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Use trace algebra to formalize the YAPI model

Use trace algebra to formalize the YAPI model. EE290N Spring2002 Alessandro Pinto Mentors: Roberto Passerone Jerry Burch. Outline. References Introduction to YAPI Implication of select Introduction to Trace Algebra Traces to model YAPI Buffer model

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Use trace algebra to formalize the YAPI model

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  1. Use trace algebra to formalize the YAPI model EE290N Spring2002 Alessandro Pinto Mentors: Roberto Passerone Jerry Burch

  2. Outline • References • Introduction to YAPI • Implication of select • Introduction to Trace Algebra • Traces to model YAPI • Buffer model • Conservative Approximation • Example • Conclusion

  3. References • K.A. Vissers et. al. “YAPI: Application Modeling for Signal Processing System”, DAC00 • J. Burch, R. Passerone, ASV “Overcoming Heterophobia: Modeling Concurrency in Heterogeneous Systems” • J. Burch, R. Passerone, ASV “Modeling Techniques in Design-by-Refinement Methodologies” • E.A. Lee, T.M. Parks “Dataflow Process Networks” • G. Kahn, “The Semantics of a Simple Language for Parallel Programming” • G. Kahn, D.B. MacQueen, “Corutines and Networks of Parallel Processes”

  4. The YAPI Model 1 • KPN (blocking read, non-blocking write) • Non-determinism

  5. The YAPI Model 2 • Selection can be done on input and output YAPI Read,Write on unbounded FIFOs Read,Write on bounded FIFOs TTL

  6. Implication of select • continuitymonotonicity determinacy i1=[1], i2=[] F(i1,i2) = [1] i1=[1,1], i2=[2] F(i1,i2) = [2,1] i=[1] F(i) = ([1],[]) i=[1,3] F(i) = ([1],[3]), ([3],[1]), ([1,3],[]) …

  7. Trace Algebra 1 Model of individual behaviors Model of agents Trace algebra C • Trace Algebra • Set of traces • Projection of traces • Renaming of traces • Concurrency Algebra • Set of agents • Parallel composition of agents • Projection of agents • Renaming of agents Trace structure algebra A i1 I1=[1,3,4,-1] I2=[0,3,5,2] O=[1,6,9,1] o + i2 Source: R. Passerone

  8. Trace Algebra 2 • Alphabet • Set of traces over alphabet A • Renaming y = rename(r) (x) where x is a trace and r is a renaming function • Projection y = proj(B)(x) where x is a trace and BA • Trace structure T=(,P) where  is the signature and P  B(A) • A set of axioms must be verified !!!!!!!

  9. Trace Algebra 3 Homomorphism h Derive Trace algebra C’ Trace structure algebra A’ “Abstract” Domain Yinv Yu Yl Trace algebra C Trace structure algebra A “Detailed” Domain Let Tspecand Timplbe trace structures in A. Then if Yu( Timpl ) Yl( Tspec ) then Timpl  Tspec Source: R. Passerone

  10. Implication of select: Trace Algebra View • Monotonicity is captured by projection • proj(I)(x) ⊑ proj(I)(y)  proj(O)(x) ⊑ proj(O)(y) • For the previous example two traces belongs to the same process:

  11. First step: Set of Traces • Which set of traces should we choose? • Streams • Synchronous • DE

  12. Streams 1 • No timing information • KPN can be expressed on the same set of traces • Continuity is a constraint on the trace structures • In this sense YAPI processes can form more trace structures then KPN YAPI KPN

  13. Streams 2 • Operation on traces • Renaming • Projection • Operation on trace structures

  14. A more detailed domain 1 • We want to capture two important things • Relative arrival time • Bounded FIFOs • Synchronous Domain

  15. A more detailed domain 2 • Renaming • Projection Two possible traces

  16. Buffers • Buffers are modeled as processes • The strict version of a sequence a is • We indicate with ak the sequence up to instant k • Definition of a buffer • Trace structure TB=((I,O),P)

  17. Buffer Constraint Two possible traces of AddSub Buf1 Gen AddSub Buf2 Bufferlength =2 T=Gen||Buf1||Buf2||AddSub Bufferlength =1

  18. Conservative Approximation 1 Nondet KPN Homomorphism h(x) Approximation Synch

  19. Conservative Approximation 2 h is in general many to one so this is an upper bound h(P) P Abstract Domain Detailed Domain

  20. Conservative Approximation 2 h(P) P Abstract Domain h(B(A) – P) h(P) - h(B(A) – P) B(A) - P Detailed Domain

  21. From Synch to ND-KPN 1 • Definition of the homomorphism h(x)

  22. From Synch to ND-KPN 2 • Let’s consider a process: • If p is true add the inputs, if p is false subtract them h(x) h(x)

  23. Applications of the Approximation • Verification Problem • Design Problem

  24. Example 1 • Norm Bufferlength=n =(I,O) =(I,O) =(I,O)

  25. Example 1 Bufferlength=n =(I,O) =(I,O) =(I,O)

  26. Example 1 Bufferlength=n =(I,O) =(I,O) =(I,O)

  27. Example 1 Bufferlength=n =(I,O) =(I,O) =(I,O)

  28. Example 2 • One possible execution Bufferlength=n

  29. Example 3 id h(x)

  30. Example 4 • One possible execution

  31. Example 4 • Design problem

  32. Conclusion • The YAPI model was a good motivation for this project • Trace algebra has been applied to describe the denotational semantics of the model at different level of abstraction • A Conservative approximation between Synchronous model and ND-KPN allows abstraction from one model to another for verification and design

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