<s>
In	O
theoretical	O
computer	O
science	O
,	O
the	O
modal	B-Application
μ-calculus	I-Application
(	O
Lμ	B-Application
,	O
Lμ	B-Application
,	O
sometimes	O
just	O
μ-calculus	B-Application
,	O
although	O
this	O
can	O
have	O
a	O
more	O
general	O
meaning	O
)	O
is	O
an	O
extension	O
of	O
propositional	O
modal	O
logic	O
(	O
with	O
many	O
modalities	O
)	O
by	O
adding	O
the	O
least	O
fixed	O
point	O
operator	O
μ	O
and	O
the	O
greatest	O
fixed	O
point	O
operator	O
ν	O
,	O
thus	O
a	O
fixed-point	O
logic	O
.	O
</s>
<s>
The	O
(	O
propositional	O
,	O
modal	O
)	O
μ-calculus	B-Application
originates	O
with	O
Dana	O
Scott	O
and	O
Jaco	O
de	O
Bakker	O
,	O
and	O
was	O
further	O
developed	O
by	O
Dexter	O
Kozen	O
into	O
the	O
version	O
most	O
used	O
nowadays	O
.	O
</s>
<s>
It	O
is	O
used	O
to	O
describe	O
properties	O
of	O
labelled	B-Application
transition	I-Application
systems	I-Application
and	O
for	O
verifying	B-Application
these	O
properties	O
.	O
</s>
<s>
Many	O
temporal	O
logics	O
can	O
be	O
encoded	O
in	O
the	O
μ-calculus	B-Application
,	O
including	O
CTL*	O
and	O
its	O
widely	O
used	O
fragmentslinear	O
temporal	O
logic	O
and	O
computational	O
tree	O
logic	O
.	O
</s>
<s>
An	O
algebraic	O
view	O
is	O
to	O
see	O
it	O
as	O
an	O
algebra	O
of	O
monotonic	O
functions	O
over	O
a	O
complete	O
lattice	O
,	O
with	O
operators	O
consisting	O
of	O
functional	B-Application
composition	I-Application
plus	O
the	O
least	O
and	O
greatest	O
fixed	O
point	O
operators	O
;	O
from	O
this	O
viewpoint	O
,	O
the	O
modal	B-Application
μ-calculus	I-Application
is	O
over	O
the	O
lattice	O
of	O
a	O
power	O
set	O
algebra	O
.	O
</s>
<s>
The	O
game	O
semantics	O
of	O
μ-calculus	B-Application
is	O
related	O
to	O
two-player	O
games	O
with	O
perfect	O
information	O
,	O
particularly	O
infinite	O
parity	O
games	O
.	O
</s>
<s>
The	O
set	O
of	O
formulas	O
of	O
(	O
propositional	O
,	O
modal	O
)	O
μ-calculus	B-Application
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
The	O
notation	O
(	O
and	O
its	O
dual	O
)	O
are	O
inspired	O
from	O
the	O
lambda	B-Language
calculus	I-Language
;	O
the	O
intent	O
is	O
to	O
denote	O
the	O
least	O
(	O
and	O
respectively	O
greatest	O
)	O
fixed	O
point	O
of	O
the	O
expression	O
where	O
the	O
"	O
minimization	O
"	O
(	O
and	O
respectively	O
"	O
maximization	O
"	O
)	O
are	O
in	O
the	O
variable	O
,	O
much	O
like	O
in	O
lambda	B-Language
calculus	I-Language
is	O
a	O
function	O
with	O
formula	O
in	O
bound	O
variable	O
;	O
see	O
the	O
denotational	O
semantics	O
below	O
for	O
details	O
.	O
</s>
<s>
Models	O
of	O
(	O
propositional	O
)	O
μ-calculus	B-Application
are	O
given	O
as	O
labelled	B-Application
transition	I-Application
systems	I-Application
where	O
:	O
</s>
<s>
Given	O
a	O
labelled	B-Application
transition	I-Application
system	I-Application
and	O
an	O
interpretation	O
of	O
the	O
variables	O
of	O
the	O
-calculus	O
,	O
,	O
is	O
the	O
function	O
defined	O
by	O
the	O
following	O
rules	O
:	O
</s>
<s>
Less	O
formally	O
,	O
this	O
means	O
that	O
,	O
for	O
a	O
given	O
transition	B-Application
system	I-Application
:	O
</s>
<s>
Additionally	O
,	O
the	O
operator	O
can	O
be	O
interpreted	O
as	O
liveness	B-Application
(	O
"	O
something	O
good	O
eventually	O
happens	O
"	O
)	O
and	O
as	O
safety	B-Application
(	O
"	O
nothing	O
bad	O
ever	O
happens	O
"	O
)	O
in	O
Leslie	O
Lamport	O
's	O
informal	O
classification	O
.	O
</s>
<s>
Satisfiability	O
of	O
a	O
modal	B-Application
μ-calculus	I-Application
formula	O
is	O
EXPTIME-complete	O
.	O
</s>
<s>
Like	O
for	O
linear	O
temporal	O
logic	O
,	O
the	O
model	B-Application
checking	I-Application
,	O
satisfiability	O
and	O
validity	O
problems	O
of	O
linear	O
modal	B-Application
μ-calculus	I-Application
are	O
PSPACE-complete	O
.	O
</s>
