composing dataflow analyses and transformations
DESCRIPTION
Composing Dataflow Analyses and Transformations. Sorin Lerner (University of Washington) David Grove (IBM T.J. Watson) Craig Chambers (University of Washington). const prop followed by unreachable code elimination. const prop again. Phase ordering problem. - PowerPoint PPT PresentationTRANSCRIPT
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Composing Dataflow Analyses and Transformations
Sorin Lerner (University of Washington)David Grove (IBM T.J. Watson)
Craig Chambers (University of Washington)
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x := 11;if (x == 11) { DoSomething();}else { DoSomethingElse(); x := x + 1;}y := x; // value of y?
Phase ordering problem
• Optimizations can interact in mutually beneficial ways, and no order exploits all of these interactions.
• Classic example: constant propagation and unreachable code elimination.
x := 11;DoSomething();y := x;// value of y?
x := 11;DoSomething();y := 11;
const prop followedby unreachablecode elimination
const propagain
true
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One known solution: Iterate individual analyses until the results don’t change
x := 11;do { if (x == 11) { DoSomething(); } else { DoSomethingElse(); x := x + 1; }} while (...)y := x; // value of y?
•Compiler is slow.•In the presence of loops in the source program, might not yield best possible results.
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Another known solution: hand writtensuper-analysis
Lose modularity:– difficult to write,
reuse, and extend such analyses
Examples:– conditional constant
propagation [Wegman and Zadeck 91]
– class analysis, splitting and inlining [Chambers and Ungar 90]
– const prop and pointer analysis [Pioli and Hind 99]
MonolithicSuper-Analysis
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Ideally...
• ... we want to:– Write analyses modularly– Exploit mutually beneficial interactions– Have a fast compiler
• We present a framework that achieves this.
Composition Framework
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The key to modular composition
• Traditionally, optimizations are defined in two parts:1. A dataflow analysis.2. Rules for transforming the program representation
after the analysis is solved.
• The key insight is to merge these two parts:– Dataflow functions return either a dataflow value OR
a replacement graph with which to replace the current statement.
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Roadmap
• Several small examples that show how flow functions work
• One large example that shows how modular analyses are automatically composed together
• Overview of the theory behind the framework• Experimental validation
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Flow function returning a dataflow value
y := 5
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Flow function returning a dataflow value
y := 5
[ ... ]
[ ..., y → 5]
PROPAGATE
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Flow function returning a replacement graph
y := x+2
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[x → 3]
Flow function returning a replacement graph
y := x+2
[x → 3]
REPLACEy := 5
Replacement
graph
Step 1: Initialize input edges with dataflow information
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Flow function returning a replacement graph
y := x+2y := x+2
[x [x →→ 3] 3]
y := 5
[x → 3]
PROPAGATE
[x → 3, y → 5]
Step 2: Perform recursive dataflow analysis on the replacement graph
Step 1: Initialize Step 1: Initialize input edges with input edges with dataflow dataflow informationinformation
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Flow function returning a replacement graph
y := x+2y := x+2
[x [x →→ 3] 3]
y := 5
[x → 3]
PROPAGATE
[x → 3, y → 5][x → 3, y → 5]
Step 2: Perform Step 2: Perform recursive dataflow recursive dataflow analysis on the analysis on the replacement replacement graphgraph
Step 1: Initialize Step 1: Initialize input edges with input edges with dataflow dataflow informationinformation
Step 3: Propagate dataflow information from output edges.
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Flow function returning a replacement graph
y := x+2
[x → 3]
[x → 3, y → 5]
Replacement graphs:– used to compute
outgoing dataflow information for the current statement.
Replacement graphs:– used to compute
outgoing dataflow information for the current statement.
– a convenient way of specifying what might otherwise be a complicated flow function.
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Flow function returning a replacement graph
y := x+2
[x → 3]
[x → 3, y → 5]
Soundness requirement:– Replacement graph must
have the same concrete semantics as the original statement, but only on concrete inputs that are consistent with the current dataflow facts.
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Flow function returning a replacement graph
y := x+2
[x → 3]
[x → 3, y → 5]
Let’s assume we’ve reached a fixed point.
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Flow function returning a replacement graph
y := x+2
[x → 3]
[x → 3, y → 5]
y := 5
Let’s assume we’ve reached a fixed point.
