Conversion
of
to
Data
Control
Dependence
Depemdemcel
J.R.
Allen
Ken Kennedy
Carrie
Porterfield
Joe Warren
Department
of Mathematical
Sciences
Rice
University
Houston,
Texas
77251
Abatraet
presented
experimental
ty.
Program
analysis
methods,
especially
those
which
support
automat
ic
vector
izat ion,
are
based
on the concept
of
inte~statement
~,
where
a dependence
holds
between
two atatemente
when one
of
the
statements
computes
vslues
needed
by
the
other.
Powerful
program
transformation
systems
that
convert
sequential
programs
to
a form
more
suitable
for
vector
or parallel
machines
have been
developed
using
this
concept
[A1lK
82, KKLW 801.
‘
1.
The
in
for
seems
Fortran
This
language
,ahould
level
[AWSI
Unfortu~ately,
without
the
benefit
effectively.
from
either
vector
with
that
Support
Corporation.
for
this
research
was
provided
by
IBM
[A1lK
Permission to make digital or hard copies of part or all of this work or
personal or classroom use is granted without fee provided that copies are
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Permission
to and
copythe
without
fee allonorthe
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thk material
granted
bear
this notice
full citation
firstof page.
To copyisotherwise,
to
provided to
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the on
copies
are not
or distributed
for direct
republish,
servers,
or made
to redistribute
to lists,
requires prior
commercial
advantage,
ACM copyright notice and the title of the
specific
permission
and/orthe
a fee.
ik date appear, and notice is given that copyin~ is by
© publication
1983 ACMand
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as
@
1983 ACM
0-89791-090-7/83/001/0177
Fartran
177
a
in
explicit
programs
in
the
veccor
a transla-
@n!w2ez
77
level
translation
operations
and
--
high
d.evelop.ing
66
vector
(or
programs
PFC).
into
Fortran
8x
82].
is
natural
assignment
vector
iterators.
be
placQ
the
in
100
$00.75
or
language
invoked
This
~I
hardware
a
be
with
of
written
operations.
to
vector
we are
lines
vector
operations.
Rice,
vector
could
use
language
Mzal.kl
of
may
program
converts
assignment
permission of the Association for Computing Machinery. To copy
otherwise, or to republish, requires a fee and/or specific permission.
write
vector
8x were
translated
implicit
At
The
tions
to
vector
to
operations
Fortran
known
fix)
operations.
millions
Fortran
is
b.e
vector
the
operations.
~
utilize
explicit
machine
replace
original
tor,
of
must
vector
language
must
to
code
it
which
fully
many
prior
existing
equivalent
1
the
developed
this
as
81].
Fortran
If
to
standard
programmer
that
In
ANSI
vector
enable
programs
explicit
operations.
next
explicit
spawned
the
array
the
referred
such
has
permit
and
that
(hereafter
hardware
systematito
data
algorithms
an
architectures
units
that
vector
clear
contain
high
computer
processing
languages
of
it
of
vector
interest
will
IF statements.
The variables
in the conexpression
of
an
IF
statement
can
be
as
inputs
to
the
statement
being
conThe result
is
thst
control
dependence
statement
become
explicit
data
depenexpreesed
through
the
definitions
and uses
controlling
logical
variablea.
implemented
in
PFC,
written
at
Rice
Universi-
Vectorization
developraent
specification
to
presents
a method
for
control
dependence
this
fashion.
The
and
powerful
fact,
One solution
ie to cenvert
all
control
dependence
to
data
dependence
by
eliminating
goto
statements
and
introducing
logical
variables
to
control
the
execution
of
statements
in
the
proIn this
scheme,
action
statements
are
congram.
This
paper
cally
converting
dependence
in
been
Dependence
with
an
have
vectorizer
Motivation
1.1.
The dependence
analyais
in
these
systems
is
based
on data
dependence.
In the presence
of complex
control
flow,
data
dependence
is
not
sufficient
to tranaform
programs
becauae
of
the
introduction
of ~
~
A control
depen.
dence
exista
between
two statements
when the
execution
of one statement
ean prevent
the
execution
of the other.
Control
dependence
do not
fig
conveniently
into
dependence-based
program
translators.
verted
ditional
viewed
trolled.
between
dence
of the
here
to
Fortran
a
DO
loop
by
for
wmtld
For
to
example,
be
the
the
100
+ C
the
vector
Ideally,
loop.
translating
DO 100 I ❑ 1,
A(I)
= A(I)
CONTINUE
translated
look
DO
statement
converted
subscripts
loop
operaeach
to
a
to
A(I:1OO)
However,
the
simple,
in
Fortran
while
loads
8X
acid
require
before
‘fetch
a
in
quite
store.”
a loop
right
of
following
SI
That
behaves
side
left
hand
as
are
illustrates
the
puted
on
next,B
tor
= 1,
intent
is
one
the
its
own
reeults;
S1
fetch
from
the
statement
of
used
cannot
the
A(l:IOO)
be
of
as
A
input
cycle)
disappears
S1
simulated
by
to
a vec-
“fetch
the
intermix
losds
its
effe(ct
one
iteration
be
a
in
that
loop
same
a
by
and
as
link
a
from
called
on
Kuck
relationship
of
the
value
graph
the
an
if
there
to
upon
Xi
upon
a
i.,
b.e
?FC
called
associates
part
iculsr
S1 on
itself
each
loop.
in
loop
does
haa
also
hence
the
run
sequenin
the
J
= A(I-1,1:10)
+
. . .
Data
a
all
zation.
Consider
of
and
follow-
OS’i~n.
by
on
on
determines
terms,
does
a
ali.ghtly
wore
~-cazx.id
dependence
For
with
the
the
to
vectori-
the
whether
themselves
the
vectorized
~
@&udeQw
of
the
test
S2
will
be
executed.
ia
considered,
both
dependence
dependence
in
When
SI
is
same
does
machinery
not
and
loop
is
not
of
associated
PFC
with
rewritten
DOIOOI=l,
N
BR1 = A(I).G’T.O
IF (BR1) A(I+l)
not
itself
Nevertheless.
outcoute
control
the
When
on
indirectly.
dependence
a
a
directly
depend
correctly
of
depeudcnce
.the
be
preclude
S1
inputs.
can
is..
into
not
= B(I)
any
as
+ 10
CONTINUE
sophisticated
dependence..
example,
S2
would
not
its
existence
unfortunately,
variable.
in
100P:
does
of
That
on
nicely
directly
it
S2.
control
because
if
801.
sufficient
Certainly
S2
any
nor
S1
fit
not
dependence
S2.
But
the
S2 depend
S1
is
following
data
for
Sl
Xl . . ...*Xn
these
S1
of
statements
In
dimensions
[Kenn
considerations
the
s2.
51
the
uses
on
of
uimn
only
of
relations.
alcne
of
neitiler
S1 might
depends
number
Ken-
vectorize
CONTINUE
of
because
S2 ~
Xi+l
vectorized
flow
S2 &r.QcLQ
computed
time.
ox
often
that
theoxy
depends
demotes
created.
to
DOIOOI=l.
