introduction to data flow analysis

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Introduction to Data Flow Analysis

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Data Flow AnalysisData Flow Analysis

• Construct representations for the structure of flow-of-data of programs based on the structure of flow-of-control of programs

• Collect information about the attributes of data at various program points according to the structure of flow-of-data of programs

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PointsPoints

• Within each basic block, a point is assigned between two adjacent statements, before the first statement, and after the last statement

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An ExampleAn Example

d1: i = m - 1d2: j = nd3: a = u1

d4: i = i + 1

d5: j = j - 1

d6: a = u2

B1

B2

B3

B4

B5 B6

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PathsPaths

• A path from p1 to pn is a sequence of points p1, p2, …, pn such that for each i, 1 i n-1, either

• pi is the point immediately preceding a statement and pi+1 is the point immediately following that statement in the same block, or

• pi is the end of some block and pi+1 is the beginning of a successor block

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d1: i = m - 1d2: j = nd3: a = u1

d4: i = i + 1

d5: j = j - 1

d6: a = u2

B1

B2

B3

B4

B5 B6d7: i = u3

if e3

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Reaching DefinitionsReaching Definitions

• A definition of a variable x is a statement that assigns or may assign a value to x

• A definition d of some variable x reaches a point p if there is a path from the point immediately following d to p such that no unambiguous definition of x appear on that path

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d1: i = m - 1;d2: j = n;d3: a = u1; dod4: i = i + 1;d5: j = j - 1; if e1 thend6: a = u2 elsed7: i = u3 while e2

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Ambiguity of Definitions Ambiguity of Definitions

• Unambiguous definitions (must assign values)– assignments to a variable– statements that read a value to a variable

• Ambiguous definitions (may assign values)– procedure calls that have call-by-reference paramet

ers– procedure calls that may access nonlocal variables– assignments via pointers

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Safe or Conservative InformationSafe or Conservative Information

• Consider all execution paths of the control flow graph

• Allow definitions to pass through ambiguous definitions of the same variables

• The computed set of reaching definitions is a superset of the exact set of reaching definitions

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Information for Reaching Information for Reaching DefinitionsDefinitions

• gen[S]: definitions generated within S and reaching the end of S

• kill[S]: definitions killed within S

• in[S]: definitions reaching the beginning of S

• out[S]: definitions reaching the end of S

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Data Flow EquationsData Flow Equations

• Data flow information can be collected by setting up and solving systems of equations that relate information at various points

out[S] = gen[S] (in[S] - kill[S])

The information at the end of a statement is either generated within the statement or enters at the beginning and is not killed as control flows through the statement

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The Iterative AlgorithmThe Iterative Algorithm

• Repeatedly compute in and out sets for each node in the control flow graph simultaneously until there is no change

in[B] = p pred(B) out[P]

out[B] = gen[B] (in[B] - kill[B])

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Algorithm: Reaching DefinitionsAlgorithm: Reaching Definitions

/* Assume in[B] = for all B */for each block B do out[B] := gen[B]change := true;while change do begin change := false; for each block B do begin in[B] := p pred(B) out[p] oldout := out[B] out[B] := gen[B] (in[B] - kill[B]) if out[B] oldout then change := true endend

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d1: i = m - 1d2: j = nd3: a = u1

d4: i = i + 1d5: j = j - 1

d6: a = u2

B1

B2

B3 B4

d7: i = u3

111 0000000 1111

000 1100110 0001

000 0010001 0000

000 0001100 1000

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BlockInitial Pass 1 Pass 2

In[B] In[B] In[B]Out[B] Out[B]Out[B]

000 0000 000 0000 111 0000 000 0000 111 0000111 0000

000 0000 111 0011 001 1110 111 1111 001 1110000 1100

000 0000 001 1110 001 0111 001 1110 001 0111000 0001

000 0000 001 1110 000 1110 001 1110 000 1110000 0010

B1

B2

B3

B4

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Conservative ComputationConservative Computation

• The computed gen set of reaching definitions is a superset of the exact gen set of reaching definitions

• The computed kill set of reaching definitions is a subset of the exact kill set of reaching definitions

• The computed in and out sets of reaching definitions is a superset of the exact in and out sets of reaching definitions

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Local Data Flow InformationLocal Data Flow Information

• The gen and kill sets for a basic block is obtained from the gen and kill sets for the statements in the basic block

• Only the in and out sets for the basic blocks are computed in the global data flow analysis

• The in and out sets for the statements in a basic block can be computed locally from the in set for the basic block if necessary

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UD-Chains and DU-ChainsUD-Chains and DU-Chains• A variable is used at statement s if its r-valu

e may be required• The reaching definitions information is ofte

n stored as use-definition chains (or ud-chains)

• The ud-chain for a use u of a variable x is the list of all the definitions of x that reach u

• The definition-use chains (or du-chains) for a definition d of a variable x is the list of all the uses of x that use the value defined at d

