Advanced Compilers Data Flow Analysis Background Theoryplas.cnu.ac.kr/courses/2017f/a_compilers/ac...

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Advanced CompilersData Flow Analysis – Background

TheoryFall. 2017

Chungnam National Univ.

Eun-Sun Cho

States and Paths

• States

– Consists of the values of all the variables

– The execution of a program can be viewed as a series of

transformations of the program state

– Each execution of an intermediate code statement transforms an input

state to a new output state

• Input state : associated with the program point before the statement

• Output state : associated with the program point after the statement

• Execution path from point p1 to point pn to be a sequence of

points p1, p2,… pn such that for each i = 1, 2, …n-1, either

– pi is the point immediately preceding a statement and pi+1 is the point

immediately following that same statement, or

– pi is the end of some block and pi+1 is the beginning of a successor

block 2

There is an infinite number of possible execution paths through a program

– There is no finite upper bound on the length of an execution path

(1,2,3,4,9) (1,2,3,4,5,6,7,8,3,4,9) (1,2,3,4,5,6,7,8,3,4,5,6,7,8,3,4,9)…3

d1 : a = 1

if read() <=0 goto B4

d2 : b = a

d3 : a = 243

goto B2

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

B1

B2

B3

B4

Data Flow Analysis – Basic Concepts

• Dataflow analysis has a unified theory.

• Basic concepts

– Data flow information:

• represented as semilattice

– Data flow functions:

• model effect of basic blocks

– Data flow equations:

• relations of control flow and effects of basic blocks

– Data flow solutions: (finally!)

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Dataflow Values

• Dataflow values

– Abstraction of set of all possible program states that can be observed

for that point

– A domain means the set of “possible dataflow values”

• Eg. The domain of dataflow values for reaching definitions is the

set of all subsets of definitions in the program

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Semilattices

Semi-lattice L for representing DFA information;

– L is an algebraic structure L , , T

– L consists of a set of values: L={x1, x2,...}

• L might have infinite number of elements

– L has a meet operator z=x y, where x, y, z L

– Two unique elements of L: , T (bottom, top)

– Height of semi-lattice is finite

– L can be an algebraic product:

• L= L1 L2 .... Lk

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Properties of Meet Operator

• For all x, y L, there exists a unique z L s.t. z=x y

(closure)

• For all x,y L: x y = y x (commutativity)

• For all x,y,z L: (x y) z = x (y z) (associativity)

• For all x L, (x ) =

• For all x L, (x T) = x

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Example

• Consider as

– x y is unique

– x y is y x

– (x y) z = x (y z)

– (x {} ) = {} ( is {})

– (x U ) = x, when U is universe (T is U)

• Consider as

– x y is unique

– x y is y x

– (x y) z = x (y z)

– (x U ) = U ( is U)

– (x {} ) = x (T is {})

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Partial Ordering and Meet Operator

• Meet operator induces a partial order ( ) on values in

L:

– x y x y = x

– Note that we usually do in the opposite way ; define (that is, g.l.b.)

from a given

• Strict partial order

– x y x y = x and x y

• Height of L

– the length of the longest strictly ascending chain

– the maximal n s.t. there ,…, exist x1, x2 , …, xn s.t.

x1 x2 … xn T9

Example

• Consider as

– x y x y = x ( is )

• Consider as

– x y x y = x ( is )

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Note: Partial Ordering

• Partial order

– If x y then value x has less information than value y.