N
IF (A(I).GT.0)
GO TO 100
A(I+l)
= B(I)
+ 10
vectorization
vertex
are
S2
maximum
important
100
rela-
data
However,
dependence,
algorithm
dependence
dependence
describe
casea
each
loop-carried
Depemience
itself.
employs
it
10
recursive
the
100
concepts
incremented
that
vectorized
= 1,
of
its
S1
S.2
variables
~
value
xun.
~f
statement
have
the
Xn=529
all
can
depend
term
in
hy
between
the
links
78].
a se.quenco
for
statement
the
if
XO=S1,
I
correctly
I
a
oth-
two
edge
these
[Keun
SZ at
exists
that
on
or
itera-
values
uses
each
use.;
S~ might
S1
input
such
to
77].
that
is
concept
Control
ve.ctorized.
which
in
and
between
WQll
be
and
statement
[Kuck
the
A(I,l:IO)
1.2.
value
another
these
flow
definitions
W
ing
&2f211h
a
be
between
definition
the
(and
use
so
directly
models
fii-~
dependence
be
developed
The
directed
represents
the
CONTINUE
permitted
definitions.
between
a
array,
tranaliterati.on;
classical~~~
tionship
ia
any
not
produce
statements
not
aemantica.
used
may
the
did
same
is
statement
of
contraa.t,
computes
statement
study
By
loop
vector
that
distinguishing
requires
the
with
vect.orized
the
Correctly
example
stores
the
the
be
erwise,
semantics.
first
statement
by
cannot
Within
S1 does
A
51
Using
+ B(I:1OO)
simulated
of
indirectly
uses
our
and
can
Thufs,
tion.
storew
in
I.
loop.
when
when
can
on
I
form
=A(O:99)
before
statement
loop
the
I
100
nedy
with
. . .
of
DO 100
the
+
location
Since
com-
on
the
of
only
in.
tially,
component
be
to
a
stored
loop
that
due
iteration
side
+ B(I)
iteration
statement
clearly
this
100
A(I)
=A(I-1]
CONTINUE
100
Since
I
ia
= A(I-l,J)
CONTINUK
CONTINUE
specific
difference.
DO 100
A(I,J)
90
100
intermixes
hand
the
loop
that
assignment
assignment
the
component
The
not
vector
before
vector
of
any
stored.
of
assignment
stores.
DO 100 I = 1, 10
DO 90 J = 1, 10
is
semantica
components
fetched
+ C
process
the
a scalar
all
is
translation
because
is,
if
=A(I:IOO)
ite.rati.on
dependence
(thereby
which
of
a
of
able
viewing.
associating
5R1)
the
the
to
these
two
flow..
Control
178
to
the
s.tatementa
data
dependence
becomes
variables
S2 a.a inputs
verted
the
problem
in
the
conditi.o.n.
statement,
ia
dependence
dependence.
now
has
xith
much
the
clear
been
the
vari-
simpler.
By
controlling
re.lationehip.
in
..terms
completely
of
of
data
con-
The
beauty
assignments
of
this
are
the
scalar
For
example,
example
scheme
is
that
straightforward
condition
are
consider
conditional
to
expanded
a
slight
execution
vectorize
into
if
arrays.
variation
on
our
This
by
version
using
could
the
be
.
Fortran
BR1(l:N)
Many
vector
ditional
machines
have
vector
to
select
tion
ia
to
the
By
generalizing
be
converting
dence.
PFC
cent a in
ormat
2.
can
idea
a
the
into
a
to
vectorize
con-
is
to
IF
IF
method
data
for
true.
The
can
in
loops
The
~
for
~
~
some
or
change
produce
ples:
(2)
-in
the
some
is
the
notion
~~
explicit
Eion
transfer
in
puted
the
goto,
crested
a
an
of
module
control
is
guard
can
be
ments
to
be
of
be
that
a block
Example:
an
call
~:
more
loops.
take
no
action
placeholdere
but
for
which
the
be
is
that
classification.
our
the
IF
to
any
the
clause
action
words,
every
viewed
as
Zhe
eliminate
Fortran
as
all
statement
reason
a qual,itier
or
or
has
that
can
be
statement,
branch
used
no
to
eit,h.er
a
the
same
the
label
phase
statements
as
(3)
of
PFC
the
identically
IF
the
condition.
If
it
true,
qualifier.
analysis.
branches
can
be
one
or
types:
terminates
that
in
but
.LE.
at
that
a
DO
a
label
forward
is
whose
the
since
loop
DEL)
GOTO 200
into
branch
(if
the
an
the
nesting
the
permitted.
the
or
outside
occurs
loop
not
after
level)
target
same
branches
are
branch
nesting
a
must
label
exit
loop
DO 100 I = 1, 10
IF (A(I).EQ.0.0)
B(I)
= B(I)
/A(I)
CONTINUE
b.adwwi
~
occurring
same
be
is
at
branch
(if
which
the
in
GOTO 100
other
be
attempts
program.
to
The
nesting
In
level.
(1)
uses
179
with
two
branebee
as
a
statement
branch
but
at
in
the
+ B(I)
100) GOTO 10
this
cl.assificat.ion,
different
within
Br.aluhzduatim
until
to
the
1=1+1
accordance
sion
branch
before
A(I)
=A(I)
IF (1 .LE.
attached
can
an
lexic,ally
10
place
inate
in
which
occurs).
we view
In
etatement
branches)
in
which
Example:
is
aa
Note
which
statement.
conversion
goto
that
branch
action
a conditional
IF
IF
The
of
branch
branch
the
in
the
three
of
state-
CONTINUE .
Notice
is
a branch
the
DO statement.
cao
action
conditional
a branch
level.
a macro-
of
computation.
replacing
. . .
CONTINUE
. . .
CONTINUE
range
within
statements
--
by
original
statement
purpose
ti
after
com-
staLement$
or
Qa@lQkX@itQwMa
evaluates
conditional
statements
the
~~:
loca-
because
aa
to
into
200
make
goto,
viewed
IF
without
100
(4)
statement
guard
transformed
than
by
a
the
100
(2)
another
Note
‘iterated.
ite
the
control
Exam-
which
statement
statement
when
original
if
with
(other
written
computation
t,o
goto.
~
another
the
which
statement
branch
cause
conditional
which
is,
DO 100 I = 1, 100
IF (ABS(A(I)-B(I))
action.
(3)
or
expression
under
only
is
conjoined
the
cause
effect.
Examples:
may
a
form:
replaced
For
call.
statements
it
the
write,
of
action
and
the
(guard)
four
which
side
--
assigned
ae
given
atate
program.
each
called
that
into
statements
read,
action
That
statements
of
be
(1)
classified
important
assignment,
ia
Boolean
conditions
program
action
groups:
(1)
main-
for
c1
statements
can
this
Conweraion
be
an
a
statement,
original
simple
guard
depen-
responsible
conversion
Statements
for
is
if
categorized
of
is
condition
condition
executed.
the
to
IF
ion.
Central
is
statements
logical
transfera.
