Foundations of Data Flow Analysis...Why Data Flow Analysis Works? Suitable initial values and boundary conditions Suitable domain of values Bounded, Finite Suitable meet operator Suitable

CS738: Advanced Compiler Optimizations

Foundations of Data Flow Analysis

Amey Karkare

[email protected]

http://www.cse.iitk.ac.in/~karkare/cs738

Department of CSE, IIT Kanpur

[email protected]

http://www.cse.iitk.ac.in/~karkare/cs738

Agenda

◮ Intraprocedural Data Flow Analysis◮ We looked at 4 classic examples◮ Today: Mathematical foundations

Taxonomy of Dataflow Problems

◮ Categorized along several dimensions


◮ Categorized along several dimensions◮ the information they are designed to provide


◮ Categorized along several dimensions◮ the information they are designed to provide◮ the direction of flow


◮ Categorized along several dimensions◮ the information they are designed to provide◮ the direction of flow◮ confluence operator



◮ Four kinds of dataflow problems, distinguished by



◮ Four kinds of dataflow problems, distinguished by◮ the operator used for confluence or divergence



◮ Four kinds of dataflow problems, distinguished by◮ the operator used for confluence or divergence◮ data flows backward or forward


Confluence →⋃ ⋂

Direction ↓

Forward

Backward



Direction ↓

Forward R D

Backward



Direction ↓

Forward R D Av E

Backward



Direction ↓

Forward R D Av E

Backward L V



Direction ↓

Forward R D Av E

Backward L V V B E

Why Data Flow Analysis Works?

◮ Suitable initial values and boundary conditions

◮ Suitable domain of values◮ Bounded, Finite

◮ Suitable meet operator

◮ Suitable flow functions◮ monotonic, closed under composition

◮ But what is SUITABLE ?

Lattice Theory

Partially Ordered Sets

◮ Posets


◮ Posets

S: a set


◮ Posets

S: a set

≤: a relation


◮ Posets

S: a set

≤: a relation

(S,≤) is a poset if for x , y , z ∈ S


◮ Posets

S: a set

≤: a relation

(S,≤) is a poset if for x , y , z ∈ S◮ x ≤ x (reflexive)


◮ Posets

S: a set

≤: a relation

(S,≤) is a poset if for x , y , z ∈ S◮ x ≤ x (reflexive)◮ x ≤ y and y ≤ x ⇒ x = y (antisymmetric)


◮ Posets

S: a set

≤: a relation

(S,≤) is a poset if for x , y , z ∈ S◮ x ≤ x (reflexive)◮ x ≤ y and y ≤ x ⇒ x = y (antisymmetric)◮ x ≤ y and y ≤ z ⇒ x ≤ z (transitive)

Chain

◮ Linear Ordering

Chain

◮ Linear Ordering

◮ Poset where every pair of elements is comparable

Chain

◮ Linear Ordering


◮ x1 ≤ x2 ≤ . . . ≤ xk is a chain of length k

Chain

◮ Linear Ordering


◮ x1 ≤ x2 ≤ . . . ≤ xk is a chain of length k

◮ We are interested in chains of finite length

Observation

◮ Any finite nonempty subset of a poset has minimal and

maximal elements

Observation

◮ Any finite nonempty subset of a poset has minimal and

maximal elements

◮ Any finite nonempty chain has unique minimum and

maximum elements

Semilattice

◮ Set S and meet∧

Semilattice


◮ x , y , z ∈ S

Semilattice


◮ x , y , z ∈ S◮ x

∧x = x (idempotent)

Semilattice


◮ x , y , z ∈ S◮ x


◮ x∧

y = y∧

x (commutative)

Semilattice


◮ x , y , z ∈ S◮ x


◮ x∧

y = y∧

x (commutative)◮ x

∧(y

∧z) = (x

∧y)

∧z (associative)

Semilattice


◮ x , y , z ∈ S◮ x


◮ x∧

y = y∧


∧(y

∧z) = (x

∧y)

∧z (associative)