• Properties of partial ordering

– Reflexivity (for all x, x x)

– Antisymmetry (if x y and y x, then x = y)

– Transitivity (if x y and y z, then x z)

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Examples of Semilattices

• Infinite number of elements

• Top : any values (UNDEF)

• Bottom: not a constant (NAC)

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{d1} {d2} {d3}

{d1, d2} {d2 , d3}{d1 , d3}

= {d1 , d2 , d3}

T = {}

• Meet operator : set union

• Top : no RD

• Bottom: all RD

RD means the elements in the domain of Reaching Definition Analysis

-2 -1 0 1 2 ……

T

= NAC

= UNDEF

Set As a Bit Vector

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<1,0,0> <0,1,0> <0,0,1>

<1,1,0> <0,1,1><1,0,1>

= <1,1,1>

T = <0,0,0>

• Meet operator : bitwise

• Top : <0,0,0>

• Bottom: <1,1,1>

{d1} {d2} {d3}

{d1, d2} {d2 , d3}{d1 , d3}

= {d1 , d2 , d3}

T = {}

• Meet operator : set union

• Top : no RD

• Bottom: all RD

previous examplebit vector

representation

Efficient representation

Then,

why do we use semilattice as dataflow values?

The answer will be shown later on...

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• Dataflow analysis is a unified theory.

• Basic concepts








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Data Flow Functions

• Flow Functions

– model “effect” of basic blocks

– a mapping from the lattice used in the analysis to itself

– eg. in Reaching Definition Analysis

for each basic block X, do

IN(X) = Y predessor(X) OUT(Y)

OUT(X) = GEN(X) + (IN(X) – KILL(X))

FX : A function takes IN(X) and yields OUT(X)

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Flow Function

for each basic block X

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receive m (val)

f0 0

f1 1

return m

m <= 1

i 2

i <= m

return f2f2 f0 + f1

f0 f1

f1 f2

i i+1

1

2

3

4

8

9

10

11

N Y

Bit

Position

Definition Basic

Block

1 m in node 1 B1

2 f0 in node 2

3 f1 in node 3

4 i in node 5 B3

5 f2 in node 8 B6

6 f0 in node 9

7 f1 in node 10

8 i in node 11

B1

B6FB1(<x1x2x3x4x5x6x7x8>)=<111x4x500x8>

FB6(<x1x2x3x4x5x6x7x8>)=<x10001111>

B3



• Basic concepts








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Equations for Iterative Analysis

• in(B) = Init , for B = entry

Q Pred(B) out(Q) , otherwise

• out(B) = FB(in(B))

or

• in(B) = Init , for B = entry

Q Pred(B) FQ(in(Q)) , otherwise

• Solution : actually undecidable

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• Basic concepts








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MOP: Meet-Over-All Paths

• The problem of deciding if an arbitrary path in a

program is executable is undecidable

– Program analysis is commonly performed under the

assumption that “all paths in the program are executable”

• MOP (meet-over-all-paths)

– for every path in the flow graph is taken,

– MOP(B) = p Path(B) Fp(Init) , for each block B

– where Fp = FBn …. FB1, for B1 =entry, … , Bn=B

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Hard to Solve the Equations for MOP : Why?

• out(B2) = FB2(in(B2)) = FB2 ( Q Pred(B2) FQ(in(Q)) )

= FB2 (FB1(in(B1)) FB3(in(B3)))

= FB2 (FB1(Init) FB3(in(B3)))

since in(B3) = Q Pred(B3) FQ(in(Q)) = FB2(in(B2))

FB2(in(B2)) = FB2 (FB1(Init) FB3(FB2(in(B2) ))

out(B2) = FB2 (FB1(Init) FB3(out(B2))

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B2

B1(Entry)

B3

“What we want to know” (the estimated state after B2)

“What-we-want-to-know” is defined by “what-we-want-to-know” itself.

Equations of Similar Types

• Examples : “x = f(x)”

x = 2 * x ..…… x = 0

x = x + 1 ….. no answer

x = x * 1…… many (the whole domain) answers

x = x ∪ {a} …. x is a set which contains a.

and

x = FB2 (FB1(Init) FB3(x)) ….. x is ??

(x is out(B2), “What we want to know” and the estimated state after B2)

Note that such out(B2) is one of the dataflow values

and we know the dataflow values is supposed to form a semilattice.

