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Structural methods for synthesis of large specifications

Structural methods for synthesis of large specifications. Thanks to Josep Carmona and Victor Khomenko. Outline. Structural theory of Petri nets Marking equation Invariants ILP methods based on the marking equation Detection of state encoding conflicts Synthesis

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Structural methods for synthesis of large specifications

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  1. Structural methods for synthesis of large specifications Thanks to Josep Carmona and Victor Khomenko

  2. Outline • Structuraltheory of Petrinets • Markingequation • Invariants • ILP methodsbasedonthemarkingequation • Detection of state encoding conflicts • Synthesis • Methods based on unfoldings

  3. State space explosion problem • Even in boundednets, thestatespace can beexponentialonthesize of the net • Concurrency (explosion of interleavings)

  4. Event-based vs. State-based model State Graph Petri Net

  5. a+ c+ a+ a- b+ b+ b- c+ c- p1 -1 0 0 0 1 -1 0 p2 1 0 -1 0 0 0 0 p3 1 -1 0 0 0 0 0 p4 0 0 0 0 0 1 -1 p5 0 0 0 -1 0 1 0 p6 0 0 1 0 -1 0 1 p7 0 1 0 1 -1 0 0 b+ a- c- b+ b- Marking equation p1 Incidence matrix p3 p4 p2 p5 p6 p7

  6. 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 a+ a- b+ b+ b- c+ c- -1 0 0 0 1 -1 0 1 0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 -1 0 1 0 0 0 1 0 -1 0 1 0 1 0 1 -1 0 0 Marking equation M’ = M + Ax p1 p2 p3 p4 p5 p6 p7 = + Necessary reachability condition, but not sufficient.

  7. Spurious markings

  8. Checking Unique State Coding z = {a+ b+ a- b-} x M0 M1 M2 • M1 and M2 have the same binary code (z must be a complementary set of transitions) • M1 and M2 must be different markings (they must differ in at least one place)

  9. Checking Unique State Coding z = {a+ b+ a- b-} x M0 M1 M2 ILP formulation: M1 = M0 + Ax M2 = M1 + Az bal(z) M1 M2 x, z, M1, M2 0 bal(z)  a: #(a+) - #(a-) = 0

  10. Some experiments (USC)

  11. a- QR(a-) ER(a-) a=0 a=1 QR(a+) ER(a+) a+ Checking Complete State Coding ILP formulation: M1 = M0 + Ax M2 = M1 + Az bal(z) M1 ER(a*) M2 ER(a*) x, z, M1, M2 0 n ILP problems must be solved (n is the number of transitions with label a*)

  12. Enabling conditions in ILP p1 p2 p3 p4 p5 a+ a+ M  ER(a+): M(p1)+M(p2) 2  M(p3)+M(p4)+M(p5)  3 M  ER(a+): M(p1)+M(p2) 1  M(p3)+M(p4)+M(p5)  2 (*formulation for safe nets only)

  13. Some experiments (CSC)

  14. Synthesis • Each signal can be implemented with asubset of the STG signals in the support(typically less than 10) • Synthesis of a signal: • Project the STG onto the support signals (hide the rest) • After projection, the STG still has CSC for the signal • Use state-based methods on each projection

  15. Synthesis example

  16. a- QR(a-) ER(a-) a=0 a=1 QR(a+) ER(a+) a+ Checking the support for a signal Let  be the set of signals and ’ a potential support for a. Let z’ be the projection of z onto ’. ’ is a valid support for a if the following model has no solution: ILP formulation: M1 = M0 + Ax M2 = M1 + Az bal(z’) M1 ER(a*) M2 ER(a*) x, z, M1, M2 0

  17. Algorithm to find the support z’ := {a}  {trigger signals of a}; forever z” := ILP_check_support (STG, a, z’); if z” = 0 thenreturn z’; z’ := z’  {unbalanced signals in z”}; end_forever

  18. Experiments (Support + Synthesis) CPU Literals ILP Petrify Petrify ILP • Petrify (Cortadella et al ): • State-based & technology mapping • BDD

  19. Design flow STG structural encoding remove internal signals and check CSC STG with CSC structural transformations optimized STG . . . support for z support for b support for a projection STG for a STG for b STG for z logic synthesis (petrify) circuit for a circuit for b circuit for z

  20. Synthesis example

  21. x3 x1 x2

  22. z x4 x5

  23. State encoding

  24. DTACK- DSw+ DSr+ LDS+ D+ LDTACK+ LDS+ D+ LDTACK- LDTACK+ D- DTACK+ LDS- DSr- DTACK+ D- DSw- Detection of conflicting states LDS+ LDTACK- LDS- LDTACK+ DSr+ DTACK- D+ D- DSr- DTACK+ • ILP • [Carmona & Cortadella, ICCAD’03] • SAT-UNFOLD • [Khomenko et al., Fund. Informaticae]

  25. DTACK- DSw+ DSr+ LDS+ D+ LDTACK+ LDS+ D+ LDTACK- LDTACK+ D- DTACK+ LDS- DSr- DTACK+ D- DSw- s+ s- Disambiguation by consistent signal insertion Disambiguate the conflicting states by introducing a new signal s: • STG Insertion of signal s must: • Solve conflict • Preserve consistency • Preserve persistency 10000 (CSC + consistency + persistency = SI-circuit) LDS+ LDTACK- 10100 10010 LDS- LDTACK+ 01110 DSr+ DTACK- 10110 10110 D+ D- 11111 10111 DSr- DTACK+ 01111

