CS412413

1
CS412/413

• Introduction to
• Compilers and Translators
• April 7, 1999
• Lecture 26 Optimizations based on
  data-flow analysis

2
Administration

• Programming Assignment 4 (Iota)is due April 28
  -- available on web.
• Prelim 2 April 16
• static semantics, IR and assembly code
  generation, object-oriented languages, data-flow
  analysis, optimization
• non-graded HW3 will be given out Friday
• No class April 19

3
Review

• Last time introduced quadruple IR for data-flow
  analysis, optimization.
• Data-flow analysis framework
• values in L to be propagated through program
  representing information about the program (e.g.
  live variables) either backward or forward
• flow functions defining effect of program nodes
  on propagated information
• Space of propagated values is a lattice with join
  (?), meet (?) operations, top (?) and bottom (?)

4
Propagating information

• Each node n applies its flow function Fn to
  entering information
• Entering information must take meet of
  information coming from all nodes that affect it
  -- least conservative information consistent with
  nodes affecting it
• Want to find least conservative assignment of
  values that is still safe -- as high in lattice
  as possible

5
Reaching definitions

• Reaching definitions output is for each node,
  list of definitions (assignments) that might
  still be in effect when the node is reached
• L is all sets of defining nodes in call flow
  graph. Least conservative assignment means
  minimal lists of reaching definitions, so
• Top (?) is the empty set, bottom (?) is the set
  of all nodes, meet (?) is set union (?) -- dual
  lattice to previous lattice example, with ? ,
  ? ?

6
Data-flow equations

• Reaching definitions
• outn genn ? (inn - killn)
• in n ?n? predn outn
• Mapping from in to out is flow function
• outn Fn(inn)
• Fn(x) genn ? (x - killn)
• Mapping from out to in is done by the combining
  operation (meet)
• inn ?n ? prednoutn ?n? prednoutn

7
Data-flow equations

• Let x1xm be outn1outnm assignments to
  nodes 1m
• Let predi be predecessors to node ni (assuming
  forward data-flow analysis)
• xi Fi(?j ? predi xj)
• Solution is point in Lm X (x1,xm)
• (x1,xm) (F1(?j ? pred1 x1),, Fm(?j ?
  predm xm))
• Total set of equations is X F(X) where F is the
  function that propagates information one step
  through control flow graph.

8
Fixed points

• A fixed point of a function is a value that is
  mapped to itself X F(X)
• Wanted maximal fixed point (least conservative
  satisfying assignment to all xi)
• Iterative data-flow analysis initialize all xi
  with top of lattice (X ?), apply F(X) until
  fixed point is reached F( F(F(F(X))))
• F is monotonic each iteration moves lower in
  lattice steps to fixed point cannot exceed
  mheight of lattice (sets of n things mn)
• Worklist algorithm efficient implementation

9
Summary

• Now have way of characterizing a data-flow
  analysis problem and producing an efficient
  solving algorithm
• Specify meet operation over value lattice
• Specify individual flow functions for different
  node types
• Compute repeated application of F iteratively
  using simple work-list algorithm presented
  earlier avoids recomputing xi unnecessarily
• Given proper lattice properties, convergence and
  maximal fixed point are guaranteed

10
Optimizations

• Dead code elimination
• Constant propagation
• Copy propagation
• Common sub-expression elimination

11
Dead code elimination

• Optimization remove statements (quadruples) that
  assign to variables that are not live-out
• Analysis live-variable analysis
• backward data-flow analysis
• values are sets of live variables? ?, ?
  ,Fn(x) usex ? (x def n)

12
Constant propagation

• Optimization observing some assignment x c,
  replace later uses of x with constant c -- allows
  more constant folding, better register
  allocation, dead code, instruction selection
• Analysis reaching definitions
• forward data-flow analysis
• values are sets of reaching definitions
• ? ?, ? , Fn(x) genn ? (x - killn)
• If only definition reaching a use of x isx c,
  can replace x with c

13
Copy propagation

• Optimization observing some assignment x y,
  replace later uses of x with variable y -- better
  register allocation, creates dead code
• Analysis starts with reaching definitions
• Let RDn be definitions reaching beginning of n.
  If only one definition of x reaches n and y is
  still available, can substitute y for x.
• Additional forward analysis availability. Values
  are sets of definitions with available RHS.
• ? ?, ? all defns,Fn(x) (genn ? x) -
  killn

14
CP gen and kill

• Let uses(y) be all nodes that use y in xy
• node n genn killn
• a b n uses(a)
• a b OP c uses(a)
• a Mb uses(a)
• Ma b
• goto L
• if a goto L1 else L2
• a f(b,c,d) uses(a)
• f(b,c,d)
• Can substitute y for x at node n if RDn
  contains only one definition of x and
  RHS-AVAILn also contains that definition.

15
CP example

• RD RHS-AVAIL
• 1. a b
• 2. d a 1 1 1
• 3. if c goto L1 else L2 1,2 1
• 4. L1 e a 1,2 1
• 5. goto L2 1,2,4 1,4
• 6. L2 a f 2 1,2 1
• 7. g a 2,6

16
Common sub-expression elimination

• Idea if same expression is computed twice(a b
  OP c), replace second computation with temporary
  that original was stored into
• a b OP c d b OP c ?t b OP c a t
  d t
• Need to know which expressions b OP c are still
  available (generalization of previous analysis)
• Available if neither b nor c have been reassigned
  and every path to node includes assignment --
  computable using data-flow

17
CSE Data-flow

• forward data-flow analysis AE(n) gives sets of
  definitions computing expressions that are
  available at n
• lattice value is ltd, a OP bgt
• ? ? , ? all valid pairs ltd, a OP bgt,
  AE(entry) ,Fn(x) (x ? EVALn) KILLn
• EVALn is either or ltn, a OP bgt if n has form
  c a OP b
• KILLn all pairs ltd, a OP bgt such that n
  updates a or b

18
Example

• y (x 1)(x 1) x w z x 1
• AE
• 1. a x 1
• 2. b x 1 lt1, x1gt
• 3. y a b lt1, x1gt, lt2, x1gt
• 4. x w lt1, x1gt, lt2, x1gt
• 5. z x 1

19
Optimizations

• t x 1
• a t
• b t
• y a b
• x w
• z x 1
• t x 1
• a t
• b t
• y t t
• x w
• z w 1

Common sub-expression elimination
Dead code elimination(assuming a, b, x not used
after)
t x 1 y t t z w 1
Copy Propagation
20
Uninitialized variables

• Can use data-flow analysis for semantic checking
  too detect uninitialized variables (use before
  def )
• Use reaching definitions analysis
• If use occurs with no reaching definition,
  variable cannot have been initialized -- likely
  an error
• More restrictive check require that all paths
  include definition (Java rule) check that
  variable is available instead

21
Summary

• Standard optimizations require data-flow analyses
  that fit into lattice analysis framework can
  also be used for semantic analyses
• Live variable analysis backward, ? ?
• Reaching definitions forward, ? ?
• RHS availability, general availability forwar
  d, ? ?