Predicate Sensitive Dataflow Analysis - Lecture Slides - Fall 2006 | EECS 583, Study notes of Electrical and Electronics Engineering

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EECS 583 – Class 9 Predicate Sensitive DataflowAnalysis

University of Michigan February 8, 2006

• 1 -

Reading Material

Y

Today’s class^ »

“Analysis Techniques for Predicated Code”, R. Johnson and M. Schlansker, MICRO-29, 1996

Y

Material for the next lecture^ »

Compilers: Principles, Techniques, and Tools

A. Aho, R. Sethi, and J. Ullman, Addison-Wesley, 1988, 9.9, 10.2, 10.3, 10.

»^

“Compiler Code Transformations for Superscalar-Based High-Performance Systems”, S. Mahlke et al., Supercomputing '92,Nov. 1992, pp. 808-817.

• 3 -

Two Types of Analysis Y

Predicate relation analysis^ »

Properties about the values thatpredicates can take on »^

Disjoint

= 2 predicates never

true at the same time »^

Superset

= 1 predicate true

more often »^

Subset

= 1 predicate true less

often »^

Predicate query system

Y

Predicate-sensitive dataflowanalysis^ »

Calculate dataflow informationon predicated code »^

GEN/KILL now occur on acertain predicate (not always)

t = pclearx =p,q = cmpp.uc.un (a<b)t = cmpp.oc(a<b)x =

if p

x =

if q

r,s = cmpp.un.uc(c!=d)

if q

t = cmpp.on(c!=d)

if q

x =

if s

= y

if r

= x

if t

= x

• 4 -

Predicate Relation Analysis Y

Execution trace

• record of boolean

values assigned to predicates duringan execution of a predicate block^ »

1 = all execution traces

Y

Execution set

for predicate p – set

of traces in which p is True^ »

Denoted as P »^

P is a subset of 1

Y

Code segment^ »

Execution traces^ y

trace

p^

q^

r^

s^

t

y^

e

1

0

0

0

1

y^

e

0

1

1

0

1

y^

e

0

1

0

1

0

»^

Execution sets^ y

P = {e1}, Q = {e2,e3}, R = {e2}, S= {e3}, T = {e1,e2}, 1 ={e1,e2,e3}

t = pclear x = p,q = cmpp.uc.un (a<b) t = cmpp.oc(a<b) x =

if p

x =

if q

r,s = cmpp.un.uc(c!=d)

if q

t = cmpp.on(c!=d)

if q

x =

if s

= y

if r

= x

if t

= x

p^

q r^

s

t

• 6 -

Class Problem – Create the Execution Sets^ r = pclear^ x =^ p,q = cmpp.un.uc (a == b)^ x =

if p

x =

if q

r,s = cmpp.on.uc(c < d)

if p

r,t = cmpp.on.uc(e > f)

if s

u,v = cmpp.un.uc

if r

x =

if s

x =

if v

= x

if r

x =

if t

= x

What is r + t??

• 7 -

Partitioning by a CMPP

Y

m,n = cmpp.un.uc (i < k) if u^ »

If m or n is true, then u must be true

»^

If m is true, then n must be false (+ the reverse)

»^

If u is true, then one of m or n is true

»^

If u is false, then both m and n are false

Y

Execution sets^ »

M union N <= U

»^

M intersect N = Null

»^

U <= M union N

Y

U is partitioned into disjoint subsets M and N, U = M | N

u m

n

• 9 -

Partition Graph Y

Partition Graph

• Directed

acyclic graph where nodesrepresent execution sets andedges represent partitionrelations between nodes^ »

U = M | N

»^

Edge from U to M »^

Edge from U to N »^

U

Æ

M and U

Æ

N denoted to

come from same partition

Y

Complete

• Contains a unique

node having no predecessorsfrom which all nodes arereachable

p = (a<b) * true q = !(a<b) * true r = (c>d) * p s = !(c>d) * p u = (e!=5) * true v = !(e!=5) * true

p^

q^

u^

v

r^

s

• 10 -

Partition Graph Properties Y

Root

= unique node with no

predecessors

Y

Leaf

= node with no

successors

Y

Ancestor

of P = any node on

path from root to P

Y

Descendant

of P = any node

reachable from P

Y

Level

= shortest-path distance

from root^ »

Level(root) = 0

Y

Lowest common ancestor

node with largest level fromthe common ancestors

p = (a<b) * true q = !(a<b) * true r = (c>d) * p s = !(c>d) * p u = (e!=5) * true v = !(e!=5) * true

p^

q^

u^

v

r^

s

• 12 -

Synthesizing New Partitions

1 p^

q r^

s

p^

q r^

s

t^

t^ lca(p,r) = 1^ 1 – t = 1 – (p + r) = 1 – p – r = s

Identify partition, U = M|N where U is not reachable from 1 If M,N are reachable from 1, then there exists a partition

lca(M,N) = U|V, where V is the relative complement

of U with respect to lca(M,N)

• 13 -

Class Problem – Create the Partition Graph^ r = pclear^ x =^ p,q = cmpp.un.uc (a == b)^ x =

if p

x =

if q

r,s = cmpp.on.uc(c < d)

if p

r,t = cmpp.on.uc(e > f)

if s

u,v = cmpp.un.uc

if r

x =

if s

x =

if v

= x

if r

x =

if t

= x

• 15 -

Implementing PQS Functions in 1-dnf

1 p^

q r^

s

is_disjoint(p,q) = if there is a partition with p on 1 side, q on the other, then disjoint is_subset(p,q) = if q is an ancestor, then p is a subset lub_sum(p, E) = return p + qi for all qi in E. (Remove redundant/subsumed terms)

t

Examples:

Are p and s disjoint? Are t and p disjoint? Is r a subset of p? Is s a subset of q? s + {q} =? r + {s} =? p + {r,s} =?

• 16 -

Implementing PQS Functions in 1-dnf (2) lub_diff(E, p) = return Sum(qi-p) for all qi in E. 4 subcases

1. qi <= p

Æ

qi removed, E += NULL

2. p <= qi

Æ

E += rel_cmpl(p,qi)

3. p,qi disjoint

Æ

E += qi

4. otherwise

Æ

E += approx_diff(p,q)

/* q – p */ approx_diff(p,q)

relative_complement(p, lca(p,q)) relative_complement(p,q)

find a path from q to p E = false for each edge r

Æ

s on path do

let R = S|Ti be the partition

containing r

Æ

s

add Ti to E return E

1 p^

q r^

s

t

Examples: {q} – r =? {q} – p =? {t} – r =? {t} – s =?

• 18 -

Predicate-Sensitive Liveness

Destination operands

GEN’

GEN’(x) = GEN(x) – Q KILL’(x) = KILL(x) + Q

x = y + 1 if q

Source (data) operands GEN’(y) = GEN(y) + Q KILL’(y) = KILL(y) - Q

GEN

Source (predicate) operands

backward analysis

GEN’(q) = GEN(q) + true = true KILL’(q) = KILL(q) – true = false

• = lub_sum - = lub_diff

• 19 -

What About UN/UC CMPP’s?

Destination operands

GEN’

GEN’(p) = GEN(p) – true = false KILL’(p) = KILL(p) + true = true

p = cmpp.un(a<1) if q

Source (data) operands GEN’(a) = GEN(a) + true = true KILL’(a) = KILL(a) – true = false

GEN

Source (predicate) operands

backward analysis

GEN’(q) = GEN(q) + true = true KILL’(q) = KILL(q) – true = false