  26. Implicit place DEF1 (Behavior): The behavior of the net does not depend on the place. DEF2 (Petri net): it never disables the firing of a transition. y+ b+ a+ a- b- x+ y- x- x- b- y- a- a+ x+ x+ y+ b+ y-

  27. y+ x- y- b+ x- x+ b- y+ ... y+ x- y- b+ x- x+ b- y+ ... ? Consistency Consecutive firings of a signal must alternate b+ y- y+ x- y- b+ x- x+ b- y+ ... x+ x- x- b- y+

  28. y=0 y=1 Implicit Places & Consistency b+ y- x+ x- x- b- y+ Theorem (Colom et al.) Places y=0 and y=1 are implicitif and only if signal y is consistent

  29. s- s+ Disambiguation by consistent signal insertion Disambiguate the conflicting states by introducing a new signal s: • Insertion of s into the STG: • s- will precede LDS+ • s+ will precede DTACK- LDS+ LDTACK- s- ; LDS+ LDS+ LDS- LDTACK+ DSr+ DTACK- D+ D- DSr- DTACK+

  30. s=0 s=1 read cycle write cycle DSr+ DSw+ DTACK- s+;DTACK- LDS+ s-;LDS+ D+ LDTACK+ LDS+ LDTACK- D+ LDTACK+ DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  31. read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ LDTACK+ LDS+ s=0 s=1 LDTACK- D+ LDTACK+ DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  32. read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ LDTACK+ LDS+ s=0 s=1 LDTACK- D+ LDTACK+ DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  33. read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ LDTACK+ LDS+ s=0 s=1 LDTACK- D+ LDTACK+ DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  34. read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ LDTACK+ LDS+ s=0 s=1 LDTACK- D+ LDTACK+ DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  35. read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ LDTACK+ LDS+ s=0 s=1 LDTACK- D+ LDTACK+ DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  36. read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ LDTACK+ LDS+ s=0 s=1 LDTACK- D+ LDTACK+ DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  37. read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ s=0 is not implicit!! LDTACK+ LDS+ s is not consistent!! s=0 s=1 LDTACK- D+ LDTACK+ DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  38. read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ s=0is implicit s=1is implicit LDTACK+ LDS+ s=0 s=1 LDTACK- D+ LDTACK+ s is consistent DTACK+ s-;D- D- LDS- DSr- DTACK+ D- DSw-

  39. s=0 s=1 read cycle write cycle DSr+ DSw+ s+;DTACK- s-;LDS+ D+ LDTACK+ LDS+ LDTACK- D+ LDTACK+ DTACK+ s-;D- LDS- DSr- DTACK+ D- DSw-

  40. read cycle write cycle s+ DSr+ DSw+ DTACK- s- D+ LDS+ LDTACK+ LDS+ LDTACK- D+ LDTACK+ s- DTACK+ D- LDS- DSr- DTACK+ D- DSw-

  41. Main algorithm for solving CSC conflicts while CSC conflits exist do (σ1,σ2):= Find traces connecting conflict (s=0,s=1):= Find implicit places to break conflict Insert s+/s- transitions connected to (s=0) or (s=1) endwhile

  42. State space explosion problem • Goal: avoid state enumeration to check implicitness of a place. • Classical methods to avoid the explicit state space enumeration: • Linear Algebra (LP/MILP) • Graph Theory • Symbolic representation (BDDs) • Partiar order (Unfoldings) Structural methods

  43. LP model to check place implicitness A place p is implicit if the following LP model is infeasible,where P’ = P – {p}: x M0 M LP formulation: M0 + Ax = M M[P’] – F[P’,p•]·s 0 M[p] – F[p,p•]·s <0 s·1 = 1 x, M, s  0 P – {p}: p . . . M : [Silva et al.]

  44. LP model to check place implicitness A place p is implicit if M0[p] is greater than or equal to the optimal value of the following LP, where P’ = P – {p}: A place p is implicit if the following LP model is infeasible,where P’ = P – {p}: DUAL LP formulation: M0 + Ax = M M[P’] – F[P’,p•]·s 0 M[p] – F[p,p•]·s <0 s·1 = 1 x, M, s  0 LP formulation: min y· M0 y·A[P’.T] ≤ A[p,T] y· F[P’, p•] ≥ F[p, p•] y≥ 0 [Silva et al.]

  45. MILP model to insert a implicit place MILP formulation: min y· M0 y·A’[P’.T] ≤ A’[p,T] y· F’[P’, p•] ≥ F[p, p•] y≥ 0 A A’ p MILP variables: y, p

  46. MILP model to find insertion points that disambiguate the conflict MILP formulation: MILP “s=0 implicit” MILP “s=1 implicit” #(σ1,s+) = #(σ1,s-) + 1 #(σ2,s-) = #(σ2,s+) + 1 M0[s=0] + M0[s=1] = 1 LDS+ LDTACK- σ1 σ2 LDS- LDTACK+ DSr+ DTACK- D+ D- If there is a solution, rows in A’ for s=0 and s=1 describe the insertion points (arcs in the net) DSr- DTACK+

  47. petrify (state-based) MILP (structural) Number of inserted encoding signals Benchmarks from [Cortadella et al., IEEE TCAD’97]

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