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Analysis of
programs with pointers
Wednesday, November 30, 2011
Simple example
•
What are the dependences in this program?
•
Problem: just looking at variable names will not give you the correct
information
–
After statement S2, program names “x” and “*ptr” are both expressions
that refer to the same memory location.
–
We say that ptr pointsto x after statement S2.
•
In a Clike language that has pointers, we must know the pointsto
relation to be able to determine dependences correctly
x := 5
ptr := @x
*ptr := 9
y := x
S1
S2
S3
S4
dependences
program
Wednesday, November 30, 2011
Program model
• For now, only types are int and int*
• No heap
– All pointers point to only to stack variables
• No procedure or function calls
• Statements involving pointer variables:
– copy:
x := y
– load:
x := *y
– store:
*x := y
• Arbitrary computations involving ints
Wednesday, November 30, 2011
Pointsto relation
• Directed graph:
– nodes are program variables
– edge (a,b): variable a pointsto variable b
• Can use a special node to represent NULL
• Pointsto relation is different at different program
points
x
ptr
y
Wednesday, November 30, 2011
• Outdegree of node may be more than one
– if pointsto graph has edges (a,b) and (a,c), it means that
variable a may point to either b or c
– depending on how we got to that point, one or the other
will be true
– pathsensitive analyses: track how you got to a program
point (we will not do this)
Pointsto graph
if (p)
then x := &y
…..
p
What does x point to here?
Wednesday, November 30, 2011
Ordering on pointsto relation
• Subset ordering: for a given set of
variables
– Least element is graph with no edges
– G1 <= G2 if G2 has all the edges G1 has and
maybe some more
• Given two pointsto relations G1 and G2
– G1 U G2: least graph that contains all the
edges in G1 and in G2
Wednesday, November 30, 2011
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View Full DocumentOverview
•
We will look at three different pointsto analyses.
•
Flowsensitive pointsto analysis
– Dataflow analysis
– Computes a different pointsto relation at each point in program
•
Flowinsensitive pointsto analysis
– Computes a single pointsto graph for entire program
– Andersen’s algorithm
• Natural simplification of flowsensitive algorithm
– Steensgard’s algorithm
• Nodes in tree are equivalence classes of variables
– if x may pointto either y or z, put y and z in the same equivalence class
• Pointsto relation is a tree with edges from children to parents rather
than a general graph
• Less precise than Andersen’s algorithm but faster
Wednesday, November 30, 2011
Example
x := &z
ptr := @x
y := @w
ptr := @y
ptr
x
z
y
w
ptr
x
z
y
w
ptr
x,y
z,w
Flowsensitive algorithm
Andersen’s algorithm
Steensgard’s algorithm
Wednesday, November 30, 2011
Notation
• Suppose S and S1 are setvalued variables.
• S
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