Dataflow Analysis

Dataﬂow Analysis
Miller Lee
June 8, 2013
Miller Lee Dataﬂow Analysis

Available Expression
Aim
An expression e is available at program point p if e have
already been computed, and not later modiﬁed, on all paths to
p.
Why we need AE analysis?
x = a + b;
y = a * b;
while y > a + b do
a = a + 1;
x = a + b;
end
x = a + b;
y = a * b;
while y > x do
a = a + 1;
x = a + b;
end
Test it at run-time is not eﬃcient.

Let’s analyze it
1 List all expressions: a + b, a * b, a + 1.
2 How long is the life span of an expression?
When is an expression be generated??
When is an expression be killed??
3 How to automate the analysis?

The Basic Idea

Available Expression
x = a + b;
y = a * b;
while y > a + b do
a = a + 1;
x = a + b;
end

Solution

Here comes the problems
Why does it work?
How can we prove that this algorithm is correct?
Do this algorithm deﬁnitely stop?
Because it works, so it is correct, this reason is not accepeted.
There are two parts we can notice.
1 The relations between data
2 The characteristic of the transfer function

Dataﬂow and Lattices

Partial Order
Deﬁnition
Giving a Pair (P, R)
P is a set of elements.
R is a relation over P (R ⊆ PxP)
R is reﬂexive, anti-symmetric, and transitive.

Example
Figure : Partial Order, R = ⊆

Least Upper Bound
Given partial order (D, ≤), for each S ⊆ D
g is an upper bound of S if x ∈ D and ∀y ∈ S : y ≤ x
g ∈ V is a least upper bound. x, y ∈ S
1 x ≤ g
2 y ≤ g
3 If z is any element that x ≤ z and y ≤ z, then g ≤ z
Join Operation V
x V y is the least upper bound of {x, y}

Greatest Lower Bound
Given partial order (D, ≤), for each S ⊆ D
g is an lower bound of S if x ∈ D and ∀y ∈ S : x ≤ y
g ∈ V is a greatest lower bound. x, y ∈ S
1 g ≤ x
2 g ≤ y
3 If z is any element that z ≤ x and z ≤ y, then z ≤ g
Meet Operation ∧
x ∧ y is the greatest lower bound of {x, y}

Examples
Figure : ub = {a,b}, which is also a lub.

Examples
Figure : The viewpoint of teelung People

Lattice
Deﬁnition
A partial order is a lattice if every two elements of P have a
unique least upper bound and greatest lower bound.
1 idempotent: x ∧ x = x
2 commutative: x ∧ y = y ∧ x
3 associative: x ∧ (y ∧ z) = (x ∧ y) ∧ z
Special Elements
top element : for all x ∈ V , ∧ x = x.
bottom element ⊥: for all x ∈ V , ⊥ ∧x =⊥.
Semilattice
A join semi-lattice (meet semi-lattice) has only the join (meet)
operator deﬁned.

Examples
R = ∪
= {a, b, c}
⊥ = {}
For any x ∈ P({a, b, c}),
{}∪x = x,
{a,b,c}∪x={a,b,c}.
Figure : P({a,b,c})

Examples
(2S
, ⊆) forms a lattice for any set S.
2S is a powerset of S, the set of all subsets of S.
if (S, ≤) is a semilattice, so is (S, ≥)
i.e., can ”ﬂip” the lattice
Lattice for constant propagation

Algorithm for a forward data-ﬂow problem
OUT[ENTRY] = VENTRY
for each basic block B other than ENTRY do
OUT[B] =
end
while changes to any OUT occur do
for each basic block B other than ENTRY do
IN[B] = ∧S a predecessor of BOUT[S];
OUT[B] = fB(IN[B]);
end
end

The Dataﬂow Equation of Available Expression
Let s be a statement
succs(s) = { immediate successor stmts of s }
pres(s) = { immediate predecessor stmts of s }
In(s) = program just before executing s
Out(s) = program just after executing s
In(s) = ∩s ∈preds(s) Out(s )
Out(s) = Gen(s)∪(In(S) - Kill(s))

Monotone Functions
Deﬁnition
If (D, ⊆) and (D , ⊆ ) are two posets and F : D → D is
function, the F is called monotone if and only if
F(x) ⊆ F(y) for any x, y ∈ D with x ⊆ y.
Observation
Functions for computing In(s) and Out(s) are monotonic
In(s) = ∩s ∈preds(s) Out(s )
Out(s) = Gen(s)∪(In(S) - Kill(s))
Extensivity
People think f is monotonic if x ≤ f (x). This is a diﬀerent
property called extensivity.