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B2

B1(Entry)

B3

Approximated Solution - MFP

• MFP -Maximal

Fixed Point solution

an iterative

algorithm

visit and evaluate

in/out of each B

from an initial

value

do again and

again until there

is no change

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initialize IN(X) = for all basic blocks X

initialize OUT(X) = GEN(X) for all basic blocks X

change = 1

while (change) do

change = 0

for each basic block X, do

old_OUT = OUT(X)

IN(X) = Y predessor(X) OUT(Y)

OUT(X) = GEN(X) + (IN(X) – KILL(X))

if (old_OUT != OUT(X)) then

change = 1

endif

endfor

endfor eg. Reaching Definition Analysis

In MFP Algorithm - Intuitive Idea

• For all X, OUT(X) is a set of RD, that is, a

semilattice!

– for all X, OUT(X) only increases (or remains the same)

while the analysis is going

• Even along the longest chain, it grows and grows until it

reaches (after a finite number of iteration, by the

definition of semilattice,) And it is not able to increase

further, so will remain the same.

• Or it will stop somewhere in the middle of the DAG, and

remains the same.

• This is why we chose the semilattice with finite height as

the domain of dataflow values, which guarantees that the

finite number of iteration will give an answer, when it only

grows.

25How can we prove it? (need more theory)

MFP – More Theory (1)

• “How can we know that OUT(B) is always increasing?”

• Consider following definitions of monotone functions

– (Monotone functions) : a function f is monotone, for all x, y, x y

implies f(x) f(y)

– “For a monotone function, f () < f f () is always true.”

• Proof) < f (), since is the bottom, the lowest element.

• f () < f f (), since f is monotone.

– “If we consider FB (X) = GEN(B) ∪ (X – KILL(B)), it is monotone.”

• Proof Sketch)

• For X1, X2, s.t. X1 < X2, it is true that C + (X1 -D) < C + (X2 -D).

• Thus FB (X1) < FB (X2) , from these two facts.

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• The algorithm repeat until there is no change is

– one of the typical solution of “fixed point problem”

• (Fixed point) a fixed point of a function f: LL is an element z L

s.t. f(z) = z.

• Examples

x = 2x ..…… 0 is the fixed point

x = x + 1 ….. no fixed point

x = x * 1…… many fixed points (the whole domain)

x = x ∪ {a} …. any set containing a is a fixed point

and

X= FB2 (FB1(Init) FB3(X)) ….. ???

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B2

B1(Entry)

B3


• (Distributive) f(x y) = f(x) f(y)

• (LF ) LF is a set of all monotone functions from L to L

• (f n ) For any f LF, f n is defined by

f 0 = id and

for n 1, f n = f f n-1 Note that f g (x) = f(g(x))

• (Fix f) Fix f is {f n | n >= 0} for distributive f LF

in other words, “if f is continuous ccpo (chain complete partial ordering)”

actually T instead of for meet operator •

• FIXED POINT THEORM “Fix f is the least fixed point of f” (..believe it!)

– it stops and it is the minimum!

– it is also true when f is FB(X) = GEN(B) ∪ (X – KILL(B))

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MFP Theory (4)

• Example 1. To find out a least fixed point of f(A) = A∪ {a}

to solve equation A = A∪ {a}

step1. apply {}

thus f({}) = {}∪{a} = {a}

step 2. apply f({})

thus f(f({})) = {a} ∪{a} = {a}

since this will not change further, {a} is the least fixed point of f(A)= A∪ {a}

• Example 2. To solve X= FB2 (FB1(Init) FB3(X)) ... when X is OUT(B2)

step1. apply {}

thus f({}) = FB2 (FB1(Init) FB3({}))

step2. apply f({})

thus f(f({})) = FB2 (FB1(Init) FB3(FB2 (FB1(Init) FB3({})) ))

step 3. apply f(f({})) .....