PFC
@
the
executed
computa-
statements
conditional
of
The
ie
IF
Fundamentals
Fortran
via
which
dependence
phaae
transf
in
+ 10
support
applied.
which
program
logical
This
statement
reaches
form
= B(l:N)
to
usually
this
original
a
O
hardware
positions
cor@rol
~
vector
A(I:N)
operations,
mask
to
= A(l:N).GT.
(BR1(l:N))
.
statement
8x WHERR statement.
wHEM
.
represents
+ 10
transformed
the
statement.
action
= B(I)
of
computing
WarJi.
~:
100
by
action
loop.
DOIOOI=l,
N
BR1 ❑ A(I).GT.O
IF (BR1) A(I)
CONTINUE
order
tained
branch
II?
transformations
the
elim-
of
loops
program.
moves
and
converto
its
branchee
target
out
are
nested
in
the
same
number
converts
wsrd
(2)
each
branch
~
of
or
branch
under
execution
on
branchea
left
action
logic.
and
by
Backward
points
following
niques
in
sections
more
present
these
two
tech-
Exit
‘A(I).LT.OW
is
branches
differ
exit
branches
sffect
ments
both
since
the
before
and
a branch
tion
of
the
out
it
from
the
after
of
loop,
loop.
the
which
branches
execution
of
the
the
the
following
guard
100
200
Once
and
the
jump
neither
is
In
ia.
execu-
statement
order
the
to
iae.ue
between
in
example:
separate
of
thereafter.
If
the
doea
time.
DO loop
is
S2 will
were
not
their
terminated
be
present,
nally
pro-
is
exit
branches
within
If
all
other
every
as
eliminate
may
somehow
is
in
branchea
the
remnval
same
with
and
elimination
!iu3.cd
expression
evaluates
would
be
the
and
clause.
PFC
data
true
executed
converting
the
for
t.o
Fortran
only
is
can
then
in
each
onl,y
in
the
original
to
a
as
test
of
by
an
fied
forward
if
logical
a
for
a
guard
in
vectorization
we
one
that
is
employ
the
simplification
external
represen-
guard
erp.a 1
allows
the
IF
us
[Quin
52,.
to
delay
con-
issues
until
Sec-
order
dependence.
introduce
180
~
if
not
have
statement
compute
a
in
a
e~i
the
br.anc.h
haa
been
generally.
Eoolean
variable
is
taken
the
original
for
logical
modi-
not
expression
been
realization
corresponding
guarded
an
branch
fla,g
.th,e
the
sxk..~s
is
particular
by
of
that
be
in
More
guarded
f&g
statement
loop
true
branch
of
T.h,e, exit
a
the
be
a
che
be
program.
will
would
to
while
movement
xeq,uir,ea.
in
will
conjunction
with
the
lc@p
statement
which
an
a
flag
a
original
program
at
rehcakicnh
of
only
the
w~ere
reached
using
each
the
is
branch
out
statement
true
in
an
of
associat.e.d
nal
By
expression
in
in
condi-
~ might
simplification
to
b.r.anch
pr,o.gram
exi.,
statement
program.
condition
realization
choose
simple
~
we
internal
representations;
will
int
of
any
expression
which
Boolean
This
the
use
of
execution
DO
then
relocation
statement.
and
it
branch
computation
if
guard
using
used
given
external
internal
implicant
a
our
inter-
the
the
a
An
example.
generate
a
guards
map
to
~
and
by
Rotationally,
7).
to
external
that
conditions
from
genexated.
Far
The
exit
each
proc,eduxe
a reason-
mirrored
the
~
prime
to
5610
taken
basis
however,
in
of
different
many
.Rsturning
an
so
will
naturally
a provably
6.
In
branches
exactly
branch
output
prob-
simpler.
the
nested
the
inter-
the
representation
is
conditi~~s
as.swne
of
tion
branches.
‘Jhe
of
si.d.eration
converted
then
much
~
can
be
guards
in
find,
logic
Section
being
haye
tation
atate-
from
distinguish
Fortran
function
concise.
McC1
branches.
target.
these
all
correctness
will
compute
represent
quite
(ace
procedure
them
of
several
statement;
we can
the
We
form
Quine-McCluskey
complicated
can
concise
of
language
suitably
the
eliminating
becomes
PFC
branch
its
more
guarda
can
backward
if
by
exited.
conversion
words,
the
branches
or
IF
loops
loop
exit
forward
of
of
the
are
since
modification
menta
unaffected
branches.
forward
requires
completely
of
representation
the
therefore
Thus.
than
that
moat
ion.
a
use
tion
we
that
duality
in
the
S1
branch.
into
imply
rep,re,een,tation
GOTO 200
S2
S3
S4
statement
lem
a particular
representation
200
con(-t).
of
represent
logic,
external
will
(X(I))
are
guard.
implemental
executed
ducing
S1
IF
opera-
negation
appearance
for
the
that
to
our
not
The
DO loop
be
The
conjunction
we
actual
guard
this
for
nor
If
past
certainly
and
issue
used
their
In
correct
the
the
the
logic
and
able
S1
jump
predicates
simplification,
the
nally
GOTO 200
taken.
a
P1 A P2 A ~P3
1,100
statement
to
on
300.W
p should
(V)
be
true
statement
statement.
applied
might
was
one
e.g.
language.
DO 100 I ❑
S1
IF (X(I))
S2
CONTINUE
S3
S4
he
at
example,
for
that
are
hold
state-
terminates
all
of
formal
may
‘A(I).LT.O
disjunction
in
That
branch.
a DO loop
affects
Consider
other
=
predicate
for
may
(A).
a
p
condition
the
guard
predicates,
Exit
that
350.
of
junction
branches
For
is
Hence
3.
program.
execution
of
system
that
the
the
a system
logical
conditions
recent
tion
detail.
in
on
this
predicate
statement
place.
based
of
most
the
conditioning
PFC are
atoms
expressing
various
part
The
in
The
possible
state-
expressions.
in
Guards
for-
branches
for
control
these
are
a
predicates
forward
expressions
their
procedure
either
branch.
eliminates
guard
This
into
a backward
~
computing
ments
DO loops.
exit
denoted
in
the
origi-
defined
to
associated.
before
be
with
control
program.
exis
variable
we
In
will
EXi
into
the
program.
EXi
condition
controlling
condition
is
will
uae
will
so
convention
represent
conditions
represent
their
used
loop
evaluated.
the
be
the
to
exit
that
~(exi)
that
lower
and
capture
each
realizations
there
entry
the
case
of
is
one
exit
to
the
loop.
loop
branch
has
of
within
the
logical
flag
of
is
that
the
A new
branch
of
EXi)
that
by
=
the
the
the
will
have
be
been
the
ita
moat
100
the
in
taken
taken
of
by
conjoining
(including
the
loop.
the
S6
outer
inner
set
Aex2A
. . . Aexn.
program
ever
so
the
program
flag
that
would
be
will
cent
an
is
exit
have
been
no
executed,
to
taken,
the
set
branch
other
even
run
arrange
ia
exit
false,
inue
to
flag
corresponding
becomes
will
effect
to
false
in
the
Thus,
the
when-
in
DO
an
the
by
which
is
is
the
The
statement
in
example
after
reloca-
100
200
.TRUE.
