O
involves	O
the	O
construction	O
of	O
crossover	O
gadgets	B-Algorithm
that	O
allow	O
two	O
independent	O
signals	O
to	O
cross	O
each	O
other	O
.	O
</s>
<s>
Additionally	O
,	O
the	O
constraint	O
graphs	O
can	O
be	O
required	O
to	O
have	O
bounded	O
bandwidth	B-Algorithm
,	O
and	O
the	O
problems	O
on	O
them	O
will	O
still	O
remain	O
PSPACE-complete	O
.	O
</s>
<s>
The	O
reduction	B-Algorithm
follows	O
from	O
QSAT	O
.	O
</s>
<s>
In	O
order	O
to	O
embed	O
a	O
QSAT	O
formula	O
,	O
we	O
need	O
to	O
create	O
AND	O
,	O
OR	O
,	O
NOT	O
,	O
UNIVERSAL	O
,	O
EXISTENTIAL	O
,	O
and	O
Converter	O
(	O
to	O
change	O
color	O
)	O
gadgets	B-Algorithm
in	O
the	O
constraint	O
graph	O
.	O
</s>
<s>
The	O
other	O
gadgets	B-Algorithm
can	O
also	O
be	O
created	O
in	O
this	O
manner	O
.	O
</s>
<s>
The	O
original	O
applications	O
of	O
nondeterministic	B-Application
constraint	I-Application
logic	I-Application
used	O
it	O
to	O
prove	O
the	O
PSPACE-completeness	O
of	O
sliding	O
block	O
puzzles	O
such	O
as	O
Rush	O
Hour	O
and	O
Sokoban	B-Device
.	O
</s>
<s>
Nondeterministic	B-Application
constraint	I-Application
logic	I-Application
has	O
also	O
been	O
used	O
to	O
prove	O
the	O
hardness	O
of	O
reconfiguration	O
versions	O
of	O
classical	O
graph	O
optimization	O
problems	O
including	O
the	O
independent	O
set	O
,	O
vertex	O
cover	O
,	O
and	O
dominating	O
set	O
,	O
on	O
planar	O
graphs	O
of	O
bounded	O
bandwidth	B-Algorithm
.	O
</s>
<s>
Given	O
a	O
3-CNF	B-Application
formula	O
and	O
two	O
satisfying	O
assignments	O
,	O
this	O
problem	O
asks	O
whether	O
it	O
is	O
possible	O
find	O
a	O
sequence	O
of	O
steps	O
that	O
take	O
us	O
from	O
one	O
assignment	O
to	O
the	O
others	O
,	O
where	O
in	O
each	O
step	O
we	O
are	O
allowed	O
to	O
flip	O
the	O
value	O
of	O
a	O
variable	O
.	O
</s>
<s>
This	O
problem	O
can	O
be	O
shown	O
PSPACE-complete	O
via	O
a	O
reduction	B-Algorithm
from	O
the	O
Non-deterministic	O
Constraint	B-Application
Logic	I-Application
problem	I-Application
.	O
</s>
