Data Flow Analysis 3 15-411 Compiler Design

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Data Flow Analysis 315-411 Compiler Design
Nov. 8, 2005
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Key Reference on Global Optimization

• Gary A. Kildall, A Unified Approach to Global
  Program Optimization, ACM Symposium on Principles
  of Programming Languages, 1973, pages 194-206.
• From the abstract
• A technique is presented for global analysis of
  object code generated for expressions. The global
  expression optimization presented includes
  constant propagation, common sub-expression
  elimination, elimination of redundant register
  load operations and live expression analysis. A
  general purpose program flow analysis algorithm
  is developed which depends on an optimizing
  function. The algorithm is defined formally using
  a directed graph model of program flow structure
  and is shown to be correct.

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Kildalls Contribution

• A number of techniques had been developed for
  compile-time optimization to
• ? locate redundant computations,
• ? perform constant computations,
• ? reduce the number of store-load
  sequences, etc.
• Some provided analysis of only straight-line
  sequences of instructions others tried to take
  program branching into account.
• Kildall gave a single unified flow analysis
  algorithm which extended all the straight-line
  techniques to include branching.
• He stated the algorithm formally and proved it
  correct in his POPL paper.

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Constant Propagation Example program

• begin
• integer i, a, b, c, d, e
• a 1 c0
• for i 1 step 1 until 10 do
• begin b 2
• d a b
• e b c
• c 4
• end
• end

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Directed Graph Representation
Nodes represent sequences of instructions with no
branches. Edges represent control flow between
nodes.
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Constant Propagation

• Convenient to associate a pool of propagated
  constants with each node in the graph.
• Pool is a set of ordered pairs which indicate
  variables that have constant values when node is
  encountered.
• The pool at node B denoted by PB consists of a
  single element (a,1) since the assignment a 1
  must occur before B.

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Constant Propagation (cont.)

• Fundamental problem of constant propagation is to
  determine the pool of constants for each node in
  an arbitrary program graph.
• By inspection of the program graph for the
  example, the pool of constants at each node is
• PA ? PB (a, 1) PC (a, 1) PD
  (a, 1), (b, 2)
• PE (a, 1), (b, 2), (d, 3) PF (a, 1), (b,
  2), (d, 3)

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Constant Propagation (cont.)

• PN may be determined for each node N in the graph
  as follows
• ? Consider each path (A, p1,p2, , pn,N). Apply
  constant propagation along path to obtain set
  of constants at node N.
• ? Intersection for each path to N is the set of
  constants which can be assumed for optimization.
• (It is unknown what path will be taken at
  execution time, so intersection is conservative
  choice)

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Global Analysis Algorithm--Informal

• Start with an entry node in the program graph,
  along with a given entry pool corresponding to
  this entry node.
• Process the entry node and produce optimization
  information for all immediate successors of the
  entry node.
• Intersect incoming optimizing pools with already
  established pools at the successor nodes.
• (First time node is encountered, assume incoming
  pool is first approximation and continue
  processing.)
• for each successor, if amount of optimizing
  information is reduced by this intersection, then
  process successor like initial entry node.

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Global Analysis Algorithm (cont)

• It is useful to define an optimizing function f
  which maps an input pool together with a
  particular node to a new output pool.
• Given a set of propagated constants, it is
  possible to examine the operation of a particular
  node and determine the set of constants that can
  be assumed after the node is executed.
• In the case of constant propagation, let V be a
  set of variables, C be a set of constants, and N
  be the set of nodes in the graph.
• The set U V C represents ordered pairs which
  may appear in any constant pool.
• In fact, all constant pools are elements of the
  power set U, denoted P(U).
• Thus, f N P(U) ! P(U), where (v, c) 2 f(N, P)
  if and only if
• 
  (cont.)
• 
  
  

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Global Analysis Algorithm (cont.)

• 1. (v, c) 2 P and the operation at node N
  does not assign a new value to the variable v.
• 2. The operation at N assigns an
  expression to the variable v, and the expression
  evaluates to the constant c.

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Constant Propagation (cont.)

• Successively longer paths from A to D can be
  evaluated, resulting in PD,3 , PD,4 , , PD,n for
  arbitrarily large n.
• The pool of constants that can be assumed no
  matter what flow of control occurs is the set of
  constants common to all PD,i , i.e.
  
