Comp 512
Spring
2011
Global Redundancy Elimination:
Computing Available Expressions
Copyright 2011, Keith D. Cooper & Linda Torczon, all rights reserved.
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Faculty from other educational institutions may use these materials for nonprofit
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Computing Available Expressions
For each block b
• Let AVAIL(b) be the set of expressions available on entry to b
• Let NKILL(b) be the set of expression not killed in b
• Let DEF(b) be the set of expressions defined in b and not
subsequently killed in b
Now, AVAIL(b) can be defined as:
AVAIL(b) = xpred(b) (DEF(x)  (AVAIL(x)  NKILL(x) ))
preds(b) is the set of b’s predecessors in the control-flow graph
This system of simultaneous equations forms a data-flow problem
 Solve it with a data-flow algorithm
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Expressions defined in b
Using Available Expressions for GCSE
The Big Picture
Expressions not killed in b
1.  block b, compute DEF(b) and NKILL(b)
2.  block b, compute AVAIL(b)
3.  block b, value number the block starting from AVAIL(b)
4. Replace expressions in AVAIL(b) with references
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Computing Available Expressions
First step is to compute DEF & NKILL
assume a block b with operations o1, o2, …, ok
KILLED  
DEF(b)  
for i = k to 1
}
}
assume oi is “x  y + z”
if (y  KILLED(b)) and (z  KILLED(b)) then
add “y + z” to DEF(b)
add x to KILLED(b)
NOTKILLED(b)  { all expressions }
For each expression e
for each variable v  e
if v  KILLED(b) then
NOTKILLED(b)  NOTKILLED(b) - e
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O(k) steps
Many data-flow problems
have initial information
that costs less to compute
O(N) steps
N is # of operations
* 4
Computing Available Expressions
The worklist iterative algorithm
Worklist  { all blocks, bi }
while (Worklist  )
remove a block b from Worklist
recompute AVAIL(b ) as
AVAIL(b) = xpred(b) (DEF(x)  (AVAIL(x)  NKILL(x) ))
if AVAIL(b ) changed then
Worklist  Worklist  successors(b )
}
• Finds fixed point solution to equation for AVAIL
• That solution is unique
• Identical to “meet over all paths” solution
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How do we know
these things?
* 5
Data-flow Analysis
Definition
Data-flow analysis is a collection of techniques for compile-time
reasoning about the run-time flow of values
• Almost always involves building a graph
Flow graph
Problems are trivial on a basic block
 Global problems  control-flow graph (or derivative)
 Whole program problems  call graph (or derivative)

• Usually formulated as a set of simultaneous equations
Sets attached to nodes and edges
 Lattice (or semilattice) to describe values

Data-flow problem
• Desired result is usually meet over all paths solution



“What is true on every path from the entry?”
“Can this happen on any path from the entry?”
Related to the safety of optimization
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Data-flow Analysis
Limitations
1. Precision – “up to symbolic execution”
 Assume all paths are taken
2. Solution – cannot afford to compute MOP solution
 Large class of problems where MOP = MFP= LFP
 Not all problems of interest are in this class
3. Arrays – treated naively in classical analysis
 Represent whole array with a single fact
4. Pointers – difficult (and expensive) to analyze
 Imprecision rapidly adds up
 Need to ask the right questions
Summary
Good news:
Simple problems can
carry us pretty far
For scalar values, we can quickly solve simple problems
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* 7
Data-flow Analysis
Semilattice
A semilattice is a set L and a meet operation  such that,
 a, b, & c  L :
1. a  a = a
2. a  b = b  a
3. a  (b  c) = (a  b)  c
 imposes an order on L,  a, b, & c  L :
1. a ≥ b  a  b = b
2. a > b  a ≥ b and a ≠ b
A semilattice has a bottom element, denoted 
1.  a  L,   a = 
2.  a  L, a ≥ 
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Data-flow Analysis
How does this relate to data-flow analysis?
• Choose a semilattice to represent the facts
• Attach a meaning to each a  L
Each a  L is a distinct set of known facts
• With each node n, associate a function fn : L  L
fn models behavior of code in block corresponding to n
• Let F be the set of all functions that the code might generate
Example — AVAIL
• Semilattice is (2E,), where E is the set of all expressions &  is 

Set are bigger than |variables|,  is 
• For a node n, fn has the form fn(x) = Dn  (x Nn)

