1. Mechanical Certification of Loop Pipelining Transformations: A Preview
Disha Puri*, Sandip Ray+, Kecheng Hao*, Fei Xie*
In the context of
16 July 2014
*Computer Science Department, Portland State University
+Strategic CAD Labs, Intel Corporation

2. Agenda Background Need for Certified Loop Pipelining Algorithm
Framework for Certified Pipelining Algorithm
Current Status & Future Work
Related Work

3. Agenda Background Need for Certified Loop Pipelining Algorithm
Framework for Certified Pipelining Algorithm
Current Status & Future Work
Related Work

4. Behavioral Synthesis High level specification of hardware design
(in C, C++, System C)
Compiler Transformations
(e.g., dead code elimination)
Behavioral
synthesis tool
(e.g.,
Vivado,
CtoSilicon, Cynthesizer)
Scheduling Transformations
(e.g., Loop pipelining)
Resource Allocation
and Control Synthesis
Implementation in hardware description language
(in VHDL, Verilog)

5. Loop in Behavioral Synthesis
Entry
Exit
c := i + 3
i′ := a′ + c X Y Z
{ a := ϕ [0,Entry] [a′,Z]
i := ϕ [0,Entry] [i′,Z] }
if [not (a < N)] goto Exit
a′ := a + 2
a = 0; i = 0; while(a < N) { a = a + 2; c = i + 3; i = a + c; }
Loop in C
Loop before pipelining

6. Loop Pipelining Sequential Execution Pipelined Execution Entry Entry XZ
Exit
Pipeline Prologue X Y X
Pipeline
Full Z Y X
Pipeline Epilogue Z
Exit
Sequential Execution
Pipelined Execution

7. Motivation behind Loop PipeliningX Y Z
Execution order before pipelining X Y Z
Execution order after pipelining
“ Pipeline interval is 1”

8. Agenda Background Need for Certified Loop Pipelining Algorithm
Framework for Certified Pipelining Algorithm
Current Status & Future Work
Related Work

9. Certification of Loop Pipelining
Entry X Y Z
Exit X Y Z
Entry
Exit ?
Mechanical reasoning of synthesis tool’s implementation
Implementation is not disclosed by vendors, so we cannot certify the transformation by theorem proving

10. Certification of Loop Pipelining
Entry X Y Z
Exit X Y Z
Entry
Exit
Sequential Equivalence Checking
Sequential equivalence checking does not work due to huge abstraction gap between the two designs

11. Certifying Loop Pipelining TransformationX Y Z
Entry
Exit
Entry X Y Z
Exit
Loop Pipelining Transformation
Pipeline parameters
Certified Pipelining Algorithm
SEC
Pipeline
Reference
Model
1. Create a correct pipeline reference model
2. Compare using sequential equivalence checking (SEC)
Hao, et al “Equivalence checking for behaviorally synthesized pipelines”. DAC 2012

12. Goal: A Certified Loop Pipelining Algo
Entry X Y Z
Exit X Y Z
Entry
Exit ?
Certified Pipelining Algo
Pipeline
Reference Model
Sequential Execution

13. Agenda Background Need for Certified Loop Pipelining Algorithm
Framework for Certified Pipelining Algorithm
Current Status & Future Work
Related Work

14. Challenges in Loop Pipelining
Correspondence??
Entry
Entry X Y Z
Exit
Pipeline Prologue X Y X
Pipeline Full Z Y X
Pipeline Epilogue Z
Exit
Sequential Execution
Pipelined Execution

15. Our Approach
Framework of certified pipelining primitives to build pipelining algorithms
Shadow register primitive
Interchange primitive
Phi-elimination primitive
Branch-handling primitive
Develop a simple loop pipelining algorithm from ground up using our framework
Certify the algorithm

16. Loop Pipelining InvariantX
Pipeline
Prologue X Y Y X Z
Pipeline
Full X Z Y X
Pipeline
Epilogue Y Z X
Executing pipeline prologue + k iterations of pipeline full =
Executing k iterations of sequential loop + m blocks of sequential loop+ (m-I) blocks of sequential loop + …

17. Loop Invariant: Base case (k=1)X'
Pipeline
Prologue X' Y Y X Z
Pipeline
Full Z Y X X Y X
Executing pipeline prologue + “1” iterations of pipeline full =
Executing “1” iterations of sequential loop + 2 blocks of first iteration + 1 blocks of second iteration

18. Loop Invariant: Inductive StepX' X'
Pipeline
Prologue X' Y Y X Y Z Z X X
Pipeline
Full Y Y Z Y X X Z Z X Y Y X X

19. Invariant Implies CompletenessX' X'
Pipeline
Prologue X' Y Y Y X Z Z
Pipeline
Full X X Z Y X Y Y
Pipeline
Epilogue Z X Z Z X

20. Agenda Background Need for Certified Loop Pipelining Algorithm
Framework for Certified Pipelining Algorithm
Current Status & Future Work
Related Work

21. Current Status Completed Work Developed framework of primitives
Certified primitives:
Phi-removal primitive
Shadow register primitive
Formalized algorithm using framework
Certified the correspondence invariant

22. Next Steps Ongoing Work Certify remaining invariants
Certify the algorithm end-to-end
Future Work
Function pipelining algorithm
There are other pipelining algortihms such as not details…

23. Agenda Background Need for Certified Loop Pipelining Algorithm
Framework for Certified Pipelining Algorithm
Current Status & Future Work
Related Work

24. Related Work
Pipelines in processors [Manolios et al., 2000] [Sawada and Hunt, 2002] [Beyer et al., 2006]
Similar in notions of correctness
Work on one instance of a pipeline, not an entire algorithm
Software pipelines [Tristan and Leroy, 2010]
Works on instances of designs
Symbolic simulation

25. Thanks !!