I = 1,100
(EX1)
S1
(EX1)
EX1
(EXI)
S2
CONTINUE
IF (.NOT.
S3
S4
EX1)
= .NOT.
loops
loop
sider
by
as
the
method
treating
a branch
following
a
out
by
couputing
(which
those
loops)
branch
out
of
outermost
more
the
complicated
of
more
multiple
than
loop.
any
has
loop
can
that
be
implicitly
that
is
branch
left
is
exit
set
ter-
branch
flags
can
for
be
the
every
the
to
The
loop
guards
the
is
one
the
the
for
of
is
called
this
nested
DO
statements
loop
guard
other
algorithm
guard
to
for
given
statement
recursively
conjoining
loop
its
relocation
on
true.
itself
then
curxent
branch
initially
computes
guard
every
in
for
statement
control.
After
will
for
guard
is
loops
the
X(I)
to
the
IF
aPPIYing
GO TO 200
extended
branch.
The
looPs
under
easily
in
have
loop
exit
1.
the
statement
is
a
scanning
algorithm
an
at
procedure
the
remain
corxec.tnesa
This
if
exit
are
flag
false.
Figure
proceeds
tion:
EX1 =
DO 100
IF
IF
IF
an
flag
would
exit
loop
iterations.
previous
exit
to
to
flags
any
an
the
respect
exit
once
that
jump
by
Y I,J)
respect
with
the
is,
other
Hence,
identified
than
Here
no
XI I,J)
applying
with
jump
that
that
(indicating
false.
minated
exit
becomes
by
jump
second
Note
loop.
falae
effected
first
the
corresponding
to
original
once
etatement
though
modified
and
taken),
the
overall
code
GOTO 300
the
exclusive;
to
been
to
loop
mutually
loop:
this
is
method
the
exit
EXI)
transformation
the
generated
each
300
eimple
statements
newly
CONTINUE
IF (.NOT.
This
original
relocation,
200
to
if
the
all
the
loop
only
in
guards
modified
exl
The
branch
.TRUE.
I =1,100
(EXI)
S1
(EX<)
EX2 ❑ .TRUE.
100 J=1,1OO
IF (EX1 .AND. EX2) S2
IF (EX1 AND.
EX2) EX1 ❑ .NOT.
IF (EX1 .AND. EX2) S3
IF (EXI
.AND. EX2) EX2 = .NOT.
IF (EXI
.AND. EX2) S4
CONTINUE
IF (EX1 .AND.
.NOT. EX2) GOTO 50
IF (EX1)
S5
an
form
branch
GO TO 50
EX1 =
DO 200
IF
IF
DO
exit
nthe
at
following
branch
are
for
After
exit
GOTO S1
of
loop
assignment)
flag
taken
effect
would
exi
been
GO TO 300
since
Each
P
condition
Finally,
within
true,
replaced
have
(.NOT.
this
program.
are
exited.
200
300
Upon
50
immediately
the
branch
branch.
associated
the
generated
exit
exit
loop,
form
not
IF
Note
a single
flaga
been
= .NOT.
execution.w
simulate
of
GOTO S1
the
would
recent
each
exit
yet
loop
capturea
branch
for
all
(P)
EXi
which
out
form
the
assignment
branchea
not
IF
is
IF X(I,J)
S3
IF Y(I,J)
S4
CONTINUS
S5
CONTINUE
S6
100
the
J=l,IOO
S2
variables.
In
=1,100
DO 100
We
variables
Fortran
I
S1
variables
case
as
50
that
= EXi.
caae
upper
DO 200
the
time
in
of
outmost
the
level,
program.
branch
on
no
To
every
exit
demonstrate
relocation.
we must
DO
branches
the
show
two
things:
one
(1)
Con-
example:
(2)
The
the
algorithm
removes
the
modified
version
same
computation
first
statement
181
point
S1
of
as
all
branches.
performs
the
follows
the
exit
algorithm.
and
the
exactly
original.
rather
from
trivially
The
body
of
loop
of
guards
then
expressions
procedure
/*
/*
/*
relocate_branches
avoiding
(x);
x is the DO statement
for
the
loop
loop_guard
will
be the
conjunction
all
exit
flags
for
the
loop
loop_guard
S1:for
*I
exit
branch
that
exits
IF
the
(P)
GOTO S1
loop
headed
by
x do
a new
unique
exit
flag
after
the
loop;
replace
the exit
branch
the
assignment
%Xi
eXi
by
= .NOT.
exit
bound
be
large
terminated
S3:for
end
vectorization
in
branch
x do
no
y contained
A loop_guard;
in
x do
tion
of
ment.
Since
the
procedure.
t’ne
same
tions
(1)
for
will
second
about
only
in
exit
branch
no
point
gram
ia
which
joined
program
that
the
all
to
its
since
the
exit
being
applied.
action
the
Each
exit
$tatement
of
ables
at
determine
in
flag
to
the
branch
branch
is
replaced
the
jump
false
if
ia
would
becomes
the
txue
pro-
lo~pa
are
have
an
condition
in
con-
asei~nnent
been
words,
taken
the
thaL
effects
nal
effects
branch
that
would
program.
A
the
the
pute.
branch
have
them
might
spe-
offset
this
ply
a
exit
Lbe
flag
on
dependence
Since
the
branch
value
of
IF
the
its
varimust
reflect
this
in
place,
The
umoved.
branches
execuand
conversion
are
is
the
the
guarda
fw#ard
to
of
is
the
process
known
all
phas.e.s
as
(guard.)
which
the
wi.1.l
from
~
current
or
.be
facts
abo.ub
as
As
in
the
the
Boo,lean
forward
or
guard
variables
branches
not
the
variables
LO generate
whether
branch
statement
Eo~Lean
These
the
pre$.ently
executed.
to
co.nditioi’i
sim-
reflecting
d.isjoiua
s.tatcxoent..
