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Reachability Analysis

Reachability Analysis. Kuang-Jung Chang Advisor : Chun-Yao Wang Date: 2008.10.30. Outline. Introduction Traditional Symbolic Image Computation Image Transfer & Experiments Future Work. Reachability Analysis. Given: A sequential circuit An initial state set Objective:

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Reachability Analysis

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  1. Reachability Analysis Kuang-Jung Chang Advisor : Chun-Yao Wang Date: 2008.10.30

  2. Outline • Introduction • Traditional Symbolic Image Computation • Image Transfer & Experiments • Future Work

  3. Reachability Analysis • Given: • A sequential circuit • An initial state set • Objective: • The reachable state set from the initial state set Finite State Machine Finite State Machine Finite State Machine R0 R1 R2 R3 Fixed point

  4. Sequential Circuit • A set of primary inputs (PI): w0~wm • A set of outputs (PO): O0~Ol • A set of flip-flops: ff0~ffn • Pseudo primary input (PPI): x0~xn • Pseudo primary output (PPO): y0~yn • Output function: • A completely specified function with domain (X  W) and range O • Transition relation: • A completely specified function with domain (X  W) and range Y Combinational part of a circuit W O X Y Flip-flops

  5. Sequential Equivalence Checking • The product machine (sequential miter) of circuit A and circuit B Combinational part of circuit A W ZA W XA YA Flip-flops A Combinational part of circuit B ZB XB YB Flip-flops B

  6. Why Reachability Analysis • State minimization • Logic optimization • Sequential ATPG • Property checking State space Unused state Don’t care Undetected error Fake bug

  7. Difficulties of Reachability Analysis • Huge state space • The large number of flip-flops in a sequential circuit • Complicated Finite State Machine • 2|flip-flop| 2|PI| → 2|flip-flop|

  8. Outline • Introduction • Traditional Symbolic Image Computation • Image Transfer & Experiments • Future Work

  9. Symbolic Image Computation • The characteristic function of PPO yi and latch transition relation TRi • The transition relation TR is the conjunction of the latch transition relations • Fi(X, W) → {0, 1} • TRi (yi, X, W) • = yi Xnor Fi(X, W) • TR(Y, X, W) • =  TRi(yi, X, W) yi Fi’ Fi TR( Y, X, W) Fi(X, W) 0 0 1 1 0 0 1 1

  10. Symbolic Image Computation • Representation of state set S • Flip-flop BDD variables: l0~ln • The characteristic representation • S(L) = 1 iff L  state set S l0 l1 l1 l2 l2 l2 l2 1 0 1 0 0 1 1 0

  11. Symbolic Image Computation • Basic symbolic image computation • Nex(Y) =  X, WTR(Y, X, W)  Pre(X) Quantification  X, W BDD AND TR( Y, X, W) Pre(X) Nex(Y) 0 1 0 1 0 1

  12. Image Computation without TR • Nex(Y) =  X, W TR(Y, X, W)  Pre(X) • Latch transition relation (TRi) • Nex(Y) =  X, W TRi(yi, X, W)  Pre(X) • Clustering and Scheduling (Early quantification) Scheduling Clustering

  13. Simplification of Image Computation • Partitioned transition relation TR(y0=0, y1=0) TR(y0=0) TR(y0=0, y1=1) y0 y1 TR(y0=1, y1=0) TR(y0=1) TR( Y, X, W) TR(y0=1, y1=1) • Splitting previous state set Pre(X) 0 1 0 1

  14. Outline • Introduction • Traditional Symbolic Image Computation • Image Transfer & Experiments • Future Work

  15. Perspective of Network TRi TR Combinational part of a circuit Combinational part of a circuit PI PI …… …… …… …… PPO PPO PPI PPI Clustered TRi Image Transfer Combinational part of a circuit Combinational part of a circuit PI PI …… …… …… …… PPO PPO PPI PPI

  16. Image Transfer X Y 0

  17. ENISLE: An Intuitive Heuristic Nearly Optimal Solution for Mincut and Mincut Partition

  18. Levelize Network to Decide the Initial Image Cut Levelize and decide #cut Find image cuts by level Combinational part of a circuit PI …… …… …… PPO PPO PPI

  19. Adjust Image Cut & Overview Overview of image transition Initial image cut PPO Adjacent gates

  20. Image cut • s1423: 74 latches, 17 PIs • Maximal level : 56 • PPI image cut : 91 • Mid image cut (level 28): 85 • Overlap : 38 (46) • Matrix : 53 x 46 • PPO image cut : 74 • Overlap : 38 (36) • Matrix : 46 x 36 38 74 38 36 91 46

  21. Reordering • Only one reordering is allowed in each image transfer • Reordering would not be triggered while operation is processing • Reordering starts when BDD size exceeds its previous maximal value • s1423, 10 timeframe, 1.68288e+09 reached state ( 772 => 309 sec)

  22. Matrix - Initial (1/3)

  23. Matrix (2/3)

  24. Sort initial order of rows and columns • Sort fore image by number of back nodes which are supported by the fore node • Sort back image by size of the back node relation Support 0 0 1 0 Relation size 1 1

  25. Matrix – sorted initial order(3/3)

  26. Clustering Based on Affinity • Merge the pair with the highest affinity • Cluster threshold size : 1000 • Affinity > 0 • s1423 • Before: 46 relations, 36 relations • After: 8 relations, 2 relations

  27. Clustering Based on Affinity (con’t)

  28. Experimental Data • s1423, 10 timeframe, 1.68288e+09 reached state

  29. Future Work • General heuristic for dynamic reordering and clustering • Experiments on image cut issues • Number of image cuts • Where should image cuts be?

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