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v
0
) to denote the inequalities
For a variable
v 2 V
we write
clk
(
v
)and
clk
(
v
0
v
0
is present.
v 6
=
?
and
=
?
, implying that the signal
v
or
A
computation
of
A
=(
V;;
) is an innite sequence
=
hs
0
;s
1
;s
2
;:::i
,
with
s
i
2
V
for each
i 2
IN, which satises
s
0
j
=
and
8i 2
IN
:
(
s
i
;s
i
+1
)
j
=
.
Denote by
kAk
the set of computations of the
sts
A
.
Before we describe how to construct an
sts
P
corresponding to a given
P
Signal
program
, let us rst describe the primitives of a
Signal
program.
3.1
A Sketch of
Signal
Primitives
Signal
supports the following primitives.
1.
v
:=
f
(
u
1
;:::;u
n
)
function extended to sequences
2.
w
:=
v
$ init
w
0
shift register
3.
v
:=
u
when
b
data dependent down-sampling
4.
w
:=
u
default
v
merging with priority
5.
PjQ
program composition
In these primitives,
denote typed
signals
, i.e., sequences of values of the
considered type extended with the special symbol
u;v;w;b
?
.Inthe
when
expression,
the
b
signal is assumed boolean. In the last instruction, both
P
and
Q
denote
Signal programs.
The assignment
v
f
u
1
;:::;u
n
) can only be applied to signals
u
1
;:::;u
n
:=
(
v
which share the same clock. It denes a new signal
with the same clock such
that, for every
i
=1
;
2
;:::;v
[
i
]=
f
(
u
1
[
i
]
;:::;u
n
[
i
]).
The assignment
w
:=
v
$ init
w
0
dened a signal
w
with the same clock as
v
and such that
w
[1] =
w
0
and, for every
i>
1,
w
[
i
]=
v
[
i−
1].
The assignment
v
:=
u
when
b
denes a signal whose values equal the value
of
u
at all instances in which
u
and
b
are both dened and
b
=
t
rue.
The assignment
w
:=
u
default
v
denes a signal
w
which equals
u
whenever
u
is dened and equals
v
at all instance in which
v
is dened while
u
is absent.
Finally,
PjQ
is the composition of the programs (set of statements)
P
and
Q
.
3.2
System Variables
V
U [ X
U
The system variables of
are given by
=
,where
are the Signal
P
X
signals
explicitly declared and manipulated in
,and
is a set of auxiliary
memorization variables
, whose role is explained below.
3.3 Initial Condition
The initial condition for
is given by
:
^
u2U
u
=
?
By convention, the initial value of all declared signal variables is
. The initial-
ization of memorization variables is explained in the following subsection.
?
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