(euch
rersoval
is
statement
atateuent
cuujoin>
nut
branch
which
expression
it
of
a. a.ueI&~o
moves
progzam
for-
providing
control
correctly
the
consicle.ration
the
and
identifica-
thus
expression,
under
the
to
makes
Zem$Z!d.
conditions
for
it
removal.
of
and
logical
under
prepaas
the
branch
that
Once
represe,lt
if
.nQt. have
place
logical
at
in
in
source
of
guard
Lhe
the
variables.
of
very
is
introduced
occurred
is
conditions
correctrmss
have
p~ssible
algorithm
taken
in
about
comp.u,tation
removal
would
save
concern
tran.s.forma.t.ion
is
a
they
of
tbe
would
be
taken).
false.
imporLant
the
branch
idea
A
One
a
gained
branches
allowa
between
isunnecessaxy
with
exit
controlling
other
the
would
of
First.
branches.
depend
Fundamental
corresponding
‘in
loop
speedup
targets,
statement
wa?JihLan.Ch
is
it
type
guard.e
modi-
statements.
for
by
provide
elegant
backward
forward
the
dependence
guard.
acts
when
to
than
an
for
simplest
the
removal
that
the
unfortunate
detection
reaaona.
and
clearly
by
observa-
contained
bra~ch
ia
of
Branches
from
pxogram,
(2)
though
times
the
the
is
necessary
S.1 is
.ruodified
flags
is
that
on
Second,
of...eliminating
two
and
statement
be
from
between
the
is
more
many
branches
target
branches.
transformations
original
created
a loop,
follows
difference
the
in
assign-
_
and
exit
an
are
branchea
targets),
contain
the
to
branches
their
each
code
exit
have
insure
between
The
brarwh
generated
as
The
The
new
(the
level
executed
fied
no
exit
even
part
running
brancb
for
Forward
Relocation
tion
a particular
a
branchea.
results
S1 converts
to
been
computation
iteration
relocation
distinction
ward
4.
Branch
may
long
have
program,
-
real
essential
will
removal
information
1:
course
no
an
loop
loop
modified
P*
relocate_branches;
Figure
its
and
it
the
the
Hopefully,
from
thereby
inefficiency.
y contained
(y);
each non-DO
statement
guard(y)
= guard(y)
is
by
condition.
Branch
each DO statement
relocate_branches
This
enough
cial
end
for
false
DO
logical
variables,
of
in
run
ia
the
evaluating
effects.
out
unexpectedly
of
aide
DO loop
but
of
purpose
of
will
flag
done.
effect
with
realization
EXi;
insert
the assignment
WXi
= .TRUE.W
prior
to x;
= loop_guard
A exi;
loop_guard
insert
the branch
‘IF
(.NOT.
EXi)
GOTO Sln
S2:
entered,
some
to
logical
branches
every
transformation,
begin
create
all
once
being
these
problems
eliminated,
= true;
each
the
Since
*I
*/
of
amounts
in
point
the
or.igi,-
such
side
The
values.
careful
original
side
to
where
ccw-
bronch.
at
diverge
from
target
the
oixliu...ry
label,
flow.
tile
ti~e
curren~
these
at
control
where
split
a.t
two
flow
flow;
and
rejoins
coudit.ion
points
flow
ccmtrol
sequential
where
Thus,
modified
affects
branch,
can
at
the
sequential
(~r
cc)
to
remove
absence
of
must.be
forward
branches.
they
program
The
forward
loc.atious:
and
(1)
&
trol
computation
182
-
~:
flow
In
changes,
the
the
statement
other
ton-
immediately
following
when
a
control
branch
not
the
predicate
the
guard
(2)
AL
the
Lku2
tially
the
and
the
branch
is
the
the
ccl
and
is
statements
tions,
if
of
the
should
the
guard
A -tp),
in
the
In
ccl.
In
the
branch,
v
A p).
other
In
the
A p)
words,
the
order
absence
BR2,
to
recognize
the
IF
of
to
the
example
DO 100
BR1
IF
IF
IF
S1
S2
tar-
path
should
make
60
A(I)
80B(I)
=A(I)
100 CONTINUE
We introduce
two
the
these
S3
ideas
flag
so
two
BR1
text
are
is
GT.1011
called
is
the
we
BR2
br2
the
in
place
cal
at
of
the
~.
The
if
and
to. .true
.in
the
first
logical
values
that
to
original
later
= ~(bz2),.
brl
In
inserting
capture
the
point
of
change
values
of
in
inat
br2,
the
the
described
in
the
loop
conventions
above.
are
we
controlled
expect.
example
the
on
a pxototype
Quine-McCluskey
5.2,..McC1
56],
we.
that
the
(and
the
sim-
thought
simplifier
how-
had
simpler)
sys-
implicant
thought
some
difWhen
PFC
prime
After
correct
is
indeed
version
of
this
is
called
on
is
initialized
and
is
first
ing
call.
call,
The
a set
of
queues
basic
queue
sions
cannot
is
5.
Backward
at
statement
by
the
a
time
in
TRUE
each
the
before
succeedof
predicate_list)
Note
at
elim-
the.existence
that
therefore
disjoined
list
get
reset
array
converted;
be
predicate
the
to
procedure
to
assumes
primitives.
are
to
(in
each
then
algorithm
used
The
cco
code.
the
conditions.
algorithm
original
assign-
logi-
the
branches.
the
only
all
target
and
must
guard
forward
the
expres-
be
for
in
that
ita
tar-
created.
Branches
controlling
for
find
S3
initially
2 outlines
While.
Following
controlling
forward-convert
program
program
+ 5
would
forward
e
the
PF.C ensures
conditions
A(I)
=A(I)
+ 10
BR2 = B(I).GT.1O
.NOT. BR2)
B(I)
= B(I)
+ 10
BR2)
A(I)
=B(I)
+A(I)
one
realized
a
the
By using
evaluation.
the
we
statements.
removal
following.
.NOT.
incorrect.
Figure
IF
and
IF
the
convert
condition
[Quin.
branches
statements,
assignments
accidentally
ments
.OR.
this
used
was
only
variables
of
aspect
the
condition.
= A(I).GT.1O
= B(I).GT.1O
IF
two
variables
the
the
❑ A(I)
what
ran
that
produced
ments:
BR1
BR2
first
ever.
to
loop.
true
by
the
from
tem
be
the
accomplished
and
in
Fortran
capture
and
brl
~
to
evaluates
= ~(brl)
this
ferent
simplif,iex
conditions
defined
We use
,BR2 to
that
Note
plifier
branch
brl
statement.
BR1 and
(BRI
B(I)
CONTINUS
100
+4(1)
variables
always
important
would
I = 1, 100
= A(I).GT.10
(.NoT.
BR1)
(.NOT.
BR1)
(.NOT.
BR1 .AND.
IF
S4
clearer.
GO TO 60
=B(I)
- 5
Boolean
variables
NA(I).
PFC
x
is
A(I)
=A(I)
+ 10
IF (B(I).GT.1O)
GO TO 80
B(I)
= B(l)
+ 10
S2
S3
S4
if
into
recogis
simplification,
in
able
logical
surely
S4
an
x
S1
Such
is
BRl
be
simplify
should
controlling
With
above
must
=
reaches
reach
condition
procedure
statement
flow
and
it
long
like
procedure
simplification
loop
of
variables
example,
conversion.
example
proliferation
logical
conversion
the
Thus,
of
the
identities
conversion
CC2
simplifies
IF
For
that
true.
target
execution
DO 100 I = 1, 100
IF (A(I).GT.10)
branch
prevent
involving
taken.
capture
brlV(wbr1Awbr2)
brlV(mbrlAbr2)V(-?br1A*br2)
to
and
nize
to
that
target
will
which
prior
(so
control
flow
of
the
the
which
if
control
regardless
An
lbrl
-tbrlA-tbr2
expressions.
assump-
on
flow
on
(ccl
via
statement
guard
can
sequen-
previous
the
control
guard
A -!p)
the
the
CC2 V (ccl
flow
or
condition
p.
will
either
statement
on
CC2.
changes
(CCI
branch
previous
is
be
control
the
Under
target
other
controlling
.51
S2
S3
S4
current
branch
following
Similarly,
target
itself.
get
branch
if
statement
only
expressions
from
branch
is
forward
~:
the
ccl
the
Thus ,
controlling
for
executed
A ap.
reach
the
is
reaches
taken.
at
ccl
branch
flow
is
condition
be
forward
forward
that
by
branch
the
the
s,tate-
quite
type
control
In
following
branch
branches
of
fact.
branch
creates
cannot
PWIWS;
thus
eliminated.
