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Appearance-Based Equivalence Checking

Appearance-Based Equivalence Checking. Speaker: Daw-Ming Li Advisor: Chun-Yao Wang 2009.02.10. Introduction. Traditional approaches of equivalence checking Building BDDs SAT solving Drawback Exponential growth of required memory and runtime.

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Appearance-Based Equivalence Checking

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  1. Appearance-Based Equivalence Checking Speaker: Daw-Ming Li Advisor: Chun-Yao Wang 2009.02.10

  2. Introduction • Traditional approaches of equivalence checking • Building BDDs • SAT solving • Drawback • Exponential growth of required memory and runtime

  3. Appearance-based approach • Given two circuits N1,N2 • Transform circuits N1 and N2 according to their appearance • Extract the same appearance of N1 and N2 to construct the circuit I

  4. Terminologies • Given two circuits N1 and N2 • Circuit I is an isomorphism between N1 and N2 • Largest Common Connected Subgraph(LCCS) • Largest Common Connected PI Subgraph (LCCPIS) • Chun Chi’s work-Rewiring using IRredundancy Removal and Addition (IRRA)

  5. Ideas • Find the LCCS between N1 and N2 • Apply IRRA, transform the LCCS to the LCCPIS • Delete the LCCPIS from circuits N1 and N2, and add the LCCPIS to circuit I • Goal • Reduce the size of circuits N1 and N2 as small as possible

  6. Flow chart Transform N1 and N2 to NAND-NOT circuit N1 and N2 are equivalent Find the LCCS between circuits N1 and N2 True Are both N1 and N2 empty? True Is N1 or N2 empty? True False False Apply the IRRA technique, transform the LCCS to the LCCPIS Redundant? False Delete the LCCPIS from N1 and N2, and add the LCCPIS to I N1 and N2 are not equivalent

  7. ... ... N1_0 N2_0 .. .. .. .. N2: N1: N1_1 N2_1 N1_2 N2_2 .. .. … … N1_3 N2_3 … … Example Find the largest common connected subgraph

  8. Example ... ... N1_0 N2_0 .. .. .. .. N2: N1: N1_1 N2_1 N1_2 N2_2 .. .. … … N1_3 N2_3 N2_4 … … Apply the IRRA technique, transform the LCCS to the LCCPIS

  9. Example ... ... N1_0 N2_0 .. .. .. .. N2: N1: N1_1 N’2_1 N1_2 N2_2 .. .. … … N1_3 N2_3 … … Put the LCCPIS to the isomorphism set

  10. Example ... ... N1_0 N2_0 .. .. .. .. .. N2: N1: N2_2 N1_1 N’2_1 I: .. … … N2_3 … The PO of LCCPIS == The new PI of N1 and N2

  11. .. .. Example ... ... N1_0 N2_0 .. .. .. I_1 N2: N1: N1_1 N’2_1 I: .. I_2 … … … Find the LCCPIS between N1 and N2, and iterate the procedure until N1 or N2 is empty

  12. Case I ... I N1: N2: I: …… Both N1 and N2 are empty Circuits N1 and N2 are equivalent

  13. Case II ... ... N2: N2 N1: I I: …… …… N1 is empty, but N2 is not empty Check whether N2 is redundant or not If N2is redundant, circuits N1 and N2 are equivalent

  14. Case III ... ... N2: N2 N1: I I: …… …… N1 is empty, but N2 is not empty Check whether N2 is redundant or not If N2is not redundant, circuits N1 and N2 are not equivalent

  15. The case of applying IRRA (1/3) ... ... N2_0 N2_0 .. .. .. .. N2: N2: N’2_1 N2_2 N2_1 N2_2 .. .. … … N2_3 N2_3 N2_4 … …

  16. The case of apply IRRA(2/3) ... ... N2_0 N2_0 .. .. .. .. N2: N2: N’2_1 N2_2 N2_1 N2_2 .. .. … … N2_3 N2_3 N2_4 … …

  17. The case of apply IRRA(3/3) ... ... N2_0 N2_0 .. .. .. .. N2: N2: N’2_1 N2_2 N2_1 N2_2 .. .. … … N2_3 N2_3 N2_4 … …

  18. Reference • “A Direct Algorithm to Find a Largest Common Connected Induced Subgraph of Two Graphs”, Graph-Based Representations in Pattern Recognition, Springer Berlin Heidelberg, pp 162-171, 2005. • “Rewiring using IRredundancy Removal and Addition”, Chun-Chi

  19. Future work • Consider the situation when N1 or N2 cannot become empty set • Study more papers

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