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Simulation and SAT-Based Boolean Matching for Large Boolean Networks

Simulation and SAT-Based Boolean Matching for Large Boolean Networks . Kuo-Hua Wang, Chung-Ming Chang, and Jung-Chang Liu Dept. of Computer Science and Information Engineering Fu Jen Catholic University Hsinchuang City, Taipei County 24205, Taiwan, R.O.C. khwang@csie.fju.edu.tw.

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Simulation and SAT-Based Boolean Matching for Large Boolean Networks

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  1. Simulation and SAT-Based Boolean Matching for Large Boolean Networks Kuo-Hua Wang, Chung-Ming Chang,and Jung-Chang Liu Dept. of Computer Science and Information Engineering Fu Jen Catholic University Hsinchuang City, Taipei County 24205, Taiwan, R.O.C. khwang@csie.fju.edu.tw

  2. Outline • Introduction • Background • S&S-Based Functional Properties Detection • Simulation-Based Boolean Matching • Experimental Results • Conclusions

  3. Introduction • Boolean Matching • to check the equivalence of two functions and under the input permutation and input/output phase assignment. • Example:

  4. X if unsatisfiable Background • Boolean Satisfiability (SAT) Problem • Given a CNF (Conjunctive Normal Form), find a variable assignment to satisfy it or prove it is equal to the constant 0. v=1 Circuit CNF mitter

  5. : And : Complement e a d c b Background – Cont. • And-Inverter Graphs (AIGs) • Nodes • Primary Input • 2-input AND • Constant 0 (1) • Edges • with complement attribute or not • Major Advantages • It is fast to simulate on AIGs. • It allows easy conversion from any circuit into AIGs. • It is more compact than BDD’s for many large functions.

  6. Phase 1: Initialization S&S-Based Functional Properties Detection - use signatures to generate an initial input mapping Phase- 2: Simulation Simulation-Based Boolean Matching - use random simulation results to distinguish inputs Phase 3: Recursion Recursive Matching Algorithm - match the remaining inputs for hard cases SAT Verification and Return Result Overview of Matching Algorithm

  7. Group Symmetry: Rotational Symmetry: Group Symmetry and Rotational Symmetry

  8. S&S Based Functional Properties Detection • Three Types of Functional Properties • 2-Variable • E (Equivalence) symmetry & NE (Non-equivalence) Symmetry • Single Variable Symmetry • 1-Variable • Functional Unateness • Detection Algorithm • Using random simulation to remove impossible properties • 2 input vectors with Hamming distance 2 (or 1) • different output values will remove some functional properties • Using SAT technique to check some properties, i.e, the equivalence of two cofactors.

  9. Notation Condition Notation Condition xi A=D 1 A=C 0 B=C xj xj B=D 0 1 0 1 A=B C=D A B C D Functional Properties • Functional Symmetries E: Equivalence Symmetry NE: Non-Equivalence Symmetry • Functional Unateness • positive unate : , i.e., • negative unate : , i.e., • binate • The inputs with different functional properties can not match to each other.

  10. Examples:Hamming_Distance(v1, v2) = 21.2. Hamming_Distance(v1, v2) = 13.4. Removal of Functional Properties

  11. Definitions and Notations • Two functions and with5 inputs.Let • Mapping Group • is unique since • Mapping Relation • It is unique if all mapping groups are unique. • Input /Output Weight • The number of 1’s in the input/output vector. • The weight of the vector 11000 is 2.