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Flow function returning a replacement graph
y := 5
[x → 3]
[x → 3, y → 5]
Replacement graphs:– used to transform the
program once a fixed point has been reached.
Let’s assume we’ve reached a fixed point.
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Iterative analysis example
y := x+2
[x → 3, y → 5]
[x → 3] [x → T]
Now, let’s assume we haven’t reached a fixed point.
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Iterative analysis example
y := x+2
[x → 3, y → 5]
PROPAGATE
[x → 3] [x → T]
[x → T, y → T]
Now, let’s assume we haven’t reached a fixed point.
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Branch folding example
if (x == 11)
F T
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Branch folding example
if (x == 11)REPLACE
[x → 11][x → 11]
F T
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Branch folding example
if (x == 11)if (x == 11)
[x [x →→ 11] 11] [x → 11]
[x → 11]FF TT [x → 11]
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Branch folding example
if (x == 11)
[x → 11]
F T [x → 11]
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Composing several analyses
x := new C;do { b := x instanceof C; if (b) { x := x.foo(); } else { x := new D; }} while (...)
class A { A foo() { return new A; }};class C extends A { A foo() { return self; }};class D extends A {};
Constant PropagationClass Analysis
Inlining
Unreachable code elimination
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x := new C
merge
b := x instanceof C
x := new D x := x.foo()
merge
while(…)
if (b)
TF
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
PROPAGATE
while(…)
PROPAGATE PROPAGATE
[x → T] [x → {C}] T
merge
merge
TF
PROPAGATE
T
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
PROPAGATE
while(…)
PROPAGATE PROPAGATE
[x → T] [x → {C}] T([x → T], [x → {C}], T, T)
merge
merge
PROPAGATE
TF
T
PROPAGATE
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
PROPAGATE
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
while(…)
merge
merge
TF
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
while(…)
PROPAGATE
[x → T, b → T]
merge
merge
TF
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
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([x → T], [x → {C}], T, T)
x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
([x → T], [x → {C}], T, T)
REPLACE
b := true
while(…)
[x → T, b → T]
merge
merge
TF
([x → T], [x → {C}], T, T)
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x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
b := true
([x → T], [x → {C}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
PROPAGATE
while(…)while(…)
mergemerge
mergemerge
TTFF
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([x → T, b → true],
[x → {C}, b → {Bool}], T, T)([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)b := true
([x → T], [x → {C}], T, T)
while(…)while(…)
mergemerge
mergemerge
TTFF
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
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([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
• Replacement graph is analyzed by composed analysis.
• When one analysis chooses a replacement graph, other analyses see it immediately.
• Analyses communicate implicitly through graph transformations
while(…)
merge
merge
TF
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
REPLACE
σ
while(…)
merge
merge
TF
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T) σσ
while(…)
merge
merge
TF
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
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x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT)) σσσσ σ
σ
while(…)while(…)
mergemerge
mergemerge
TTFF( , , , )
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
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x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)
σ
σσ
while(…)while(…)
mergemerge
mergemerge
TTFF
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT)) σσ
( , , , )( , , , )
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
while(…)
merge
merge
TF
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
( , , , )
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
while(…)
merge
merge
TF
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
( , , , )
REPLACE
( , , , )
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x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT))
while(…)while(…)
mergemerge
mergemerge
TTFF
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT))
( , ,( , , , ), )
( , , , )
( , , , )( , , , )
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x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
while(…)
merge
merge
TF
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
( , , , )
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
( , , , )
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σ
x := new C
b := x instanceof C
x := new D x := x.foo()
if (b)([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
REPLACE
x := C::foo(x)
while(…)
merge
merge
T( , , , )
( , , , )
F
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
σ
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x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT))
x := C::foo(x)
σREPLA
CE
x := x
σ
class C extends A { A foo() { return self; }}
while(…)while(…)
mergemerge
mergemerge
FF TT σσ( , ,( , , , ), )
( , ,( , , , ), )
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT))
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x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)