183
from
struct
can
it
dependence
backward
e.liminat.ed
conditions.
removal
handily,
a
an
be
backward
eliminate
cannot
-
backward
branches
cannot
program,
because
implicit
simulated
branches
forward
remove
loop.
a
directly
backward
looping
guarded
cannot
last
branches.
be
A
with
the
be
constate-
directly
100
is
taken.
One
procedure
forward_convert
returns
(x,
avoids
cc-o)
condition;
to
/*
x
/*
/*
cco
ccl
/*
1*
/*
predicate_list(x)
is a queue
of all
predicate
that
must be disjoined
at x becauae
of branchea
to x.
ccl
the stat~~ent
is
under ~on~id~ration
ia the
condition
prior
will
be the
condition
to x.
guarding
x
*/
their
a)
*/
*/
*I
implicitly
bits
S1
);
S1
is
in
These
/* GOT’O y
(forward
to same level)
begin
add_ko_queue
(predicate_list(y).
set
paas
a
of
);
removal
in
DOL
handled
be
because
of
branch
the
statements
would
by
code
*/
Branch
of
the
more
problems
Forward
backward
a~gorithm
than
branches
in
One
branch
if
are
two
Forward
~
executed
branch
set
the
following:
GO TO 200
becauae
when
X is
it
aa
taken.
In
other
the
using
first
backward
the
words,
realization
jump
idea
to
the
a
BB)
will
has
been
previous
and
example
Guard
-tbrl
v
true
?br ~
true
(brl
A bbl)
mbrl
v
(bz~
= .FALSE.
A bbl)
true
THEN
point
a branch
is
back
that
BB1
is
set
to
true
occurs.
to
ways
can
our
that
be
two
the
reached
EdlthKough:
alternatives.
target
from
y
the
to
-tX.
prevent
the
in
This
S1
backward
Figure
guard
from
the
2
(7-)
is
of
atart
branch
by
there
a
of
backward
the
pro-
BackMad
bbj)
184
path
the
y.
is
fall
through
The
condition
taken
ccunplete.ly
is
on
Curs.nLmditizn
from
under
exit
from
predecessor.
krau.11:
control
iterative
branch
to
can
before
this
implicitly
being
control
statement
encoded
illustrated
S1
would
true
been
the
whenever
= X
which
for
true
by
on
gram.
GO TO 100
removal
false
flag
to
y.
region
bIi
and
The
condition
branch
can
by
enter
a branch
backward
under
a
passes.
established
ia
(with
firat
and
subsequent
and
this
BR1
. . .
BB1
S1
. . .
S2
the
guard
be
the
region
region
that
Corresponding
2.
...
would
fact
noteworthy
only
(1)
(Y)
guard
can
bb
approach:
guard
can-
Figure
““”
S1
. . .
S2
IF
to
which
flag
to
BBI = .TRUE.
GOTO 100
ENDIF
branch
200
the
backward
a generalized
iterative
used
has
. . .
IF (Y)
Removal.
hawever.
the
like
(X)
that
Statement
create
presence
through
produce:
200
the
we
x
Forward
IF
100
is
the
denote
time
used
implicitly
conditions
k@2.k
forward_convert;
loops.
ao
A guard
pass
any
ia
variable
bx.ax&h
first
suggest
through
100
branches
severe,
carefully.
false.
executed
an
guard
pasa
the
X is
conditions
Boolean
(ccl)
implicit
an
taken.
set
These
*/
CC1
ia
through
esac;
Backward
inhiinto
taken.
/* All
other
type
begin
guard
x by ccl
end
just
alternatives:
on
if
Applying
2.
two
ahtaya
different
end
Figure
quite
more
alternativea
one
backward
end
under
approach
branches
seems
executed
only
branch
/* IF(P)
GOTO y (forward
to same level)
*/
begin
create
a new branch
flag
bri
with
realization
BRi;
replace
x
with
‘IF
(~(ccl))
BRi = P;”;
add_to_queue
(predicate_list(y),
CC1 A bri);
ccl
+ ccl
A -tbrl
end
return
ia
code
~
this
problem
reflect
ia
code
(2)
ccl
+- false;
delete
statement
the
region.
the
S1 must
that
branches
~
forward
limitation
consider
an
course,
any
conversion
) do
( predicate–list(x)
statement_type(x)
of
IF
backward
leaving
as
Of
iterative
This
end
case
(known
removal
(1)
P ‘- get-from-queue
ccl
+ ccl
v p
to
of
branchea,
untouched.
for
( predicate_list(x)
these
control
must
not_empty
approach
complications
isolate
*/
*/
+ Cco;
while
begin
possible
the
which
the
with
(flag
this
can
happen
is
true
when
the
the
bri
condition
from
and
backward
Hence,
the
branch
is
the
guard
at
there
ia
region.
of
more
the
the
the
each
backward
forward
branch
condition
was
the
target
‘l’he
alao
be
one
jump
procedmre
returns
true.
the
into
should
backward
the
be
the
at
modified.
the
x
cco
ccl
the
bbj.
end
c*ee
must
following
(X) GO TO 200
S1
GO TO 300
S2
IF (Y) GO TO 100
100
200
300
The
53
correct
S2
ia
to
200
been
ing
guard
executed
wae
S2,
the
ward
branch
regions
the
term
into
the
target
pasa
to
after
aignifiea
complete
branch
( predicate_list(x)
x
*/
*/
) do
( predicate_liat
(x)
);
in
to
same
(x.
y,
same
level)
*/
bri
BRi
= PW;
A bri);
) *I
level
ccl);
x
GOTOY
bb.
wit~
insert
let
TPk
replace
~Pk
‘IF
‘IF
‘IF
each
in
may
first
example
(~
to
same
level)*/
be branch
back
flag
aeaociated
this
branch
(realization:
BBj );
%Bj-=
.FALSE-.N before
y;
be a new eemporary
variable;
x with
the
atatementa
=
(CC1)W
(TP ! ) TPk = Pw
(TPk)
BBj ❑ ..TRUE.n
(TPk)
GO TO yn
end;
becomes
f*--kl4 other
begZn
guard
(x)
ad
esac;
return
(ccl)
remove
branches;
=x
the
:urrent
with.
the
result
rors
the
after
fact
Figure.
condition.of
target
major
Figure
2
the
51
*/
Y)
THEN
end
Figure
3:
statements
*/
+guard
(x)
Conplete
A ccl
branch
removal
Also.
note
rated
at
that
defeat
the
the
of
which
alwaya
be
executed.
these
4),
implicitly
block
purpose
equivalent
.of
IF
statements
branch,
IF
whether
,conv.er.sion.
assignment
not
is
branch
into
the
.c.u.rrent
conditi~n.
simplifies
target
citly
gene-
of
generated.
entry
185
the
of
last
all
the
current
forward
the
that
afiter
condition
backwards
another
back
the
flag
the
the
impli-
simplification
condition
last
.a
that
after
into
This
the
of
of
are
ia
enters
branch
branch
the
cases
control
never
the
in E!PC has
there
prea,ence
flag
region.
fact
Statements
to
the
the
the
back
Second,
iterative
reflects
a
Without
out
of
special.
unless
under
branch
used
no
b.ranchea
region
branch.
in
ptocedur.e
First.
backward
.Che
svould
Ina.tead,
for
branch.
forward
these
needed
in
regions.
are.
