ECE540S Optimizing Compilers

1
ECE540SOptimizing Compilers

• http//www.eecg.toronto.edu/voss/ece540/
• Data Flow Analysis, Jan. 20, 2004

2
Outline

• Quick Overview / Motivation
• Reaching Definitions
• Data Flow Frameworks
• Live Variables
• Available Expressions
• Very Busy Expressions

3
Data Flow Analysis

• Goal make assertions about the data usage in a
  program
• Use these assertions to determine if and when
  optimizations are legal
• Local within a single basic block
• Analyze effect of each instruction
• Compose effect at beginning/end of BB
• Global within a procedure (across BBs)
• Consider the effect of control flow
• Inter-procedural across procedures
• References
• Muchnick, Chapter 8.
• Dragon Book, 608-611, 624-627, 631.

4
Data Flow Analysis

• Compile-time reasoning about the run-time flow of
  values in the program
• Represent facts about the run-time behavior
• Represent effect of executing each basic block
• Propagate facts around the control flow graph

5
Data Flow Analysis

• Formulated as a set of simultaneous equations
• Sets attached to the nodes and edges
• Lattice to describe the relation between values
• Usually represented as a bit or bit vectors
• Solve equations using iterative framework
• Start with initial guess of facts at each node
• Propagate until stabilizes at maximal fixed
  point.
• Would like meet over all paths (MOP) solution

6
Basic Approach
Must be conservative!
7
Example Reaching Definitions
Problem statement for each basic block b find
which of all definitions in the program reach the
boundaries of b.
Definition A definition of a variable x is an
instruction that assigns (or may assign) a value
to x.

• Reaches A definition d of variable x reaches a
  point p in the program if there exists a path
  from the point immediately following d to p such
  that d is not killed by another definition of x
  along this path.

8
Reaching Definitions Gen Set
Gen(b) the set of definitions that appear in a
basic block b and reach its end.
Entry
BB 1
Gen (BB 1) d2, d3
a 5 c 1 a a 1 c gt a?
Gen (BB 2) d4
Gen (BB 3) d5, d6
c c c
BB 2
BB 3
a c - a c 0
Exit
Finding Gen(b) is doing local reaching
definitions analysis.
9
Reaching Definitions Kill Set
Kill(b) Set of definitions in other basic blocks
that are killed in b (i.e., by instructions in
b). For each variable v defined in b, the kill
set contains all definitions of v in other basic
blocks.
Entry
Kill (BB 1) d5, d4, d6
BB 1
a 5 c 1 a a 1 c gt a?
Kill (BB 2) d2, d6
Kill (BB 3) d1, d3, d2, d4
c c c
BB 2
BB 3
a c - a c 0
Exit
10
Reaching Definitions Data Flow Equations
RDin(b) Set of definitions that reach the
beginning of b. RDout(b) Set of definitions
that reach the end of b.
11
Reaching Definitions - Solving the Data Flow
Equations
F
RDin(BB 1) RDout(Entry) È RDout(BB
2)d3,d4 RDout(BB 1) Gen(BB 1) È RDin(BB
1)-Kill (BB 1) d2,d3 È d3,d4-d4,d5,d6
d2,d3
Repeat!
no change Þ done
Where do we start? Why?
12
Reaching Definitions Solver
RDout(Entry) F RDout(b) F " b Î N-Entry,
Exit // Gen(b) is better change true while
(change) change false for each b Î
N-Entry, Exit // Depth-first order
oldout RDout(b) if (RDout(b) ¹
oldout) change true
13
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
14
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
d1,d2,d3
d4,d5
d7
d6
15
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d4,d5
d7
d6
16
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d4,d5
d7
d6
17
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d4,d5
d7
d6
18
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5
d7
d6
19
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d7
d6
20
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d7
d6
21
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d7
d6
22
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d7
d4,d5,d6
23
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d7
d4,d5,d6
24
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d7
d4,d5,d6
25
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
26
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
CHANGED!
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
27
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
28
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
29
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d4,d5,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
30
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d4,d5,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
31
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d4,d5,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
32
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
F
F
d1,d2,d3
d1,d2,d3,d4,d5,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
33
Reaching Definitions - Example
d1 i m-1 d5 j j-1 d2 j n d6 a
u2 d3 a u1 d7 a u3 d4 i i1
NO CHANGE!
F
F
d1,d2,d3
d1,d2,d3,d4,d5,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5, d3,d6
d4,d5,d6
d4,d5,d7
34
Reaching Definitions Questions

• Does the algorithm terminate?
• Does it produce correct answers?
• In what sense is it conservative?