x := C::foo(x)x := C::foo(x)
σσ
x := x
σ
σPROPAGATE
while(…)while(…)
mergemerge
mergemerge
FF TT([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT)) σσ( , ,( , , , ), )
( , ,( , , , ), )
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT))
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x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)
x := C::foo(x)x := C::foo(x)
σσ
x := x
σ
σσ
while(…)while(…)
mergemerge
mergemerge
FF TT([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT)) σσ( , ,( , , , ), )
( , ,( , , , ), )
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT))
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x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)
x := C::foo(x)
σ
σσ
while(…)while(…)
mergemerge
mergemerge
FF TT
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT))
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT)) σσ( , ,( , , , ), )
( , ,( , , , ), )
![Page 48: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/48.jpg)
x := new Cx := new C
b := x instanceof Cb := x instanceof C
x := new Dx := new D x := x.foo()x := x.foo()
if (b)if (b)
x := C::foo(x)
σ
σ
while(…)while(…)
mergemerge
mergemerge
FF TT([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT)) σσ( , ,( , , , ), )
( , ,( , , , ), )
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ TT], [x ], [x →→ {C}], {C}], TT, , TT))
([x ([x →→ T, T, b b →→ true], true],
[x [x →→ {C}, b {C}, b →→ {Bool}], {Bool}], TT, , TT))
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
![Page 49: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/49.jpg)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
x := new C
b := x instanceof C
x := x.foo()
if (b)
while(…)
merge
merge
T([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
( , , , )
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
( , , , )
x := new D
F
![Page 50: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/50.jpg)
x := new C
b := x instanceof C
x := x.foo()
if (b)
PROPAGATE([x → T, b → true],
[x → {C}, b → {Bool}], T , T)
while(…)
merge
merge
T
x := new D
F
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
( , , , )
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
( , , , )
![Page 51: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/51.jpg)
x := new C
b := x instanceof C
x := x.foo()
if (b)
PROPAGATE
([x → T], [x → {C}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
while(…)
merge
merge
T
x := new D
F
([x → T, b → true],
[x → {C}, b → {Bool}], T , T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
( , , , )
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
( , , , )
![Page 52: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/52.jpg)
x := new C
b := x instanceof C
x := x.foo()
if (b)
while(…)
merge
merge
T
x := new D
F
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T , T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
([x → T, b → true],
[x → {C}, b → {Bool}], T, T)
( , , , )
([x → T], [x → {C}], T, T)
([x → T], [x → {C}], T, T)
( , , , )
([x → T], [x → {C}], T, T)
![Page 53: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/53.jpg)
x := new C
b := x instanceof C
x := x.foo()
if (b)
x := x
b := true
while(…)
merge
merge
T
x := new D
F
![Page 54: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/54.jpg)
x := new C
b := true
x := x
x := new C;do { b := x instanceof C; if (b) { x := x.foo(); } else { x := new D; }} while (...)
x := new C;do { b := true; x := x;} while (...)while(…)
merge
merge
![Page 55: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/55.jpg)
x := new C;do { b := x instanceof C; if (b) { x := x.foo(); } else { x := new D; }} while (...)
x := new C;do { b := true; x := x;} while (...)
•Analyses are defined modularly and separately.
•Combining them achieves the results of a monolithic analysis.
• If the analyses were run separately in any order any number of times, no optimizations could be performed.
![Page 56: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/56.jpg)
Theoretical foundation
• Definition: used abstract interpretation to define precisely how the framework composes analyses.
• Soundness theorem: if the individual analyses are sound, the composed analysis is sound.
• Termination theorem: composed analyses are guaranteed to terminate, under reasonable conditions.
• Precision theorem: if the composed flow function is monotonic, the composed analysis is guaranteed to produce results as precise as running the individual analyses in sequence, any number of times, in any order.
![Page 57: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/57.jpg)
Experimental validation
• Implemented and used our framework in the Vortex and Whirlwind compilers for 5+ years.– composed: class analysis, splitting, inlining, const
prop, CSE, removal of redundant loads and stores, symbolic assertion prop
• Compiled a range of big programs in Vortex.– largest benchmark: Vortex compiler with ~60,000 lines
• Our framework vs. iteration:– compiles at least 5 times faster
• Our framework vs. monolithic super-analysis:– same precision– compiles at most 20% slower
![Page 58: Composing Dataflow Analyses and Transformations](https://reader035.vdocuments.us/reader035/viewer/2022062410/56815d34550346895dcb344f/html5/thumbnails/58.jpg)
Conclusions
• We developed and implemented a new approach for defining dataflow analyses.
• Our approach allows analyses to be written modularly:– easier to write and reuse analyses.
• Our approach allows analyses to be automatically combined into a monolithic analysis.