◮ Partial order for semilattice

Semilattice


◮ x , y , z ∈ S◮ x


◮ x∧

y = y∧


∧(y

∧z) = (x

∧y)

∧z (associative)

◮ Partial order for semilattice◮ x ≤ y if and only if x

∧y = x

Semilattice


◮ x , y , z ∈ S◮ x


◮ x∧

y = y∧


∧(y

∧z) = (x

∧y)

∧z (associative)

◮ Partial order for semilattice◮ x ≤ y if and only if x

∧y = x

◮ Reflexive, antisymmetric, transitive

Border Elements

◮ Top Element (⊤)

Border Elements

◮ Top Element (⊤)◮ ∀x ∈ S, x

∧⊤ = ⊤

∧x = x

Border Elements


∧⊤ = ⊤

∧x = x

◮ (Optional) Bottom Element (⊥)

Border Elements


∧⊤ = ⊤

∧x = x

◮ (Optional) Bottom Element (⊥)◮ ∀x ∈ S, x

∧⊥ = ⊥

∧x = ⊥

Familiar (Semi)Lattices

◮ Powerset for a set S, 2S



◮ Meet∧

is ∩



◮ Meet∧

is ∩

◮ Partial Order is ⊆



◮ Meet∧

is ∩


◮ Top element is S



◮ Meet∧

is ∩


◮ Top element is S

◮ Bottom element is ∅





◮ Meet∧

is ∪



◮ Meet∧

is ∪

◮ Partial Order is ⊇



◮ Meet∧

is ∪


◮ Top element is ∅



◮ Meet∧

is ∪


◮ Top element is ∅

◮ Bottom element is S

Greatest Lower Bound (glb)

◮ x , y , z ∈ S


◮ x , y , z ∈ S

◮ glb of x and y is an element g such that


◮ x , y , z ∈ S

◮ glb of x and y is an element g such that◮ g ≤ x


◮ x , y , z ∈ S

◮ glb of x and y is an element g such that◮ g ≤ x◮ g ≤ y


◮ x , y , z ∈ S

◮ glb of x and y is an element g such that◮ g ≤ x◮ g ≤ y◮ if z ≤ x and z ≤ y then z ≤ g

QQ

◮ x , y ∈ S

QQ

◮ x , y ∈ S

◮ (S,∧) is a semilattice

QQ

◮ x , y ∈ S

◮ (S,∧) is a semilattice

◮ Prove that x∧

y is glb of x and y .

Semi(?)-Lattice

◮ We can define symmetric concepts

Semi(?)-Lattice

◮ We can define symmetric concepts◮ ≥ order

Semi(?)-Lattice

◮ We can define symmetric concepts◮ ≥ order◮ Join operation (

∨)

Semi(?)-Lattice

◮ We can define symmetric concepts◮ ≥ order◮ Join operation (

∨)

◮ Least upper bound (lub)

Lattice

◮ (S,∧,∨) is a lattice

Lattice


iff for each non-empty finite subset Y of S

Lattice



both∧

Y and∨

Y are in S.

Lattice



both∧

Y and∨

Y are in S.

◮ (S,∧,∨) is a complete lattice

Lattice



both∧

Y and∨

Y are in S.


iff for each subset Y of S

Lattice



both∧

Y and∨

Y are in S.


iff for each subset Y of S

both∧

Y and∨

Y are in S.

Lattice

◮ Complete lattice (S,∧,∨)

Lattice


◮ For every pair of elements x and y , both x∧

y and x∨

y

should be in S

Lattice



y and x∨

y

should be in S◮ Example : Powerset lattice

Lattice



y and x∨

y


◮ We will talk about meet semi-lattices only

Lattice



y and x∨

y


◮ We will talk about meet semi-lattices only◮ except for some proofs

Lattice Diagram

◮ Graphical view of posets

Lattice Diagram


◮ Elements = the nodes in the graph

Lattice Diagram



◮ If x < y then x is depicted lower than y in the diagram

Lattice Diagram



◮ If x < y then x is depicted lower than y in the diagram

◮ An edge between x and y (x lower than y ) implies x < y

and no other element z exists s.t. x < z < y (i.e.

transitivity is excluded)

Lattice Diagram

∅

{c}

{b}

{a}{b, c}

{a, b}

{c, a}

{a, b, c}

Lattice Diagram for ({a, b, c},∩)

Lattice Diagram

∅

{c}

{b}

{a}{b, c}

{a, b}

{c, a}

{a, b, c}

Lattice Diagram for ({a, b, c},∩)

x∧

y = the highest z for which there are paths downward from

both x and y .