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A Taxonomy of Data Flow A Taxonomy of Data Flow ProblemsProblems

in[B] = p pred(B) out[p]


out[B] = s succ(B) in[s]

in[B] = gen[B] (out[B] - kill[B])

in[B] = p pred(B) out[p]



in[B] = gen[B] (out[B] - kill[B])

Any

path

All

path

Forward-Flow Backward-Flow

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Available ExpressionsAvailable Expressions

• An expression x+y is available at a point p if every path from the initial node to p evaluates x+y, and after the last such evaluation prior to reaching p, there are no subsequent assignments to x or y

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t2 = 4 * i

t1 = 4 * i

i = …t0 = 4 * i

t2 = 4 * i

t1 = 4 * i

?

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The gen and kill SetsThe gen and kill Sets

• A block kills expression x+y if it possibly assign x or y and does not subsequently reevaluate x+y

• A block generates expression x+y if it definitely evaluates x+y and does not subsequently redefine x or y

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The gen Set for a BlockThe gen Set for a Block

• No expressions are available at the beginning

• Assume set A of expressions is available before statement x = y+z. The set of expressions available after the statement is formed by– adding to A the expression y+z– deleting from A any expression involving x

• At the end, A is the set of generated expressions

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The kill Set for a BlockThe kill Set for a Block

• All expressions y+z such that either y or z is defined and y+z is not generated by the block

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Statements Available Expressions

……………….…………nonea = b + c ……………….…………only b + cb = a - d ……………….…………only a - dc = b + c …………….……………only a - dd = a - d …………….……………none

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The in and out SetsThe in and out Sets

in[B] = , for B = initial

in[B] = p pred(B) out[p], for B initial


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Initialization of the in SetsInitialization of the in Sets

B1

B2

Oj+1 = G (Ij - K)Ij+1 = out[B1] Oj+1

I0 = O1 = GI1 = out[B1] GO2 = G

I0 = UO1 = U - KI1 = out[B1] - KO2 = G (out[B1] - K)

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Algorithm: Available ExpressionsAlgorithm: Available Expressions/* Assume in[B1] = and in[B] = U for all B B1 */in[B1] = ;out[B1] = gen[B1];for each block B B1 do out[B] := U - kill[B];change := true;while change do begin change := false; for each block B B1 do begin in[B] := p pred(B) out[p] oldout := out[B] out[B] := gen[B] (in[B] - kill[B]) if out[B] oldout then change := true endend

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• The computed gen set of available expressions is a subset of the exact gen set of available expressions

• The computed kill set of available expressions is a superset of the exact kill set of available expressions

• The computed in and out sets of available expressions is a subset of the exact in and out sets of available expressions

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Live VariablesLive Variables

• A variable x is live at a point p if the value of x at p could be used along some path in the control flow graph starting at p; otherwise, x is dead at p

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The def and use SetsThe def and use Sets

• def[B]: the set of variables definitely assigned values in B

• use[B]: the set of variables whose values are possibly used in B prior to any definition of the variable

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in[B] = use[B] (out[B] - def[B])

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Algorithm: Live VariablesAlgorithm: Live Variables/* Assume in[B] = for all B */for each block B do in[B] := change := true;while change do begin change := false; for each block B do begin out[B] := s succ(B) in[s] oldin := in[B] in[B] := use[B] (out[B] - def[B]) if in[B] oldin then change := true endend

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• The computed use set of live variables is a superset of the exact use set of live variables

• The computed def set of live variables is a subset of the exact def set of live variables

• The computed in and out sets of live variables is a superset of the exact in and out sets of live variables

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Busy ExpressionsBusy Expressions

• An expression is busy at a point p if along all paths from p to the final node its value is used before the expression is killed

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The use and kill SetsThe use and kill Sets

• use[B]: the set of expressions that are used before they are killed in B

• kill[B]: the set of expressions that are killed before they are used in B

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out[B] = , for B = final

out[B] = s succ(B) in[s], for B final

in[B] = use[B] (out[B] - kill[B])

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Algorithm: Busy ExpressionsAlgorithm: Busy Expressions/* Assume out[Bn] = and out[B] = U for all B Bn */out[Bn] = ;in[Bn] = use[Bn];for each block B Bn do in[B] := U - kill[B];change := true;while change do begin change := false; for each block B Bn do begin out[B] := s succ(B) in[s] oldin := in[B] in[B] := use[B] (out[B] - kill[B]) if in[B] oldin then change := true endend

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• The computed use set of busy expressions is a subset of the exact use set of busy expressions

• The computed kill set of busy expressions is a superset of the exact kill set of busy expressions

• The computed in and out sets of busy expressions is a subset of the exact in and out sets of busy expressions

introduction to data flow analysis

Documents

j 1d6

variableambiguous definitions

structure of flow

point p

outb outb

block b

outb oldout

basic block