35
Data Flow Frameworks

• A monotone data flow framework (L, Ù,F) consists
  of
• a domain of values L.
• a confluence operator Ù L L. (meet)
• a set of transfer functions F. If f Î F, then f
  L L.
• The semilattice (L, Ù) satisfies the conditions
• Ù defines a partial order on L.
• L contains a top element T such that T Ù X X
  " X Î L.
• L contains a bottom element ? such that ? Ù X ?
  " X Î L.
• Ù is associative X Ù (Y Ù Z) (X Ù Y) Ù Z.
• Ù is commutative X Ù Y Y Ù X.
• Ù is idempotent X Ù X X.
• F is a family of transfer functions that satisfy
  
• F is monotone X Y Þ f(X) f(Y) " f Î
  F.
• F is closed under composition " f,g Î F, f(g(X))
  Î F.

36
Partial Ordering of Lattice
T (F)
d1
d3
d1,d3
d2
d1,d2
d2,d3
? d1,d2,d3
The meet operation creates a partial order of
L. For ?, ? X T is ? ? X ? T A monotonic
framework must terminate if the height of the
Lattice is finite.
37
Meet, partial orders and height

• Ù defines a partial order on L
• x y iff x Ù y x
• reflexive x x
• antisymmetric if x y and y x then x y
• transitive if x y and y z then x z
• For L, Ù is
• idempotent x Ù x x
• commutative x Ù y y Ù x
• associative x Ù (y Ù z) (x Ù y) Ù z
• Important examples of meet operators
• Union, U
• Intersection, n
• The height of a lattice is the largest number of
  relations that fit in a descending chain.

38
Transfer Functions

• F is a family of transfer functions
• f L ? L
• F is closed under composition iff
• " f,g Î F, f(g(X)) Î F
• A framework (L, Ù , F) is monotone iff
• X Y Þ f(X) f(Y) " f Î F
• A framework is also monotone iff
• f(X Ù Y) f(X) Ù f(Y) " f Î F

39
Data Flow Frameworks (cont)

• Is Reaching Definitions a monotone data flow
  framework?
• L is the set of definitions in the program.
• The confluence operator Ù is set union.
• The partial order is defined as X Y Þ X Ê Y.
• There is a top element (F) and bottom element.
• Set union is associative, commutative and
  idempotent.
• The transfer function is and is monotone.
• The transfer function is closed under
  composition.
• The iterative algorithm will terminate!

40
Proof in words

• Set initial conditions at all points to T
• A points value cannot go up (new old)
• first application of function/meet result T
• If input to meet gets smaller, result old
  result
• If input to transfer function gets smaller,
  result old result
• Values go down due to constraints in equations
• Framework iterates until equations are satisfied
• The result is the largest (closest to T) solution

41
What does the solution mean?

• Given a path P b0 b1 b2 bn-1 bn
• the transfer function at entry of bn is fp(bn)
  fn-1(fn-2(f0(T)) ).
• We cant know at compile-time which path might
  possibly execute, so we want to compute the
  meet-over-all-path (MOP) solution
• MOP(b) Ù
  fp(bn)
  all paths P to b
• Iteration computes the Maximal Fixed Point (MFP).
• If F is distributive (" f Î F, f(X Ù Y) f(X)
  Ù f(Y))
• then MFP MOP. Otherwise MFP MOP
  (conservative).
• FP MFP MOP Ideal FP, MFP and MOP are
  safe.
• If distrubtive, FP MFP MOP Ideal
• Most data flow problems we encounter are
  distributive.

42
Proofs

• Gary Kildall, A Unified Approach to Global
  Program Optimization, Proceedings of the First
  POPL, Boston, MA, USA October 1973. If all flow
  functions are monotone and distributive,
  iterations yields MFP MOP.
• J.B. Kam and J.D. Ullman, Monotone Data Flow
  Analysis Frameworks, Acta Informatica,
  7305-317, 1977. If all flow functions are
  monotone, iterations yields MFP.

43
Data Flow Frameworks (cont)
a a 1
c c 1
d1
d2
d1,d2
All Definitions that may reach a point.
44

• Reaching definitions is an example of an any-path
  forward-analysis data flow problem.