,ai.n.c.o
removal
advantages.
backward
al.gor.ithm
iterative
branch
several
,remo.val
encapsulated
on
.The
,mir -
observations.
the
check.
backward
branch.
dis-
A bx.1..
true
to
(Figure
jump .into
is
v bb.1
is
incorp.or.ate.s
the
A brl
-tbrl
genezal
modification
is
pr.oceas._branc.h
branches
S3 should
contains
which
only
-tbbl
condition
.simplif.ication
that
3
algorithm
sequence
~F--(p)
and
the
to
x
S3
The
The
on
previous
(.NOT.BR1
.C)R. BBI.AND.BRI)
GO TO 300 has been eliminated
(.NOT.
BBl .AND. BRI)
S2
(.NOT.
BB1 .AND. BR1 .AND.
BB1 = .TRUE.
GO TO 100
ENOIF
300
S3,
removal
+*
begin
let
coutr.ol
only
The
condi-
flags
*/
prior
after
end
a for-
for
back
that
statement
regions.
not
flag
IF
/*
IF
IF
200
joined
flag
condition
the
consideration.
condition
condition
ccl
+ falee;
delete
statement
iterative
branch
back
target
for
the
branch
these
preced-
implicitly
of
of
BR 1
At
the
baa
target
under
current
current
/* GOTO y
(forward
beg in
proceaa~bzanch
since
branch
the
statement
the
the
statement_type(x)
branch
branch
condition
branch
of
100
in
negationa
the
iteration
be
multiple
of
The
A -tbbI,
forward
the
target
conjunction
negation
bzl
the
remove
must
the
be
if
backwards
to
mbbl
the
only
the
order
general,
.is
region.
and
In
In
S2 must
and
taken
taken,
tion.
for
if
the
is
is
CCO)
/* IF (P) GOTO y (forward
to
begin
create
a new logical
guard
with
realization
BRi;
‘IF
(y(ccl))
replace
x with
process_branch
(x.
y, ccl
CC1 + CC1 A -tbri
end
example.
IF
ia
(x,
P ‘get-from-queue
ccl
+ ccl
v p
target
Conaider
/*
ccl
+ Cco;
while
not_empty
begin
iterative
with
remove_branches
condition;
/*
/*
disjunction
conjoined
generated
slightly
of
was
branch
A bbj)
condition
condition
to
branch
the
term
branch
is set
J
incorporates
of
than
second
bb.
it
target
CCY V (bri
If
Since
occurs,
that
reached
the
A bbj.
branch
for
p.o.ssible
branch
execution
external
should
be
independent
wards
of
any
branch.
ability
of
cal
specific
Most
the
iteration
algorithm
combination
of
important,
of
the
to
handle
backwards
is
any
branches
there
back-
however.
the
the
exists
literals
pathologi-
with
Hence
minimal
three
no
shorter
in
@ that
the
conjunction
aleo
of
implies
Quine-McCluskey
a subset
of
~.
procedure
contains
phases:
effort.
(1)
6.
Boolean
In
developing
conversion.
of
have
issues
two
representations.
for
one
basic
to
we need
some
mechanism
put
program
will
be
for
we
the
need
to
by
and
find
the
that
are
cants
to
(3)
Phase
the
number
out-
simple
simplification
function
external
representations.
are
maintained
forming
val
the
--
be
possi-
at
a
McCluskey
McC1
branch
applying
prime
the
the
branch
remo-
actual
implicant
merging
the
simplifier
ing
discussion.
and
negations
Ln
of
~
a
a
Any
(~
Boolean
of
if
no
of
is
variable
of
is
an
ensu-
known
tial
since
give
rise
the
as
in
This
a
Y
be
fundamental
variable
are
each
a
aet
eet
of.
of.
all
i.t
formula
in
t. is.a
Y occurs
n,var.i.ab.les
in
in..Y,
minterm
can
mi,n@nn
it,
are
denote
Y
PF.C,
alternation,
fo.rmula,
of
miuterms;
rep.r.esentat.ion.
as..,
. .
~..~
McCluakey
there
or
reducing
t.hers
then
finding
for
the
is-~
aet
t.o
of
we
.written,
refer
the
~..
.simpli.fies
to
.The
.Boolean
canonical
disjunc.tive..
a.miuimal
set
rointerms,
of
,aa
~
.
pzo.cedur.e
and
cam,be,
A
a
as
Z
ita
active
of
At
at
the
which
such
one
exponen-
m variables
time
that
removal
1
can
only
Quine-
is
take
branch
created.
advantage
are
two
the
of
guard
to
c.o.ndit.ions
in
the
the.set
the
us
of
maintaining
over
allowa
a
forward
created
and
modified
of
dur-
ways:
it
b.y
(2)
by
form
formula
1
branch
two
a.
new
conjoining
current
condition
this
result
the
flag
in.
the
. The, one,e
current
and
its
the.
branch.
be
its
with
by
the
con-
negation
negation
the
to
col-
comprise
next
etatement
unnegated
attached
condition
to
Inter-
condition
the
for
is
formed
negation
effected
current
with
ones
the
flag
are
at
and
condition
the
bxanch
it
can
new
mintern.
new,
conditions
flag
to
the
branch.
the
At.
@
and
186
a
target
must
be
dled
by
same
set
union
of
tions.
@ a
is
How -
PFC
minterms
representation
while.
Qr
Qu.ine-
normal
if
it
nine.
is
in
phase
lection..
this,
formulas
fundamental
the
requirements.
formulas
internally
a set
the
.of..-,~
formula
ia
methods
potentially
in
comprise
Boolean
m
so
also
by
every
negation,
Every
good
However,
or
slower
avoid
joining
.s.ince
a variable
and
eight
we
nally,
A
each
If
a
82].
technique
short
best
Phase
2 can
to
nl.5~
storage
is
the
the
storage,
than
slightly
1
in
since
[AIKW
procedure
branch
by
if
.rmce,,
either
that
2m minterma.
ing
(1)
Y.
2? minterms,.