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B2

B1(Entry)

B3

Properties of MFP

• MFP is not always

equal to MOP

• eg. constant analysis

– clearly the value

assigned to w is 3

– but, at entry to B3 all

we know is that

neither u’s value nor

v’s value is a constant

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entry

w > 0

u 1

v 2

u 2

v 1

w u + v

exit

B1

B0

B2

B3

MOP = MFP only when FB is a monotone and distributive

function

Properties of MFP (More)

• Distributive) f(x y) = f(x) f(y)

• FB:

– If s is not an assignment statement, then FB is the identity

function

– If s is an assignment (x = ..) ,

• RHS of s is constant … emit x c

• RHS of s is y + z is emit x const if both y and z are const

– NAC, if one of them is NAC (… )

– UNDEF, otherwise (… T)

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FB3(FB1 (m0) FB2 (m0)) FB3 (FB1 (m0)) FB3 (FB2 (m0))

FB3(FB1 (m0) FB2 (m0)) < FB3 (FB1 (m0)) FB3 (FB2 (m0))

Levels of Approximation

• MFP MOP IDEAL

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= IDEALMOP

MFP

Universe

Exact set of behaviors

IDEAL

UNSAFE

Under-estimation is erroneous.

Example: Constant Propagation

• Aims

– Proves that a variable always has a known value

– Specializes codes around that value

• Moves some computations to compile time

• Exposes some unreachable blocks

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Dataflow Values

• The set of data-flow values is a product lattice

• The lattice for a single variable

– All constants appropriate for the type of the variable

– (NAC) : not a constant

– T (UNDEF) : undefined

– The semilattice for a typical integer-valued variable

• A dataflow value for this framework is a map from each

variable in the program to one of the values in the constant

semilattice.

• The value of a variable v in a map m is denoted by m(v)

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• fs : the transfer function of statement s

• If s is not an assignment statement, then fs is simply the

identity function

• If s is an assignment to variable x, then

• m’(v) = m(v), for all variables v x, where m’ = fs (m)

(a) If the RHS of the statement s is a constant c, then m’(x) = c

(b) If the RHS is of the form y+z ,then

m’(x) = m(y) + m(z) if m(y) and m(z) are constant values

NAC if either m(y) or m(z) is NAC

UNDEF otherwise

(c) If the RHS is any other expression (e.g. a function call or assignment

through a pointer), then m’(x) = NAC

Transfer Function

Monotonicity of Transfer Function

• In case (b), each possible

input value of y, the value of x

does not get bigger as the

value of z gets smaller :

monotone

• Otherwise, fs either does not

change the value of m(x), or it

changes the map to return a

constant or NAC : monotone

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m(y) m(z) m’(x)

UNDEF

(T)

UNDEF UNDEF

c2 UNDEF

NAC NAC

c1 UNDEF UNDEF

c2 c1+c2

NAC NAC

NAC

()

UNDEF NAC

c2 NAC

NAC NAC

Equation

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UNDEF

(T)

c1 NAC

()

UNDEF (T) UNDEF c1 NAC

c1 c1 c1 NAC

c2 c1 c2 NAC NAC

NAC () NAC NAC NAC

• in(s) = Init , for s = entry

q Pred(s) out(q) , otherwise

• out(s) = fs(in(s))

Notes

• Initial value : m0(v) = T (UNDEF) for all variable v

– 프로그램이진행해가면서 NAC() 가많아진다

– 더이상변화가없을때까지반복한다.

• As long as there exists a path that defines a variable

reaching a program point, the variable will not have an

UNDEF value

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review Nondistributivity

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fB3(fB1 (m0) fB2 (m0)) < fB3 (fB1 (m0)) fB3 (fB2 (m0))

Nondistributivity

f(x y) f(x) f(y)

• clearly the value

assigned to w is 3

• but, at entry to B3 all we

know is that neither u’s

value nor v’s value is a

constant

• MFP but not MOP!

B1

Advanced Compilers Data Flow Analysis Background Theoryplas.cnu.ac.kr/courses/2017f/a_compilers/ac...

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