45
Data Flow Analysis Review

• What is Data Flow Analysis
• Reaching Definitions
• Simple Example Solver
• More Realistic Example
• Characteristics of Data Flow Framework
• What is sufficient to show that they (a)
  terminate (b) generate correct results and (c)
  generate MOPMFP.

46
General Data Flow Equations

• Any-path forward-flow (Reaching Definitions)

• Any-path backward-flow (Live Variable Analysis)

• All-path forward-flow (Available Expressions)

• All-path backward-flow (Very Busy Expressions)

47
Live Variables

• A variable v is said to be live at a point p in
  the program if the current value of v will be
  used before being assigned a new value, or before
  the program ends. Otherwise it is dead.

• Problem statement determine the set of variables
  that are live at the boundaries of each basic
  block in the program.

48
Live Variables - Use Set

• Use (b) Set of variables that are used before
  (possibly) being defined in b.

Use (BB 1) F
Use (BB 2) a, b
Use (BB 3) a, c
This is doing LV analysis in a BB (local
analysis).
49
Live Variables - Def Set

• Def (b) Set of variables defined in b.

Def (BB 1) a, b
Def (BB 2) c
Def (BB 3) F
50
Live Variables - Data flow Equations

• LVin(b) Set of live variable at the beginning
  of b.
• LVout(b) Set of live variables at the end of b.

51
LV - Solving the Data Flow Equations
LVout(BB 1) LVin(BB 2) a,b LVin(BB 1) F
È LVout(BB 1)-Def(BB 1) F È a,b-a,b
F
LVout(BB 2) LVin(BB 3) È LVin(BB 2) a,c È
a,b a,b,c LVin(BB
2) Use(BB 2) È LVout(BB 2)-Def(BB 2)
a,b È a,b,c-c a,b
F
LVout(BB 3) LVin(Exit) F LVin(BB 3)
Use(BB 3) È LVout(BB 3)-Def(BB 3) a,c È
F-F a,c
Repeat!
no change Þ done
Where do we start? Why?
52
Live Variables Solver
LVin(Exit) F LVin(b) F " b Î N-Entry, Exit
// USE(b) is better change true while
(change) change false for each b Î
N-Entry, Exit // Reverse DFO
oldin LVin(b) if (LVin(b) ¹ oldin)
change true
53
Live Variables Solver - Questions

• Does this algorithm terminate?
• Does it compute the correct answer?
• In what sense is it conservative?

54
Live Variables - Example
55
Live Variables - Example
F
56
Live Variables - Example
c
b,d
a,b,d,e,g
b,d,e,g
F
57
Live Variables - Example
c
b,d
a,b,d,e,g
b,d,e,g
F
58
Live Variables - Example
c
b,d
a,b,d,e,g
b,d,e,g
F
F
59
Live Variables - Example
c
b,d
a,b,d,e,g
b,d,e,g
F
F
60
Live Variables - Example
c
b,d
a,b,d,e,g
b,d,e,g
b,d,e,g
F
F
61
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
F
F
62
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
F
F
63
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
64
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
65
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
66
Live Variables - Example
Some LVin has changed! Repeat!
c
b,d,e,g
a,b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
67
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
68
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
69
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
70
Live Variables - Example
c
b,d,e,g
a,b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
71
Live Variables - Example
No LVin has changed! Done!
c
b,d,e,g
a,b,d,e,g
a,b,d,e,g
b,d,e,g
b,d,e,g
b,d,e,g
F
F
72
Live Variables Analysis

• Example of any-path backward-analysis data flow
  problem
• any-path?
• backward-analysis?
• What is Live Variable Analysis good for?

73
Available Expressions

• An expression expr is said to be available at the
  entry to a basic block b if along every
  control-flow path from Entry to block b there is
  an evaluation of expr that is not killed by
  having one or more of its operands assigned a new
  value.

• Problem statement for each basic block b
  determine the set of available expressions at its
  boundaries.

74
Entry
f d d
d 5 e d d
c 5 w x y

Is the expression d d available here?
Whats different between this and reaching
definitions?
75
Available Expressions - Eval Set
Eval(b) Set of expressions evaluated in b and
are still available at its exit.
Eval (BB 1) xb, df
Eval (BB 2) F
Eval (BB 3) k1
Eval (BB 4) df, xb
This is doing AE analysis in a BB (local
analysis).
76
Available Expressions - Kill Set
Kill(b) Set of expressions killed in b.
Kill (BB 1) F
Kill (BB 2) xb
Kill (BB 3) F
Kill (BB 4) F
77
AE - Data flow Equations

• AEin(b) Set of available expression at the
  beginning of b.
• AEout(b) Set of available that reach the end
  of b.

