Constraint Systems used in Worst-Case Execution Time Analysis

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Constraint Systems used in Worst-Case Execution
Time Analysis
Andreas Ermedahl andreas.ermedahl_at_it.uu.se Dept.
of Information Technology Uppsala University
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Definition of WCET
actual BCET
actual WCET
possible execution times
safe BCET estimates
safe WCET estimates
0
tighter
tighter

• WCET Worst possible execution time for a
  program running on target hardware
• One program in isolation
• No interrupts or context switches
• Other estimates
• Best Case ET Best case, Inverse of WCET, hard
• Average Case ET Soft real-time, not hard

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Uses of WCET Estimates

• Hard real-time systems
• Scheduling
• Schedulability analysis
• System dimensioning
• Formal verification
• Program performancetuning

void foo(int j, int a) int i
for(i100, igt0 i--) if(jgt50)
ai 2 i else
ai i
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Static WCET Analysis

• Dont run the program - analyze it!
• Guaranteed safe WCET
• Trying to be as tight as possible
• Provided all input is correct

actual BCET
actual WCET
possible execution times
safe BCET estimates
safe WCET estimates
0
tighter
tighter
Measurement will give results in the unsafe area
Static analysis will give results in the safe area
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Static WCET Analysis
program

• Flow analysis
• Determine the dynamic behavior of program
• Low level analysis
• Determine execution time for program parts on the
  hardware
• Calculation
• Combine flow and low-level times to give a WCET
  estimate.

Compiler
ObjectCode
Target Hardware
ActualWCET
Reality
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WCET Analysis Architecture

• Modularization of WCET analysis
• Several separate analysis steps
• Make the analysis more retargetable

• All analysis results are converted to
  constraints

Program
WCET
Compiler
Simulator
ObjectCode
Cache Analysis
Cache Info
IntermediateCode
Pipeline Analysis
Flow Analysis
Constraint Problem
FlowInfo
Calculation
Flow FactConversion
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Flow Info Characteristics
do if(...) do
if(...) ...
else ...
if(...) ... else
... while(...)
else ... while(...) ...
WCET found here desired result
WCET found here overestimation
Example program
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Count Variables

• Program model
• Basic block graph nodes and edges
• Execution count variable (xentity) holds number
  of times entity gets executed

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Constraints Generated

• Constraints

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Pipeline Analysis

• Time for a basic block
• Time from first instruction enters pipeline to
  last instruction leaves pipeline

1
2
3
4
5
6
7
tA 7
A
tB 5
B
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Pipeline Analysis

• Pipeline overlap between basic blocks
• Timing effect of going from A to B is negative
  to indicate pipeline overlap
• We use a general purpose simulator to extract
  times for nodes and edges

1
2
3
4
5
6
7
tA 7
A
?AB -2
tB 5
B
tAB 10
?AB tAB - tA - tB 10 - 7 - 5 -2
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Pipeline Analysis Result
Result
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IPET Calculation

• WCETmax ?(xentity tentity)
• Where each xentity satisfies all constraints

Xfoo1
XCXF100
XAlt100
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Calculation methods

• Solution methods
• Integer linear programming
• Constraint satisfaction
• Solution
• Counts for each individual node and edge
• The value of the WCET

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Complicating Flow Information

• Some flow information complicates the picture
• Examples
• Local semantics For each entry of inner
  loop...
• Partially valid flow info During iterations 5
  to 10 of inner loop it holds that...
• Loop dependencies Number of iterations of
  inner loop depends on current iteration of
  outer
• Non-linear constraints If D was taken then K
  will be taken once

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Local semantics

• Create entry count variable xentry(scope) holds
  number of times loop is entered
• Local flow info are raised to the global level
• For example loop XH ? 10
• gets converted to
• XH ? 10 Xentry(loop)

A
xentry(loop)
B
C
For each entry of the loop block H will be
executed at most 10 times
D
E
F
G
H
I
J
Block H can not be executed more than 10 the
number of times the loop is entered
Scope graph
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Partial Flow Information

• Facts can (partially) overlap
• Virtual scopes scoperange let facts be valid
  for complete range of iterations

A
B
C
Loop bound 20 loop 1..5 XC5 (f1) loop
3..10 XC XF ? 8 (f2)
D
F
E
G
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Partial Flow Information

• Graph is unrolled according to overlapping flow
  info
• Can generate large graphs
• Flow information often local (but can stretch
  over loop borders)
• Dependent flow info can be used to consider
  subpart of graph in isolation

A
B
C
D
F
E
Fact f1 and f2 only overlaps iterations 1..10
G

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Non-linear Constrains

• Some flow info generates non-linear constraints
• Number of iterations of inner loop depends on
  current iteration of outer outer XB ? 55 or
  outer XA XA ? XB
• If D was taken then K will be taken once if
  XD gt 0 then XK gt 0 and if XD 0 then XK 0
• More powerful solver needed?

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WCET Tool Prototype

• Prototype tool implemented
• Works over whole program or use flow info to work
  bottom up over smaller program parts
• Fast solution times when testing with CPLEX or
  similar solver (a network flow problem?)
• We use integer linear programming (ILP) and
  lp_solve()
• Not all flow information can be handled
• Rather fast calculation times
• Flow information sometimes generates large graphs
  and solution times
• Other WCET research Expressing hardware effects
  using constraints generates huge constraint
  systems and large solution times

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Experimental Results

• Small programs fast calculation (even with
  complex structure and flow)
• Larger programs Flow information slows down but
  increase precision
• Example program Nsichneu
• Automatically generated program with massive
  amount of if-statements (gt 250)

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Questions

• What constraint solver should we use?
• Are the generated constraints of a certain type?
• More comments....

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The End!