Where Dn is DEF(n) and Nn is NKILL(n)
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Concrete Example: Available Expressions
A
B
E = {a+b,c+d,e+f,a+17,b+18}
m  a + b
n  a + b
2E is the set of all subsets of E
C
p  c + d
r  c + d
D
e  b + 18
s  a + b
u  e + f
F
G
y  a + b
z  c + d
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2E = [
q  a + b
r  c + d
E
v  a + b
w  c + d
x  e + f
e  a + 17
t  c + d
u  e + f
{a+b,c+d,e+f,a+17,b+18},
{a+b,c+d,e+f,a+17},
{a+b,c+d,e+f,b+18},
{a+b,c+d,a+17,b+18},
{a+b,e+f,a+17,b+18},
{c+d,e+f,a+17,b+18}, {a+b,c+d,e+f},
{a+b,c+d,b+18}, {a+b,c+d,a+17},
{a+b,e+f,a+17},
{a+b,e+f,b+18},{a+b,a+17,b+18},
{c+d,e+f,a+17}, {c+d,e+f,b+18},
{c+d,a+17,b+18},{e+f,a+17,b+18},
{a+b,c+d},{a+b,e+f},{a+b,a+17},
{a+b,b+18},{c+d,e+f},{c+d,a+17},
{c+d,b+18},{e+f,a+17},{e+f,b+18},
{a+17,b+18},{a+b}, {c+d}, {e+f}, {a+17},
{b+18}, {} ]
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Concrete Example: Available Expressions
The Lattice
{a+b,c+d,e+f,a+17,b+18},
Comparability
(transitive)
{a+b,c+d,e+f,a+17}
{a+b,c+d,e+f,b+18}
{a+b,c+d,a+17,b+18}
{a+b,e+f,a+17,b+18}
{c+d,e+f,a+17,b+18}
{a+b,c+d,e+f}
{a+b,c+d,b+18}
{a+b,c+d,a+17}
{a+b,e+f,a+17}
{a+b,e+f,b+18}
{a+b,a+17,b+18}
{c+d,e+f,a+17}
{c+d,e+f,b+18}
{c+d,a+17,b+18}
{e+f,a+17,b+18},
{a+b,c+d}
{a+b,a+17}
{c+d,e+f}
{c+d,b+18}
{e+f,b+18}
{a+b,e+f}
{a+b,b+18}
{c+d,a+17}
{e+f,a+17}
{a+17,b+18}
{a+b}
{c+d}
{e+f}
{a+17}
{b+18}
meet
{}
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* 11
Data-flow Analysis
What does this have to do with the iterative algorithm?
Worklist  { all blocks, bi }
while (Worklist  )
remove a block bi from Worklist
recompute AVAIL(bi ) as
AVAIL(b) = xpred(b) (DEF(x)  (AVAIL(x)  NKILL(x) ))
if AVAIL(bi ) changed then
Worklist  Worklist  successors(bi )
We can use a lattice-theoretic formulation to prove
• Termination – it halts on an instance of AVAIL
• Correctness – it produces the desired result for AVAIL
• Complexity – it runs pretty quickly
(d(CFG)+3 passes)
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Data-flow Analysis
Termination
• If every fn  F is monotone, i.e., f(xy) ≤ f(x)  f(y), and
• If the lattice is bounded, i.e., every descending chain is finite
Chain is sequence x1, x2, …, xn where xi  L, 1 ≤ i ≤ n
 xi > xi+1, 1 ≤ i < n  chain is descending

Then
• The iterative algorithm must halt on an instance of the problem
• Set at each block can only change a finite number of times 
• Any finite semilattice is bounded
• Some infinite semilattices are bounded
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Data-flow Analysis
Correctness
• Does the iterative algorithm compute the desired answer?
Admissible Function Spaces
1.  f  F,  x,y  L, f (xy) = f (x)  f (y)
Not distributive  answer
may not be unique
2.  fi  F such that  x  L, fi(x) = x
3. f,g  F   h  F such that h(x ) = f (g(x))
4.  x  L,  a finite subset H  F such that x = f  H f ()
If F meets these four conditions, then the problem (L,F,) has a
unique fixed point solution
 LFP = MFP = MOP
 order of evaluation does not matter
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* 14
Data-flow Analysis
Sets stabilize in two
passes around a loop
Complexity
• For a problem with an admissible function space & a bounded
semilattice,
• If the functions all meet the rapid condition, i.e.,
f,g  F,  x  L, f (g()) ≥ g()  f (x)  x
then, a round-robin, reverse-postorder iterative algorithm
Each pass does O(E )
will halt in d(G)+3 passes over a graph G
meets & O(N ) other
operations
d(G) is the loop-connectedness of the graph w.r.t a DFST
 Maximal number of back edges in an acyclic path
 Several studies suggest that, in practice, d(G) is small
(<3)
 For most CFGs, d(G) is independent of the specific DFST
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* 15
Data-flow analysis
What does this mean?
• Reverse postorder


Number the nodes in a postorder traversal
Reverse the order
• Round-robin iterative algorithm
Visit all the nodes in a consistent order (RPO)
 Do it again until the sets stop changing

So, these conditions are easily met
 Admissible framework, rapid function space
 Round-robin, reverse-postorder, iterative algorithm
 The analysis runs in (effectively) linear time
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Data-flow Analysis
How do we use these results?
 Prove that data-flow framework is admissible & rapid



Its just algebra
Most (but not all) global data-flow problems are rapid
This is a property of F
 Code up the iterative algorithm
World’s simplest data-flow algorithm
 Other versions (worklist) have similar behavior

This lets us ignore most of the other data-flow algorithms in 512
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Global CSE
(replacement step)
Managing the name space
Need a unique name  e  AVAIL(b)
1. Can generate them as replacements are done
(Fortran H)
2. Can compute a static mapping
3. Can encode value numbers into names
(Briggs 94)
Strategy
1. This works; it is the classic method
2. Fast, but limits replacement to textually identical expressions
3. Requires more analysis (VN), but yields more CSEs
Assume, w.l.o.g., solution 2
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Computing Available Expressions
The Big Picture
1. Build a control-flow graph
2. Gather the initial (local) data — DEF(b) & NKILL(b)
3. Propagate information around the graph, evaluating the equation
4. Post-process the information to make it useful
(if needed)
Most data-flow problems are solved, essentially, this way
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