  12. Simulation-Based Boolean Matching • Main Concept • to incrementally refine the mapping relation through many simulation rounds • until no improvement is made under a threshold ( the maximum number of simulation rounds) • Each Simulation Round • Generate input vectorsto simulate on AIGs. • Check their output valuesto distinguish the inputs in non-unique mapping groups. • Three Types of Input Vectors • Type-1, Type-2, and • Type-3 (only for Boolean functions with NE-symmetries)

  13. Incompatible Match • Two functions can not be matched if • the output weights of two functions are different w.r.t. the same input vector set for simulation. • Example: f (a, b, c) = a + b · c g (x, y, z) = x · y · z Input vectors: 100, 010, 001output vector of f : 1 0 0 (weight = 1) output vector of g : 0 0 0 (weight = 0) ∵ 1≠0  f and g can not be matched

  14. Type-1 Input Vectors • x1,.............,xnf(X) 1 0 0…..0 0 0 1 the signature of x10 1 0…..0 0 0 0 the signature of x2 . . . . . .0 0 0…..0 0 11 the signature of xn Input weight = 1 • The output value of f w.r.t. each input vector (row) can be used as the signature of the corresponding input variables. • It can divide the input set into 2 groups with signature 0 and 1, respectively.

  15. Type-1 - a Simple Example • f (a, b, c, d ) f a1 0 0 0 1b 0 1 0 0 1c 0 0 1 0 0d 0 0 0 11 • g (w, x, y, z ) g w1 0 0 0 1x 0 1 0 0 0y 0 0 1 0 1z 0 0 0 11 f : output=1: {a, b, d }output=0: {c} g: output=1:{w, y, z }output=0: {x} {a, b, d } map to {w, y, z } and {c} map to {x}  R = {{a, b, d}{w, y, z}, {c}{x}}

  16. Type-2 Input Vectors • For each input , consider the following input vectors with input weight 2 and . 10…10…0 001…10…0 1 . . . . 00…10.. 11 • The inputs with different output weights can not match to each other. • If we can find a unique mapping group, we can further partition the remaining input variables. Output Weight : the # of 1’s output values The signature of xi

  17. Type-2: a Simple Example • Step 1:use the output weights to get some mapping groups • R = {{a, c}{w, x}, {b}{y}}(Unique) • f (a,b,c) g (w,x,y) a b c fw x y g 11 0 0 11 0 0 1 0 1010 10 ab cw x y 11 0 111 0 0 0 1 1 0 0 110 a b cwx y 0 110 0 11 0 1 0 1010 11 w a 0 0 1 0 1 0 Step 2:use the results by groups b and y to get more input mappings  R = {{a}{w},{c}{x},{b}{y}} b x y c Output Weight Output Weight

  18. : with input weight w1 w1 ≠w2 O1 S1 : with input weight w2 S2 O2 … … Sm Om Type-3 Input Vectors • Consider f which has mNE-symmetric sets with k input variables. • Using the vectors : s1 s2 … smf w1 w1 w1 S1 S1 S2 S2 Oi: the signature of set Si k = 34 Patterns : {000, 001, 011, 111 } weight: { 0, 1, 2, 3 } Sm

  19. Non-unique mapping group {a, b, c}{x, y, z} . Recursive Matching – An Example

  20. Matching and Comparison Results Table 1: Matching Results on 112 Circuits 112 109 Table 2: Comparison Results on Three Phases

  21. Circuit Orig Unate +Symm First All First All First All SAT s4863 * * 2.56 * 1.87 1.88 0.01 s5378 1.31 * 3.38 * 2.42 2.42 0.00 s6669 6.30 * 2.83 * 4.08 50.54 46.50 s38584.1 76.31 * 210.13 * 457.82 457.82 0.03 s38417 91.81 * 324.57 * 998.53 998.53 0.13 0.37 0.03 1.00 1.03 0.15 Ratio Matching Results for Large Networks Table 3: Benchmarking Results for Circuits in ISCAS89 $: memory explosion Table 3: Cont. *: CPUtime > 5000 sec.

  22. Conclusions • Our Boolean Matching Algorithm • Integrate Simulation and SAT techniques • Exploit Functional Properties • Propose Three Types of Input Vectors • Apply a Recursive Matching Algorithm • Handle Large Boolean Networks • Future Work • Input/Output Phase Assignment • More Effective Random Simulation Vectors

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