What if there is a large number of elements?

◮ Combine simple lattices to build a complex one



◮ Superset lattices for singletons

∅ ∅ ∅

{a} {b} {c}



◮ Superset lattices for singletons

∅ ∅ ∅

{a} {b} {c}

◮ Combine to form superset lattice for multi-element sets

Product Lattice

◮ (S,∧) is product lattice of (S1,

∧1) and (S2,

∧2) when

Product Lattice


∧1) and (S2,

∧2) when

S = S1 × S2 (domain)

Product Lattice


∧1) and (S2,

∧2) when

S = S1 × S2 (domain)For (a1,a2) and (b1,b2) ∈ S

Product Lattice


∧1) and (S2,

∧2) when


(a1, a2)∧(b1, b2) = (a1

∧1 b1, a2

∧2 b2)

Product Lattice


∧1) and (S2,

∧2) when


(a1, a2)∧(b1, b2) = (a1

∧1 b1, a2

∧2 b2)

(a1, a2) ≤ (b1, b2) iff a1 ≤1 b1 and a2 ≤2 b2

Product Lattice


∧1) and (S2,

∧2) when


(a1, a2)∧(b1, b2) = (a1

∧1 b1, a2

∧2 b2)

(a1, a2) ≤ (b1, b2) iff a1 ≤1 b1 and a2 ≤2 b2

≤ relation follows from∧

Product Lattice


∧1) and (S2,

∧2) when


(a1, a2)∧(b1, b2) = (a1

∧1 b1, a2

∧2 b2)

(a1, a2) ≤ (b1, b2) iff a1 ≤1 b1 and a2 ≤2 b2


◮ Product of lattices is associative

Product Lattice


∧1) and (S2,

∧2) when


(a1, a2)∧(b1, b2) = (a1

∧1 b1, a2

∧2 b2)

(a1, a2) ≤ (b1, b2) iff a1 ≤1 b1 and a2 ≤2 b2



◮ Can be generalized to product of N > 2 lattices

Product Lattice


∧1) and (S2,

∧2) when


(a1, a2)∧(b1, b2) = (a1

∧1 b1, a2

∧2 b2)

(a1, a2) ≤ (b1, b2) iff a1 ≤1 b1 and a2 ≤2 b2



◮ Can be generalized to product of N > 2 lattices

◮ (S1,∧

1), (S2,∧

2), . . . are called component lattices

Product Lattice: Example

∅ ∅ ∅

{a} {b} {c}

× ×

Product Lattice: Example

∅ ∅ ∅

{a} {b} {c}

× × =

∅

{c}

{b}

{a}{b, c}

{a, b}

{c, a}

{a, b, c}

Height of a Semilattice

◮ Length of a chain x1 ≤ x2 ≤ . . . ≤ xk is k



◮ Let K = max over lengths of all the chains in a semilattice



◮ Let K = max over lengths of all the chains in a semilattice

◮ Height of the semilattice = K − 1

Data Flow Analysis Framework

◮ (D,S,

∧,F )


◮ (D,S,

∧,F )

◮ D: direction – Forward or Backward


◮ (D,S,

∧,F )


◮ (S,∧): Semilattice – Domain and meet


◮ (D,S,

∧,F )


◮ (S,∧): Semilattice – Domain and meet

◮ F : family of transfer functions of type S → S (see next

slide)

Transfer Functions

◮ F : family of functions S → S. Must Include

Transfer Functions

◮ F : family of functions S → S. Must Include◮ functions suitable for the boundary conditions (constant

transfer functions for Entry and Exit nodes)

