<s>
The	O
X	O
in	O
"	O
X-machine	B-Application
"	O
represents	O
the	O
fundamental	O
data	O
type	O
on	O
which	O
the	O
machine	O
operates	O
;	O
for	O
example	O
,	O
a	O
machine	O
that	O
operates	O
on	O
databases	O
(	O
objects	O
of	O
type	O
database	O
)	O
would	O
be	O
a	O
database-machine	O
.	O
</s>
<s>
The	O
X-machine	B-Application
model	O
is	O
structurally	O
the	O
same	O
as	O
the	O
finite-state	B-Architecture
machine	I-Architecture
,	O
except	O
that	O
the	O
symbols	O
used	O
to	O
label	O
the	O
machine	O
's	O
transitions	O
denote	O
relations	B-Algorithm
of	O
type	O
X	O
→	O
X	O
.	O
</s>
<s>
Crossing	O
a	O
transition	O
is	O
equivalent	O
to	O
applying	O
the	O
relation	O
that	O
labels	O
it	O
(	O
computing	O
a	O
set	O
of	O
changes	O
to	O
the	O
data	O
type	O
X	O
)	O
,	O
and	O
traversing	O
a	O
path	O
in	O
the	O
machine	O
corresponds	O
to	O
applying	O
all	O
the	O
associated	O
relations	B-Algorithm
,	O
one	O
after	O
the	O
other	O
.	O
</s>
<s>
Eilenberg	O
's	O
original	O
X-machine	B-Application
was	O
a	O
completely	O
general	O
theoretical	O
model	O
of	O
computation	O
(	O
subsuming	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
,	O
for	O
example	O
)	O
,	O
which	O
admitted	O
deterministic	O
,	O
non-deterministic	O
and	O
non-terminating	O
computations	O
.	O
</s>
<s>
His	O
seminal	O
work	O
published	O
many	O
variants	O
of	O
the	O
basic	O
X-machine	B-Application
model	O
,	O
each	O
of	O
which	O
generalized	O
the	O
finite-state	B-Architecture
machine	I-Architecture
in	O
a	O
slightly	O
different	O
way	O
.	O
</s>
<s>
In	O
the	O
most	O
general	O
model	O
,	O
an	O
X-machine	B-Application
is	O
essentially	O
a	O
"	O
machine	O
for	O
manipulating	O
objects	O
of	O
type	O
X	O
"	O
.	O
</s>
<s>
Suppose	O
that	O
X	O
is	O
some	O
datatype	O
,	O
called	O
the	O
fundamental	O
datatype	O
,	O
and	O
that	O
Φ	O
is	O
a	O
set	O
of	O
(	O
partial	O
)	O
relations	B-Algorithm
φ	O
:	O
X	O
→	O
X	O
.	O
</s>
<s>
An	O
X-machine	B-Application
is	O
a	O
finite-state	B-Architecture
machine	I-Architecture
whose	O
arrows	O
are	O
labelled	O
by	O
relations	B-Algorithm
in	O
Φ	O
.	O
</s>
<s>
Each	O
recognised	O
path	O
through	O
the	O
machine	O
generates	O
a	O
list	O
φ1	O
...	O
φn	O
of	O
relations	B-Algorithm
.	O
</s>
<s>
We	O
call	O
the	O
composition	O
φ1	O
o	O
...	O
o	O
φn	O
of	O
these	O
relations	B-Algorithm
the	O
path	O
relation	O
corresponding	O
to	O
that	O
path	O
.	O
</s>
<s>
The	O
behaviour	O
of	O
the	O
X-machine	B-Application
is	O
defined	O
to	O
be	O
the	O
union	O
of	O
all	O
the	O
behaviours	O
computed	O
by	O
its	O
path	O
relations	B-Algorithm
.	O
</s>
<s>
For	O
practical	O
purposes	O
,	O
an	O
X-machine	B-Application
should	O
describe	O
some	O
finite	O
computation	O
.	O
</s>
<s>
Once	O
the	O
initial	O
state	O
of	O
X	O