78
Available Expressions Solver
AEout(Entry) F AEout(b) Eval (b) U U
-Kill(b) " b Î N-Entry, Exit change
true while (change) change false for
each b Î N-Entry, Exit oldout
AEout(b) if (AEout(b) ¹ oldout)
change true
Why does this work?
79
Available Expressions Solver Questions(as an
exercise)

• Does this algorithm terminate?
• Does it compute the correct answer?
• In what sense is it conservative?

80
Available Expressions - Example
U ab, ac, c2, dd, i1
81
Available Expressions - Example
F
U ab, ac, c2, dd, i1
82
Available Expressions - Example
F
ab,ac,dd
ab,c2,dd,i1
U ab, ac, c2, dd, i1
U
U
U
83
Available Expressions - Example
F
ab,ac,dd
ab,c2,dd,i1
U ab, ac, c2, dd, i1
U
U
U
84
Available Expressions - Example
F
F
ab,ac,dd
ab,c2,dd,i1
U ab, ac, c2, dd, i1
U
U
U
85
Available Expressions - Example
F
F
ab,ac,dd
ab,c2,dd,i1
U ab, ac, c2, dd, i1
U
U
U
86
Available Expressions - Example
F
F
ab,ac,dd
ab,c2,dd,i1
U ab, ac, c2, dd, i1
U
U
U
87
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,c2,dd,i1
U ab, ac, c2, dd, i1
U
U
U
88
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
U ab, ac, c2, dd, i1
U
U
U
89
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
U ab, ac, c2, dd, i1
U
U
U
90
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
U
U
U
91
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
U
ac,ab, dd
U
92
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
U
ac,ab, dd
U
93
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
U
ac,ab, dd
U
94
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ac,ab, dd
U
95
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ac,ab, dd
U
96
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
U
97
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
98
Available Expressions - Example
An AEout has changed! Repeat!
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
99
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
100
Available Expressions - Example
F
F
ab,ac,dd
ab,ac, dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
101
Available Expressions - Example
F
F
ab,ac,dd
ab,dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
102
Available Expressions - Example
F
F
ab,ac,dd
ab,dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
103
Available Expressions - Example
F
F
ab,ac,dd
ab,dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
104
Available Expressions - Example
F
F
ab,ac,dd
ab,dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
105
Available Expressions - Example
No AEout set has changed! Were Done.
F
F
ab,ac,dd
ab,dd
ab,dd
ab,dd
ab,dd
U ab, ac, c2, dd, i1
ab,dd
ab,dd
ac,ab, dd
ab,dd
106
Available Expression Analysis

• An all-paths forward-analysis data flow problem
• all-paths?
• forward-analysis?
• What is available expressions analysis good for?

107
Very-Busy Expressions

• An expression is said to be very-busy at point p
  if the the expression is evaluated along all
  paths before the expression is killed, or Exit.

• Problem statement determine very-busy
  expressions at the boundaries of each basic block.

108
Very Busy Expressions - Eval Set
Eval(b) Set of expressions evaluated before any
of their operands are (possibly) assigned to in b.
Eval (BB 1) F
Eval (BB 2) cd
Eval (BB 3) ac, cd
Eval (BB 4) ab, ac
Eval (BB 5) ab, ad
This is doing VBE analysis in a BB (local
analysis).
109
Very Busy Expressions - Kill Set
Kill(b) Set of expressions killed in b.
Kill (BB 1) F
Kill (BB 2) ad,cd
Kill (BB 3) F
Kill (BB 4) F
Kill (BB 5) F
110
VBE - Data flow Equations

• VBEin(b) Set of very busy expressions at the
  beginning of b.
• VBEout(b) Set of very busy expressions at the
  end of b.

111
Very Busy Expressions Solver
VBEin(Exit) F VBEin(b) Eval(b) U U
-Kill(b) " b Î N-Entry, Exit change
true while (change) change false for
each b Î N-Entry, Exit oldin
VBEin(b) if (VBEin(b) ¹ oldin)
change true
112
Very Busy Expressions Solver Questions(as an
exercise)

• Does this algorithm terminate?
• Does it compute the correct answer?
• In what sense is it conservative?