Transfer Functions


transfer functions for Entry and Exit nodes)◮ Identity function I:

I(x) = x ∀x ∈ S

Transfer Functions


transfer functions for Entry and Exit nodes)◮ Identity function I:

I(x) = x ∀x ∈ S

◮ Closed under composition:

f ,g ∈ F , f ◦ g ⇒ h ∈ F

Monotonic Functions

◮ (S,≤): a poset

Monotonic Functions


◮ f : S → S is monotonic iff

∀x , y ∈ S x ≤ y ⇒ f (x) ≤ f (y)

Monotonic Functions



∀x , y ∈ S x ≤ y ⇒ f (x) ≤ f (y)

◮ Composition preserves monotonicity

Monotonic Functions



∀x , y ∈ S x ≤ y ⇒ f (x) ≤ f (y)

◮ Composition preserves monotonicity◮ If f and g are monotonic, h = f ◦ g, then h is also monotonic

Monotone Frameworks

◮ (D,S,∧,F ) is monotone if the family F consists of

monotonic functions only

f ∈ F , ∀x , y ∈ S x ≤ y ⇒ f (x) ≤ f (y)

Monotone Frameworks



f ∈ F , ∀x , y ∈ S x ≤ y ⇒ f (x) ≤ f (y)

◮ Equivalently

f ∈ F , ∀x , y ∈ S f (x∧

y) ≤ f (x)∧

f (y)

Monotone Frameworks



f ∈ F , ∀x , y ∈ S x ≤ y ⇒ f (x) ≤ f (y)

◮ Equivalently

f ∈ F , ∀x , y ∈ S f (x∧

y) ≤ f (x)∧

f (y)

◮ Proof? : QQ in class

Knaster-Tarski Fixed Point Theorem

◮ Let f be a monotonic function on a complete lattice(S,

∧,∨). Define

Then,



∧,∨). Define

◮ red(f ) = {v | v ∈ S, f (v) ≤ v}, pre fix-points

Then,



∧,∨). Define

◮ red(f ) = {v | v ∈ S, f (v) ≤ v}, pre fix-points◮ ext(f ) = {v | v ∈ S, f (v) ≥ v}, post fix-points

Then,



∧,∨). Define

◮ red(f ) = {v | v ∈ S, f (v) ≤ v}, pre fix-points◮ ext(f ) = {v | v ∈ S, f (v) ≥ v}, post fix-points◮ fix(f ) = {v | v ∈ S, f (v) = v}, fix-points

Then,



∧,∨). Define


Then,◮

∧red(f ) ∈ fix(f ). Further,

∧red(f ) =

∧fix(f )



∧,∨). Define


Then,◮


∧red(f ) =

∧fix(f )

◮∨

ext(f ) ∈ fix(f ). Further,∨

ext(f ) =∨

fix(f )



∧,∨). Define


Then,◮


∧red(f ) =

∧fix(f )

◮∨

ext(f ) ∈ fix(f ). Further,∨

ext(f ) =∨

fix(f )◮ fix(f ) is a complete lattice

Application of Fixed Point Theorem

◮ f : S → S is a monotonic function



◮ (S,∧) is a finite height semilattice




◮ ⊤ is the top element of (S,∧)





◮ Notation: f 0(x) = x , f i+1(x) = f (f i(x)), ∀i ≥ 0





◮ Notation: f 0(x) = x , f i+1(x) = f (f i(x)), ∀i ≥ 0

◮ The greatest fixed point of f is

f k (⊤), where f k+1(⊤) = f k (⊤)

Fixed Point Algorithm

// monotonic function f on a meet semilattice



x := ⊤;



x := ⊤;while (x 6= f(x)) x := f(x);



x := ⊤;while (x 6= f(x)) x := f(x);

return x;

Foundations of Data Flow Analysis...Why Data Flow Analysis Works? Suitable initial values and boundary conditions Suitable domain of values Bounded, Finite Suitable meet operator Suitable

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