<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
algebraic	B-Application
semantics	I-Application
is	O
a	O
form	O
of	O
axiomatic	O
semantics	B-Application
based	O
on	O
algebraic	O
laws	O
for	O
describing	O
and	O
reasoning	O
about	O
program	B-Application
specifications	I-Application
in	O
a	O
formal	B-Architecture
manner	O
.	O
</s>
<s>
The	O
syntax	O
of	O
an	O
algebraic	B-Application
specification	I-Application
is	O
formulated	O
in	O
two	O
steps	O
:	O
(	O
1	O
)	O
defining	O
a	O
formal	B-Architecture
signature	O
of	O
data	O
types	O
and	O
operation	O
symbols	O
,	O
and	O
(	O
2	O
)	O
interpreting	O
the	O
signature	O
through	O
sets	O
and	O
functions	O
.	O
</s>
<s>
The	O
signature	O
of	O
an	O
algebraic	B-Application
specification	I-Application
defines	O
its	O
formal	B-Architecture
syntax	O
.	O
</s>
<s>
The	O
word	O
"	O
signature	O
"	O
is	O
used	O
like	O
the	O
concept	O
of	O
"	O
key	B-General_Concept
signature	I-General_Concept
"	O
in	O
musical	O
notation	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
the	O
signature	O
of	O
integer	O
stacks	B-Application
,	O
we	O
define	O
two	O
sorts	O
,	O
namely	O
,	O
and	O
,	O
and	O
the	O
following	O
family	O
of	O
operation	O
symbols	O
:	O
</s>
<s>
With	O
respect	O
to	O
the	O
signature	O
of	O
integer	O
stacks	B-Application
,	O
we	O
interpret	O
the	O
sort	O
as	O
the	O
set	O
of	O
integers	O
,	O
and	O
interpret	O
the	O
sort	O
as	O
the	O
set	O
of	O
integer	O
stacks	B-Application
.	O
</s>
<s>
Semantics	B-Application
refers	O
to	O
the	O
meaning	O
or	O
behavior	O
.	O
</s>
<s>
An	O
algebraic	B-Application
specification	I-Application
provides	O
both	O
the	O
meaning	O
and	O
behavior	O
of	O
the	O
object	O
in	O
question	O
.	O
</s>
<s>
The	O
semantics	B-Application
of	O
an	O
algebraic	B-Application
specifications	I-Application
is	O
defined	O
by	O
axioms	B-Algorithm
in	O
the	O
form	O
of	O
conditional	O
equations	O
.	O
</s>
<s>
With	O
respect	O
to	O
the	O
signature	O
of	O
integer	O
stacks	B-Application
,	O
we	O
have	O
the	O
following	O
axioms	B-Algorithm
:	O
</s>
<s>
The	O
mathematical	B-Application
semantics	I-Application
(	O
also	O
known	O
as	O
denotational	B-Application
semantics	I-Application
)	O
of	O
a	O
specification	B-Application
refers	O
to	O
its	O
mathematical	O
meaning	O
.	O
</s>
<s>
The	O
mathematical	B-Application
semantics	I-Application
of	O
an	O
algebraic	B-Application
specification	I-Application
is	O
the	O
class	O
of	O
all	O
algebras	O
that	O
satisfy	O
the	O
specification	B-Application
.	O
</s>
<s>
takes	O
the	O
initial	O
algebra	O
(	O
unique	O
up	O
to	O
isomorphism	O
)	O
as	O
the	O
"	O
most	O
representative	O
"	O
model	O
of	O
the	O
algebraic	B-Application
specification	I-Application
.	O
</s>
<s>
The	O
operational	O
semantics	B-Application
of	O
a	O
specification	B-Application
means	O
how	O
to	O
interpret	O
it	O
as	O
a	O
sequence	O
of	O
computational	O
steps	O
.	O
</s>
<s>
The	O
operational	O
semantics	B-Application
of	O
an	O
algebraic	B-Application
specification	I-Application
refers	O
to	O
how	O
ground	O
terms	O
can	O
be	O
transformed	O
using	O
the	O
given	O
equational	O
axioms	B-Algorithm
as	O
left-to-right	O
rewrite	O
rules	O
,	O
until	O
such	O
terms	O
reach	O
their	O
normal	O
forms	O
,	O
where	O
no	O
more	O
rewriting	O
is	O
possible	O
.	O
</s>
<s>
Consider	O
the	O
axioms	B-Algorithm
for	O
integer	O
stacks	B-Application
.	O
</s>
<s>
An	O
algebraic	B-Application
specification	I-Application
is	O
said	O
to	O
be	O
confluent	O
(	O
also	O
known	O
as	O
Church-Rosser	O
)	O
if	O
the	O
rewriting	O
of	O
any	O
ground	O
term	O
leads	O
to	O
the	O
same	O
normal	O
form	O
.	O
</s>
<s>
It	O
is	O
said	O
to	O
be	O
terminating	B-Application
if	O
the	O
rewriting	O
of	O
any	O
ground	O
term	O
will	O
lead	O
to	O
a	O
normal	O
form	O
after	O
a	O
finite	O
number	O
of	O
steps	O
.	O
</s>
<s>
The	O
algebraic	B-Application
specification	I-Application
is	O
said	O
to	O
be	O
canonical	O
(	O
also	O
known	O
as	O
convergent	B-Application
)	O
if	O
it	O
is	O
both	O
confluent	O
and	O
terminating	B-Application
.	O
</s>
<s>
Given	O
any	O
canonical	O
algebraic	B-Application
specification	I-Application
,	O
the	O
mathematical	B-Application
semantics	I-Application
agrees	O
with	O
the	O
operational	O
semantics	B-Application
.	O
</s>
<s>
As	O
a	O
result	O
,	O
canonical	O
algebraic	B-Application
specifications	I-Application
have	O
been	O
widely	O
applied	O
to	O
address	O
program	O
correctness	O
issues	O
.	O
</s>
<s>
For	O
example	O
,	O
numerous	O
researchers	O
have	O
applied	O
such	O
specifications	B-Application
to	O
the	O
testing	O
of	O
observational	O
equivalence	O
of	O
objects	O
in	O
object-oriented	B-Language
programming	I-Language
.	O
</s>
