Notes on Finish Basic Dataflow Analysis Predicate Relation Analysis | EECS 583, Study notes of Electrical and Electronics Engineering

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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.

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Reaching vs Available Definitions^ 1:r1 = r2 + r3^ 2:r6 = r4 – r

1,2 reach^ 1,2 available^ 3:r4 = 4^ 4:r6 = 8 5:r6 = r2 + r36:r7 = r4 – r5^ 1,2,3,4 reach1 available 1,2 reach 1,2 available

1,3,4 reach 1,3,4 available

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Available Definition Analysis (Adefs)^ Y^ A definition d is available

at a point p if along all

paths

from d to p, d is not killed Y Remember, a definition of a variable is killed

between 2

points when there is another definition of that variablealong the path^ »^ r1 = r2 + r3 kills previous definitions of r1 Y Algorithm^ »^ Forward dataflow analysis as propagation occurs from defsdownwards^ »^ Use the Intersect function as the meet operator to guarantee theall-path requirement^ »^ GEN/KILL/IN/OUT similar to reaching defs^ y^ Initialization of IN/OUT is the tricky part

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Compute IN/OUT Sets for all BBs (Adefs)

U = universal set of all operations in the Procedure IN(0) = 0 OUT(0) = GEN(0) for each basic block in procedure, W, (W != 0), do^ IN(W) = 0^ OUT(W) = U – KILL(W) change = 1 while^ (change) do^ change = 0^ for^ each basic block in procedure, X, do^ old_OUT = OUT(X)^ IN(X) = Intersect(OUT(Y)) for all predecessors Y of X^ OUT(X) = GEN(X) + (IN(X) – KILL(X))^ if^ (old_OUT != OUT(X)) then^ change = 1^ endif^ endfor endfor

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Example Adefs Calculation^ 1: r1 = MEM[r2+0]^ 2: r2 = r2 + 1^ 3: r3 = r1 * r4^ 4: r1 = r1 + 5^ 5: r3 = r5 – r1^ 6: r8 = r3 * 2

7: r7 = 0 8: r1 = r1 + 5 9: r7 = r1 - 6 10: r8 = r7 + r8 11: r1 = r3 – r8 12: r3 = r1 * 2

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Class Problem - Aexprs Calculation^ 1: r1 = r6 * r9^ 2: r2 = r2 + 1^ 3: r5 = r3 * r4^ 4: r1 = r2 + 1^ 5: r3 = r3 * r4^ 6: r8 = r3 * 2

7: r7 = r3 * r4 8: r1 = r1 + 5 9: r7 = r1 - 6 10: r8 = r2 + 1 11: r1 = r3 * r4 12: r3 = r6 * r

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Efficient Calculation of Dataflow^ Y^ Order basic blocks are visited is important (fasterconvergence)^ Y^ Forward analysis – DFS order^ »^ Visit a node only when all its predecessors have been visited^ »^ Visit(n)^ y^

mark n as visited y for each successor of n not yet visited, s, do^ X^ Visit (s) y postorder(n) = count y count += 1 » count = 1; Visit(entry) Y Backward analysis – PostDFS order » Visit a node only when all of its successors have been visited

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Two Types of Analysis^ Y^ Predicate relation analysis^ »^ Properties about the values thatpredicates can take on^ »^ Disjoint

= 2 predicates nevertrue at the same time » Superset^ = 1 predicate truemore often » Subset = 1 predicate true lessoften » 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

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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 e1 (^1 0 0 0 1) y e2 (^0 1 1 0 1) y e3 (^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

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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??

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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

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Generating Partition Relations

symbol^ string

source 0 1 2 3 4 5 6 Predicate computations reordered to follow the paper p = !(a < b) * true q = (a < b) * true t1 = false t2 = t1 + !(a<b) * true t3 = t2 + (c != d) * q s = !(c != d) * q r = (c != d) * q

symbol^ string

source 0 false

false,t 1 true^

true 2 !(a<b)*

p,t 3 (a<b)*

q^

4 !(c=d)*

r 5 (c=d)*

s^

6 2+^

t3^ 6 = 2|

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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^ »^ UÆM and U

ÆN denoted tocome from same partition

Y^ Complete

• Contains a uniquenode 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^1 p^ q^

u^ v r^ s