is	O
populated	O
,	O
the	O
X-machine	B-Application
runs	O
to	O
completion	O
,	O
and	O
the	O
outputs	O
are	O
then	O
observed	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
more	O
recent	O
research	O
has	O
focused	O
on	O
deterministic	O
X-machines	B-Application
,	O
whose	O
behaviour	O
can	O
be	O
controlled	O
and	O
observed	O
more	O
precisely	O
.	O
</s>
<s>
These	O
relations	B-Algorithm
are	O
only	O
enabled	O
if	O
X	O
contains	O
the	O
domain	O
values	O
(	O
subtrees	O
)	O
on	O
which	O
they	O
operate	O
.	O
</s>
<s>
The	O
remaining	O
relations	B-Algorithm
SkipIncrement	O
and	O
SkipSubExpr	O
are	O
nullops	O
(	O
identity	O
relations	B-Algorithm
)	O
enabled	O
in	O
the	O
complementary	O
cases	O
.	O
</s>
<s>
When	O
referring	O
to	O
Eilenberg	O
's	O
original	O
model	O
,	O
"	O
X-machine	B-Application
"	O
is	O
typically	O
written	O
with	O
a	O
lower-case	O
"	O
m	O
"	O
,	O
because	O
the	O
sense	O
is	O
"	O
any	O
machine	O
for	O
processing	O
X	O
"	O
.	O
</s>
<s>
'	O
X-machines	B-Application
as	O
a	O
basis	O
for	O
dynamic	O
system	O
specification	B-Application
 '	O
,	O
</s>
<s>
69-76.xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx1	O
who	O
noticed	O
that	O
the	O
model	O
was	O
ideal	O
for	O
software	O
formal	B-Application
specification	I-Application
purposes	O
,	O
because	O
it	O
cleanly	O
separates	O
control	O
flow	O
from	O
processing	O
.	O
</s>
<s>
Provided	O
one	O
works	O
at	O
a	O
sufficiently	O
abstract	O
level	O
,	O
the	O
control	O
flows	O
in	O
a	O
computation	O
can	O
usually	O
be	O
represented	O
as	O
a	O
finite-state	B-Architecture
machine	I-Architecture
,	O
so	O
to	O
complete	O
the	O
X-machine	B-Application
specification	B-Application
all	O
that	O
remains	O
is	O
to	O
specify	O
the	O
processing	O
associated	O
with	O
each	O
of	O
the	O
machine	O
's	O
transitions	O
.	O
</s>
<s>
'	O
Formal	O
methods	O
in	O
the	O
specification	B-Application
of	O
the	O
human-machine	O
interface	O
 '	O
,	O
</s>
<s>
X-machines	B-Application
have	O
been	O
applied	O
to	O
lexical	O
semantics	O
by	O
Andras	O
Kornai	O
,	O
who	O
models	O
word	O
meaning	O
by	O
`pointed	O
 '	O
machines	O
that	O
have	O
one	O
member	O
of	O
the	O
base	O
set	O
X	O
distinguished	O
.	O
</s>
<s>
The	O
X-machine	B-Application
is	O
rarely	O
encountered	O
in	O
its	O
original	O
form	O
,	O
but	O
underpins	O
several	O
subsequent	O
models	O
of	O
computation	O
.	O
</s>
<s>
The	O
most	O
influential	O
model	O
on	O
theories	O
of	O
software	O
testing	O
has	O
been	O
the	O
Stream	O
X-Machine	B-Application
.	O
</s>
<s>
NASA	O
has	O
recently	O
discussed	O
using	O
a	O
combination	O
of	O
Communicating	B-Application
Stream	I-Application
X-Machines	I-Application
and	O
the	O
process	O
calculus	O
WSCSS	O
in	O
the	O
design	O
and	O
testing	O
of	O
