A PolynomialTime Algorithm for Global Value Numbering

1
A Polynomial-Time Algorithm for Global Value
Numbering

• SAS 2004
• Sumit Gulwani George C. Necula

2
Global Value Numbering

• Goal Discover equivalent expressions in
  procedures
• Applications
• Compiler optimizations
• Copy propagation, Constant propagation, Common
  sub-expression elimination, Induction variable
  elimination etc.
• Program verification
• Discover loop invariants, verify program
  assertions
• Discover equivalent computations across programs
• Translation validation, Plagiarism detection
  tools

3
Global Value Numbering

• Challenge Undecidable Problem
• Simplification Assumptions
• Operators are uninterpreted (will not discover u
  x)
• Conditionals are non-deterministic (will not
  discover v x)
• Will discover x y

4
Non-trivial Example

x b y b z F(b)
x a y a z F(a)
assert(x y) assert(z F(y))
5
Related Work

• Algorithms that work on SSA form of the program
• AWZ Algorithm (POPL 1988)
• Polynomial, Incomplete
• RKS Algorithm (SAS 1999)
• Polynomial, Incomplete, Improvement on AWZ
• Dataflow analysis or Abstract interpretation
  based algorithm
• Kildalls Algorithm (POPL 1973)
• Exponential, Complete
• Our Algorithm (this paper)
• Polynomial, Complete

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Non-trivial Example
x ?(a,b) y ?(a,b) z ?(F(a),F(b)) F(y)
F(?(a,b))

x b y b z F(b)
x a y a z F(a)
assert(x y) assert(z F(y))

• AWZ Algorithm ? functions are uninterpreted
• fails to discover second assertion
• RKS Algorithm uses rewrite rules for
  normalization
• Does not discover all assertions in little more
  involved examples.
• Rewrite rules not applied exhaustively (exp
  applications o.w.)
• Rules are pessimistic in handling loops

7
Outline

• Strong Equivalence DAG (SED)
• The Assignment Operation
• The Join Operation
• Pruning an SED
• Fixed Point Computation

8
Representing Equivalences
a 1 b 2 x F(1,2)
a,1 b,2 x, F(1,2)
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Representing Equivalences
a 1 b 2 x F(1,2)
a,1 b,2 x, F(1,2), F(a,2), F(1,b),
F(a,b)
Such an explicit representation can be
exponential.
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Strong Equivalence DAG (SED)

• A data structure for representing equivalences.
• Nodes n ltSet of variables, Typegt
• Type c, ?, F(n1,n2)
• For every variable x, exactly one node ltV,tgt s.t.
  x 2 V
• called Node(x)
• Terms(n) set of equivalent expressions
• Terms(ltV, ?gt) V
• Terms(ltV, cgt) V c
• Terms(ltV, F(n1,n2)gt) V
• F(e1,e2) e1 2 Terms(n1),
  e2 2 Terms(n2)

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SED Example
e, F
d,c, F
a, 2
b, ?

• Terms(n1) a, 2
• Terms(n2) b
• Terms(n3) c, d, F(a,b), F(2,b)
• Terms(n4) e, F(c,b), F(d,b), F(F(a,b),b),
  F(F(2,b),b)
• Note that e F(d,b) since F(d,b) 2
  Terms(Node(e))

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Abstract Interpretation based algorithm
G
Assignment Node
x e
G Assignment(G,x e)
G
Conditional Node


G2 G
G1 G
G1
G2
Join Node
G Join(G1, G2)
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Example
x 1 y 1 z F(1,1)
x 2 y 2 z F(2,2)
L1
L2
L3
u F(x,y)
L4
Assert(u z)
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Outline

• Strong equivalence DAG (SED)
• The assignment operation
• The join operation
• Pruning an SED
• Fixed point computation

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The Assignment Operation

• Assignment(G, x e)
• It is the strongest postcondition of equivalences
  represented by G w.r.t the assignment x e
• Delete label x from Node(x) in G
• Let nltV,tgt be the node in G s.t. e 2 Terms(n)
• (Add such a node to G if it does not already
  exists)
• Add x to node n.

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Example The Assignment Operation
F
u, F
G0 Assignment(G, u F(z,x))
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Outline

• Strong Equivalence DAG (SED)
• The Assignment Operation
• The Join Operation
• Pruning an SED

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The Join Operation

• Join(G1, G2)
• Product Construction of G1 and G2
• If nltV1,t1gt 2 G1 and mltV2,t2gt 2 G2, then
• (n,m) ltV1 Å V2, t1 t t2gt 2 Join(G1,G2)
• Definition of t1 t t2
• c t c c
• F(n1,n2) t F(m1,m2) F ((n1,m1),(n2,m2))
• t1 t t2 ?, otherwise

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Example The Join Operation
y1, F
y2, F
F
F
y6,?
y7,?
y4,y5 ?
y3,?
G1
G2
G Join(G1,G2)
G
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Outline

• Strong equivalence DAG (SED)
• The assignment operation
• The join operation
• Pruning an SED
• Fixed point computation

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Motivation The Prune Operation

• If GJoin(G1,G2), then Size(G) can be Size(G1)
  Size(G2)
• There are programs with n joins such that size of
  the SEDs after joins is exponential in n.

Discovering equivalences among all expressions
Discovering equivalences among program expressions
vs.
For the latter, it is sufficient to discover
equivalences among all terms of size at most t at
each program point (where t variables size
of program). Thus, SEDs can be pruned to have a
small size (k t)
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The Prune Operation

• Prune(G,k)
• For each node ltV,tgt such that V ? , check if it
  is a root of some DAG with all leaves labelled
  with at least one variable of size k.
• If not, then delete all the nodes that are
  reachable from only ltV,tgt

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Example The Prune Operation
G
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Outline

• Strong equivalence DAG (SED)
• The assignment operation
• The join operation
• Pruning an SED
• Fixed point computation

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Fixed Point Computation and Complexity

• The lattice of sets of such equivalences has
  height at most k
• Complexity
• Dominated by the cost of join operations
• Each join operation O(k2 N)
• This requires doing pruning while computing join
• of join operations O(j k)
• Total cost O(k3 N j)
• k of variables
• N size of program
• j of join points in program

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Conclusion

• Discovering all