113
Very Busy Expressions - Example
U cd, ad, ac, ab
114
Very Busy Expressions - Example
U cd, ad, ac, ab
F
115
Very Busy Expressions - Example
U
cd,ac,ab
U
U
U
U cd, ad, ac, ab
F
116
Very Busy Expressions - Example
U
cd,ac,ab
U
U
U
U cd, ad, ac, ab
F
117
Very Busy Expressions - Example
U
cd
U
F
U
U
U cd, ad, ac, ab
F
118
Very Busy Expressions - Example
U
cd
U
F
U
U
U cd, ad, ac, ab
F
119
Very Busy Expressions - Example
U
cd
U
F
U
U
U cd, ad, ac, ab
F
F
120
Very Busy Expressions - Example
U
cd
U
F
ab,ac
U
U cd, ad, ac, ab
F
F
121
Very Busy Expressions - Example
U
cd
U
F
ab,ac
U
U cd, ad, ac, ab
F
F
122
Very Busy Expressions - Example
U
cd
U
F
ab,ac
U
U cd, ad, ac, ab
F
F
U
123
Very Busy Expressions - Example
U
cd
U
F
ab,ac
U
U cd, ad, ac, ab
F
F
U
124
Very Busy Expressions - Example
U
cd
U
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
125
Very Busy Expressions - Example
U
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
126
Very Busy Expressions - Example
U
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
127
Very Busy Expressions - Example
U
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
128
Very Busy Expressions - Example
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
129
Very Busy Expressions - Example
VBEins have changed! Repeat!
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
130
Very Busy Expressions - Example
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
131
Very Busy Expressions - Example
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
132
Very Busy Expressions - Example
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
U
133
Very Busy Expressions - Example
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
ab,ac,cd
134
Very Busy Expressions - Example
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
ab,ac,cd
135
Very Busy Expressions - Example
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
ab,ac,cd
136
Very Busy Expressions - Example
No VBEins have changed! Were all done.
cd
cd
cd
ab,ac,cd
F
ab,ac
ab,ac
U
U cd, ad, ac, ab
F
F
ab,ac,cd
137
Very Busy Expressions

• An example of a all-paths backward-analysis data
  flow problem
• all-paths?
• backward-analysis?
• What is Very Busy Expressions useful for?

138
Other data flow abstractions
139
Use-Definition (UD) Chains
(Muchnick Section 8.10)

• The set of definitions reaching a use of a
  variable is called the use-definition chain
  (ud-chain) for the variable.
• UD chains are derived from reaching definitions

• Gen(b) Set of definitions that appear in b and
  reach its end.
• Kill(b) Set of definitions in other basic
  blocks that are killed in b.
• RDin(b) Set of definitions that reach the
  beginning of b.
• RDout(b) Set of definitions that reach the end
  of b.

140
Example
141
Definition-Use (DU) Chains

• The set of uses reached by the definition of a
  variable is called the definition-use chain
  (du-chain) for the variable.
• UD can also be solved as a data flow problem

• Use(b) Set of variable uses in b with no prior
  definition of
• the variable in b.
• Kill(b) Set of variable uses that are killed
  by b. The set of
• variables defined in b and used
  in other BBs.
• In(b) Set of uses available at the beginning
  of b.
• Out(b) Set of uses of available at the end of b.

142
Example
143
Single Static Assignment Form

• Optimization is often complicated by the fact the
  variables can be assigned in multiple places.
• Reaching Definitions is all definitions that
  might reach the point
• Available Expressions look for all expressions
  that might be killed by the assignment to a
  variable
• Wouldnt it be nice if variables were only
  assigned to in one place?
• a singe unique definition point
• In SSA Form, each static definition of a
  variable, x, becomes the definition of a new
  variable xi
• If there are n definitions then there will be an
  x1 - xn
• Uses of the variable must be replaced with the
  use of some xi

Muchnick Section 8.11
144
? Functions Dealing with joins
X2 5
X1 5
X2 5
X1 5
X?
X3 ? (X1 , X2 ) X3

• Control flow cannot be predicted at compile-time,
  so there may be more than 1 definition that might
  reach a use.
• So, insert ? (Phi) functions.
• create a new variable that is defined by a phi
  function joining the reaching definitions
• The new variable is assigned the value of the
  definition that is reached by the runtime control
  flow path, i.e. if the X1 block were executed
  then ? (X1 , X2) X1.

145
Example
X 1
Y X 1
X 5
Z X
X 100
W X X