  
• Åi
  PD,i
• This procedure is not effective since the number
  of such paths may have no finite bound, and the
  procedure would not halt.

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Optimization Function for Example

• The optimizing function can be applied to node A
  with an empty constant pool resulting in
• f(A, ) (a,1).
• The function can be applied to B with (a, 1) as
  the constant pool yielding
• f(B, (a, 1)) (a, 1),
  (c, 0).

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Extending f to Paths in the Graph

• Given a path from entry node A to an arbitrary
  node N, optimizing pool for path is determined by
  composing the function f.
• For example, f(C, f(B, f(A, ))) (a, 1), (c,
  0), (b, 2) is the constant pool for D for this
  path.

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Constant Propagation (cont.)

• The pool of propagated constants at node D can be
  determined as follows
• A path from entry node A to the node D is (A, B,
  C, D). For this path the first approximation to
  the pool for D is
• PD,1 (a, 1), (b, 2), (c,
  0).
• A longer path from A to D is (A, B, C, D, E, F,
  C, D) which results in the pool
• PD,2 (a, 1), (b, 2), (c, 4),
  (d, 3), (e, 2).

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Computing the Pool of Optimizing Information.

• The pool of optimizing information which can be
  assumed at node N in the graph, independent of
  the path taken at execution time, is
• PN Å x x 2
  FN.
• Here FN f(pn, f(pn-1, , f(p1, P))) (p1,
  p2, , pn, N) is a path from an entry node p1
  with corresponding entry pool P to node N.

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Directed Graphs and Paths

• A finite directed graph G ltN,Egt is an arbitrary
  finite set of nodes N and edges E ½ N N.
• A path from node A to node B in G is a sequence
  (p1, p2, , pk ) such that p1 A and pk B
  where (pi, pi1) 2 E for 16 i lt k.
• The length of the path is k 1.

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Program Graphs

• A program graph is a finite directed graph G with
  a non-empty set of entry nodes I ½ N.
• Given N 2 N we assume there exists a path (p1,
  p2, , pn) such that p1 2 I and pn N.
• (i.e., there is a path to every node in the graph
  from an entry node.)

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Successors and Predecessors of a Node

• The set of immediate successors of a node N is
  given by
• I(N) N 2 N 9 (N,N) 2 E.
• The set of immediate predecessors of N is given
  by
• I-1(N) N 2 N 9 (N, N) 2 E.

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Meet-Semilatticies

• Let the finite set L be the set of all possible
  optimizing pools for a given application.
• Let Æ be a meet operation with the properties
• Æ L L ! L
• x Æ y y Æ x
• x Æ (y Æ z) (x Æ y) Æ z
• where x, y z 2 L. The set L and the Æ operation
  define a finite meet-semilattice.

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Ordering on Meet-Semilattices

• The Æ operation defines a partial ordering on L
  by
• x 6 y if and only if x Æ y x.
• Similarly,
• x lt y if and only if x 6y and x ? y.

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Generalized Meet Operation

• If X ½ L, the generalized meet operation Æ X is
  defined as the pairwise application of Æ to the
  elements of X.
• L is assumed to have a zero element 0 such that
  0 6 x for all x 2 L.
• An augmented set L is constructed from L by
  adding a unit element 1 such that 1 is not in L
  and 1 Æ x x for all x in L.
• The set L L 1. It follows that x lt1 for
  all x in L.

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Optimizing Function

• An optimizing function f is defined
• f N L ! L .
• It must have the homomorphism property
• F(N, x Æ y) f(N, x) Æ f(N, y) for all N 2 N and
  x, y 2 L.
• Note that f(N, x) lt 1 for all N 2 N and x 2 L.

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Global Analysis Algorithm

• Global analysis starts with an entry pool set EP
  ½ I L, where (e, x) 2 EP if e 2 I is an entry
  node with optimizing pool x 2 L.
• A1 initialize L EP.
• A2 terminate ? If L then halt.
• A3 select node Let L 2 L, L (N, Pi) for
  some N 2 N and Pi 2 L.
• Then L L L.
• A4 Traverse Let PN be the current
  approximate pool for node N
• (Initially PN 1). If
  PN 6 Pi the go to step A2.
• A5 set pool PN PN Æ Pi, L L (N,
  f(N, PN)) N 2 I(N).
• A6 Loop Go to step A2.