Examples
Monotonic functions
x → {}
x → x ∪ {a}
x → x − {a}
Not monotone
x → {a} − x
Extensivity
x → x ∪ {a}
x → {a} − x
Figure : P({a,b,c})

Chain
Chain
Given a poset (D, ⊆), a chain in D is an inﬁnite sequence
d0 ⊆ d1 ⊆ d2 ⊆ · · · ⊆ dn ⊆ . . . of elements in D, also written
using set notation as {dn|n ∈ N}.
Stationary
A chain is called stationary when there is some n ∈ N such
that dm = dm+1 for all m > n.

Examples
The sets N, Z, Q, R of natural numbers, integers,
rationals, and real numbers form chains under their usual
order.

The Fixed-Point Theorem
Theorem
Let (D, ⊆, ⊥) be a semilattice, let
F : (D, ⊆, ⊥) → (D, ⊆, ⊥) be a continuous function, and let
fix(F) be the lub of the chain {Fn
(⊥)|n ∈ N}. Then fix(F) is
the least fix-point of F.

Examples
For D = P({a,b,c}), so ⊥ is {}
Identity function: sequence is {},{},{}. . . so least
fixpoint is {}. And all the elements are fixpoints.
x → x ∪ {a}: sequence is {},{a},{a},{a},. . . so least
fixpoint is {a}.{a},{a,b},{a,b,c} are all fixpoints.
x → {a} − x: no fixpoints.

Observation
1 If Algorithm converges, the result is a solution to the
data-ﬂow equations.

Observation
2 If the frame work is monotone, then the solution found is
the maximum ﬁxedpoint of the data-ﬂow equations.

Observation
2 If the frame work is monotone, then the solution found is
the maximum fixedpoint of the data-flow equations.
3 If the semilattice of the framework is monotone and of
finite height, then the algorithm is guaranteed to
converge.

Distributive
Deﬁnition
f (x ∧ y) = f (x) ∧ f (y) for all x and y in V and f in F.
Beneﬁt
Joins lose no information
k(h(f(T)∧g(T)))
= k(h(f(T))∧h(g(T)))
= k(h(f(T)))∧k(h(g(T)))

Meaning of a Data-Flow Solution
The Ideal Solution
The Meet-Over-Paths Solution
Maximun Fixedpoint: the result of the iterative algorithm.

The Ideal Solution
Property
IDEAL[B] =∧P, a possible path from ENTRY to BfP(vENTRY )
In terms of the lattice-theoretic partial order ≤ for the
framework in question.
Any answer that is greater than IDEAL is incorrect.
Any value smaller than or equal to the ideal is
conservative, i.e., safe.

The Meet-Over-Paths Solution
Property
MOP[B]=∧P, a path from ENTRY to BfP(vENTRY ).
For all B we have MOP[B]≤IDEAL[B].
Too much information
If the transfer functions are distributive, produce the MFP
solution.

MFP v.s. MOP
The number of paths considered is unbounded if the ﬂow
graph contains cycles.
→ the MOP does not lend itself to a direct algorithm.

MFP v.s. MOP
The number of paths considered is unbounded if the ﬂow
graph contains cycles.
→ the MOP does not lend itself to a direct algorithm.
MOP ≤ IDEAL and MFP ≤ MOP, we know that
MFP ≤ IDEAL (SAVE)
MFP ≤ MOP, because the meet operator is applied at
the end in the deﬁniton of MOP.

Non-distributive Example
Constant Propagation
In general, analysis of what the program computes is not
distributive.
MOP ≤ iterative dataﬂow solution.

The Drawback of Dataﬂow Analysis
We need to keep track of lots of irrelevant details at every
program point.
We need to consider all the path that the program will
execute.
Hard to analyze to pointers.

Example

Dataflow Analysis

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