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CS 201Compiler Construction

Lecture 4Data Flow Framework

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Data Flow Framework

The various problems considered have things in common:– Transfer functions

– Confluence Operator

– Direction of Propagation

These problems can be treated in a unified way data flow framework is an algebraic structure used to encode and solve data flow problems.

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Monotone Data Flow Framework

Components of the framework:1.Information Set: L2.Effect of joining paths: ∧ (meet operator)3.Effect of basic blocks: fn (monotone transfer func.)4.Iterative Solution: can be shown to terminate

(L,∧) is a semilattice st ∨ a,b,c εL1.a ∧ a = a (idempotent)2.a ∧ b = b ∧ a (commutative)3.a ∧ (b ∧ c) = (a ∧ b) ∧ c (assocative)

Bottom Element st ∨ a ε L, a ∧ = Top Element Τ st ∨ a ε L, a ∧ Τ = aIf top & bottom elements exist, they are unique.

Contd..

Relation ≤ is a partial order on L

a ≤ b ≅ a ∧ b = a

Can similarly define <, >, ≥ relations

A semilattice is bounded iff ∨ a εL there exists a constant ca st length of chain beginning at a is at most ca.

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Max ca

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Monotonic Functions

Effect of each basic block is modeled by a transfer function f: L L. Function f must be monotonic.

A total function f: LL is monotonic iff ∨ a,b ε L f(a∧b) ≤ f(a) ∧ f(b)

Distributive function: f(a∧b) = f(a) ∧ f(b)

For monotonic functions: a ≤ b => f(a) ≤ f(b)

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Contd..

For monotonic functions: a ≤ b => f(a) ≤ f(b)

Proof:

f(a∧b) ≤ f(a) ∧ f(b) Defn. of Monotonicityf(a∧b)∧f(a)∧f(b) = f(a∧b) Defn. of ≤:(a∧b=a)f(a)∧f(a)∧f(b)= f(a) Given a ≤ b: a ∧ b = af(a) ∧ f(b) = f(a) f(a)∧f(a) = f(a) idempotencef(a) ≤ f(b) Defn. of ≤

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Fixpoint

A fixpoint of a monotonic function f: L L is a valuea ε L such that f(a) = a

Τ > f (Τ) > f ( f (Τ) ) > f ( f ( f (Τ) ) ) ……..

There exists t such that f ( ft (Τ) ) = ft (Τ)

ft (Τ) is the greatest fixpoint of f.

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Monotone Function SpaceA monotone function space for a semilattice is a set F of monotonic functions which:

1.Contains the identity function (id)-- basic blocks may not modify information

2.Is closed under function composition-- to model the effects of paths

3.For each a ε L, there exists fεF st f( ) = a-- to model gen functions

A distributive function space is a monotone function space in which all functions are distributive.

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A Monotone Data Flow System

A monotone data flow system is a tuple < L, ∧, F, G, FM >

1.(L,∧) is a bounded semilattice with Τ & 2.F is the monotone function space3.G = (N, E, s) is the program flow graph4.FM: N F is a total function that associates a function from F with each basic block.

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Meet Over All Paths SolutionMeet over all paths solution (MOP) of a data flow system – MOP: N L

MOP(s) = NULL (NULL is the element in L which represents “no information”)

Ffπ is composition of functions from nodes along path π excluding node n. n1n2n3….nk-1nk fnk o fnk-1 o….o fn2 o fn1

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MOP SolutionFinding MOP solution is undecidable, i.e. there does not exist a general algorithm that computes MOP solution for all monotone data flow systems.

Let X: N L denote a total function that associates nodes with lattice elements.X is conservative or safe iff ∨n εN, X(n) ≤ MOP(n)

Iterative algorithm computes conservative approximation of MOP. For distributive data flow systems, it computes solution that is identical to MOP solution.

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Iterative Algorithm

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Reaching Definitions

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Contd…

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Dominators

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Constant Propagation

f (X)={(a,2),(b,3),(c,5)}f (Y)={(a,3),(b,2),(c,5)}f (X) ∧ f (Y) = {(a,not-const), (b, not-const),

(c,5)}

X ∧ Y = {(a,not-const),(b,not-const),(c,undef)}f (X∧ Y) = {(a,not-const),(b,not-const),(c,not-

const)}

f (X ∧ Y) ≤ f(X) ∧ f(Y)