I
0(3m)
m larger
a few
to
observation
called
over
over
exactly
there
contain
We
formulaa
such
is
fundamental
are
variables.
normal
S CE
minterms
in
original
the
the.
set
proportional
used
Phase
from
Let
= ~.
necessary,
of
smaller
the
&rlU..
nf(Y)
W
acceptable.
Simplification
guards
as
formulaa
alternation
then
complexity
time
exist
561.
flags
to
appears
fundamental
and
it
the
variables
literala
implialterna-
the
52,
b.e referred
).~iimaula
which
for
formula.
will
alternation
“
formulaa
notation
variable.e
Quaul.a
twice.
of
A conjunction
.
canonical
prime
is
but
is
requires
much
In
a bit
E,
strictly
S
number
for
McCluskey
with
of
not
variables
there
[McC1
Quine-
[Quin
the
method
iea
We begin
to
all
V~
implicants.
in
is
of
6.2.
Prelimin*r
If
shortest
select
McCluskey~s
two
56]c
6.1.
E of
combinatorial
is
n
have
simplification
of
of
to
ever,
of
version
a
the
prime
impractical
guards
per-
and
set
members
implemented
the
to
quickly
flag
The
label.
by
into
for
operations
a new
the
formula.
of
is
of
heurietic
representations
suitable
fundamental
performed
built
all
simplification
other
as
Internally,
a form
creating
conditions
is
in
is
internal
maps
of
3
number
which
1
guards
representations
as
the
Selection
where
~
formula
VP e! $.
ble.
The
the
of
for
tion
correct,
In
readable.
external
which
operator
of
Construction
conver-
be
insure
(2)
exter-
IF
that
IF
factor
one
will
to
clean
conditions
referred
and
program
but
worda,
We have
insures
output
for
to
constructed
method
the
basis
attempted
internal
conditions
The
s ion.
attached
conceptual
purposely
representation.
the
to
form.
the
we
out
nal,
Reduction
Simplification
collection
some
disjoined..
extending
of
all
This
the
variablesminterms
of
conditions
disjunction
minterms
then
in
the
to
is.
be
simply
various
over.
taking
hanthe
the
collec-
Conditions
in
parts.
The
list
of
a
set
the
minterms
simple
used.
in
the
the
branches
second
condition.
should
clarify
At
statement
is
disjoined
200,
expression
the
current
brl
br2
A
condition,
br3
A
giving:
(IbrlAmbr2Awbr3)V(wbrlAbr2Abr3)V
(mbrlAbr2A=br3)V(-tbrlAmbr2Abr3)
these
The
the
the
with
part
of
actual
(-sbr1A-rbr2A=br3)v(*brlAbr2Abr3)v(-tbrlAbr2A-br3)
a
condition,
disjunction
following
IF (X)
. . .
IF (Y)
. . .
IF (Z)
two
is
forward
The
The
by
m)
the
the
encountered.
example
Consider
~
that
minterma.
repreaenta
A
represented
~
present
order
were
of
(the
flags
in
represent
is
actually
part
branch
maintained
they
PFC are
first
first
and
abr2.
method
The
*brl
code:
A
=brl.
GOTO 300
minterma
and
br2.
These
Finally,
at
simplified
GOTO 100
laat
second
out.
simplify
third
minterms
300,
the
-tbrl
A
simplify
combine
statement
resetting
to
minterms
to
the
produce
flag
current
to
-tbrl
is
condition
to
true.
GOTO 200
This
example
Once
a
leada
to
several
new
obsena-
tions.
100
““”
CONTINUE
200
““”
CONTINUE
. . .
CONTINUE
(1)
300
each
of
condition
is
produce
the
the
dition.
a
to
be
A br2.
which
expand
the
(-tbrl
not
in
Qf
A brz
joined
The
be
transformation
Thus
the
by
current
current
in
it
A br2
no
than
after
(2)
The
as
the
A br3)
is
order
reverse.
of
the
is
the
/*
x
y is the target
br is the condition
before
condition).
100
is
The
proof
but
not
itera~ive
let
bb.
be
the
con t rolling
xj be.the
x jumps
branch
the
target
back
of
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Note
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increases
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to
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noted,
directly,
from
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identification
jumps
block
conversion
that
separate
of
optimization
conversion
easy
run
“of
variables
Additionally,
is
to
increments
induction
allow
statement
the
loops
by
loop
determination
list
all
bound
new
which
every
branch
Implementation
IF
upper
controlled.
back-
into
modifies
some
altered.
flag
branch
normalization
doing
handle
branch
-
required
backward
first
be
Quine
are
to
conditions
iterative
the
condition
expanded
adding
when
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of
phases
current
also
flaga
and
condition
2
These
actual
can
flag
?.
simplified
phases
to
focus
control
dependence
--
tedious
details
Second,
this
not
created
the
approach.
greatly
it
unless
not
are
a scalar
arrays
for
is
that
two
the
of
not
they
recurrences
Scalar
however,
in
in
PFC;
by
any
tested
expansion
U. Banerjee,
~ata
dependence
nary
programs,n
Report
76-837.
vec-
ment of
Illinois
Illinois,
conversion
[BohJ
661
dependence
may
dependent
IF
conversion
Vectorization
of
Qtherwise
of
here,
with
not
IF
are
be
unwieldy,
the
run
January
short
we
code
IF
to
D. Harel,
‘On folk
theorems.w
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5, July
1980,
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[Kenn
801
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476-029-4,
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October
1980.
[Kuck
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‘A
machine
organization
~
29-59.
[KKLP
811
D.J.-Kuck.
R.H.
Padua,
and
transformation
c.onverwion
to
trol
depend.ences
sion
is
useful
languages,
More
code
has
to
in
can
structured
known
as
66,
well
as
briefly
explored
in
far
beyond
IF
as
and
goto
deumnstrates
system
that
This
801,
practically
a
801
the
lM2s2.
[AIKW
821
821
R.H.
Kuhn,
~he
structure
AGM d
Vs..
for
B.
of
January
pipelined
Leasure.
and
an advanced
processors.”
*CalWt&LSWiS4~
Sd&alX%
and
IEEE ,
Chicago,
GQUL. *
flow
October
any
intellectually
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practical
56]
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though
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52]
well
1980.
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Boolean
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~he
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1952,
pleas-
useful.
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and K. Kennedy,
~FC:
a program
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Fortran
to
parallel
form,w
Report
MASC TR 82-6,
Department
of
Mathematical
Sciences~
Rice
University,
Houston,
Texas,
March.
1982.
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nSimplifieatien
PFG,n Dept.
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B. Leasure,
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‘Compiler
of
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graphs,”
conver-
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fe-ulas
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Sciences,
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U
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of
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521-531.
of
M.
simplifying
Mn.thly
599
[Towl
76]
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and
data
depenR.A..
Towk!,
program
transformations,”
dence
for
Ph-.D .
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ation.
Dept.
of
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University
Vrbana-Champa
ign.
of
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at
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1976.
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78]
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~eehniques
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ence,
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As
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however,
IF
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recognize
vector
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are
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PFC.
exact
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phase
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is
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the
at
results.
is
some
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torization
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