swarm	O
satellite	O
systems	O
.	O
</s>
<s>
'	O
X-machines	B-Application
and	O
the	O
Halting	O
Problem	O
:	O
Building	O
a	O
super-Turing	O
machine	O
 '	O
.	O
</s>
<s>
The	O
most	O
commonly	O
encountered	O
X-machine	B-Application
variant	O
is	O
Gilbert	O
Laycock	O
's	O
1993	O
Stream	O
X-Machine	B-Application
(	O
SXM	B-Application
)	O
model	O
,	O
</s>
<s>
The	O
Stream	O
X-Machine	B-Application
differs	O
from	O
Eilenberg	O
's	O
original	O
model	O
,	O
in	O
that	O
the	O
fundamental	O
data	O
type	O
X	O
is	O
of	O
the	O
form	O
Out*	O
×	O
Mem	O
×	O
In*	O
,	O
where	O
In*	O
is	O
an	O
input	O
sequence	O
,	O
Out*	O
is	O
an	O
output	O
sequence	O
,	O
and	O
Mem	O
is	O
the	O
(	O
rest	O
of	O
the	O
)	O
memory	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
complex	O
software	O
systems	O
may	O
be	O
decomposed	O
into	O
a	O
hierarchy	O
of	O
Stream	O
X-Machines	B-Application
,	O
designed	O
in	O
a	O
top-down	O
way	O
and	O
tested	O
in	O
a	O
bottom-up	O
way	O
.	O
</s>
<s>
COMX	O
:	O
A	O
methodology	O
for	O
the	O
formal	O
design	O
of	O
computer	O
systems	O
using	O
Communicating	B-Application
X-machines	I-Application
.	O
</s>
<s>
</ref>	O
Earlier	O
versions	O
of	O
this	O
work	O
were	O
not	O
fully	O
formal	O
and	O
did	O
not	O
show	O
full	O
input/output	O
relations	B-Algorithm
.	O
</s>
<s>
The	O
ability	O
to	O
reassign	O
channels	O
meant	O
that	O
some	O
of	O
the	O
testing	O
theorems	O
from	O
Stream	O
X-Machines	B-Application
did	O
not	O
carry	O
over	O
.	O
</s>
<s>
These	O
variants	O
are	O
discussed	O
in	B-Application
more	I-Application
detail	I-Application
on	O
a	O
separate	O
page	O
.	O
</s>
<s>
Full	O
input/output	O
relations	B-Algorithm
can	O
be	O
shown	O
.	O
</s>
<s>
This	O
compositional	O
model	O
was	O
proven	O
equivalent	O
to	O
a	O
standard	O
Stream	O
X-Machine	B-Application
,	O
so	O
leveraging	O
the	O
earlier	O
testing	O
theory	O
developed	O
by	O
Holcombe	O
and	O
Ipate	O
.	O
</s>
<s>
This	O
X-machine	B-Application
variant	O
is	O
discussed	O
in	B-Application
more	I-Application
detail	I-Application
on	O
a	O
separate	O
page	O
.	O
</s>
<s>
Kirill	O
Bogdanov	O
and	O
Anthony	O
Simons	O
developed	O
several	O
variants	O
of	O
the	O
X-machine	B-Application
to	O
model	O
the	O
behaviour	O
of	O
objects	O
in	O
object-oriented	O
systems	O
.	O
</s>
<s>
This	O
model	O
differs	O
from	O
the	O
Stream	O
X-Machine	B-Application
approach	O
,	O
in	O
that	O
the	O
monolithic	O
data	O
type	O
X	O
is	O
distributed	O
over	O
,	O
and	O
encapsulated	O
by	O
,	O
several	O
objects	O
,	O
which	O
are	O
serially	O
composed	O
;	O
and	O
systems	O
are	O
driven	O
by	O
method	O
invocations	O
and	O
returns	O
,	O
rather	O
than	O
by	O
inputs	O
and	O
outputs	O
.	O
</s>
<s>
'	O
CCS-Augmented	O
X-Machines	B-Application
'	O
,	O
</s>
