Boolean Satisfiability (SAT) Algorithms

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Let’s do some math Suppose a circuit has 100 inputs (this is considered tiny in modern design) • Total number of input combinations = 2 100 = 10 30 = 10 24 M = 1.6 mole Mega • Requires (in the worse case) 10 24 M test patterns to exhaust all the input scenarios • Let alone the sequential combinations • For an 1 MIPs simulator runtime = 10 24 seconds = 3 * 10 16 years Boolean SAT Algorithms / FLOLAC 2009 Prof. Chung-Yang (Ric) Huang http://dvlab.ee.ntu.edu.tw 20
Like Finding a Bug in an Ocean… Boolean SAT Algorithms / FLOLAC 2009 Prof. Chung-Yang (Ric) Huang http://dvlab.ee.ntu.edu.tw 21
Page 1 and 2: Boolean Satisfiability (SAT) Algori
Page 3 and 4: What is Verification What’s the p
Page 5 and 6: #Transistors in Intel’s CPUs Year
Page 7 and 8: What is Verification What’s the p
Page 9 and 10: And of course, don’t forget the i
Page 11 and 12: Improvement on Floating-Point Execu
Page 13 and 14: A 3-D plot of the ratio 4195835/314
Page 15 and 16: Aftermath... Intel hired 1000+ PhD
Page 17 and 18: How do you verify your design Bool
Page 19: Simulation-Based Verification Testb
Page 23 and 24: Then why is simulation still the ma
Page 25 and 26: And of course, simulation approach
Page 27 and 28: Using hardware to speed up simulati
Page 29 and 30: Verification vs. Testing Objective
Page 31 and 32: Observability Problem --- Solved In
Page 33 and 34: The Fact More than 90% of properti
Page 35 and 36: ABV Example --- OVL module assert_n
Page 37 and 38: Any Problem Whose Problem “My big
Page 39 and 40: Think --- If you simulate your desi
Page 41 and 42: Simulation and Assertion-Based Veri
Page 43 and 44: Formal Verification Technologies De
Page 45 and 46: Proving “assert_always(p)” In
Page 47 and 48: Set of Reachable States t0 t1 t2 t3
Page 49 and 50: Combinational Invariance It is wor
Page 51 and 52: In this topic, we will use SAT engi
Page 53 and 54: Boolean Satisfiability (SAT) Fundam
Page 55 and 56: Complexity of SAT solver We have le
Page 57 and 58: Something like --- A Decision Tree
Page 59 and 60: Boolean Satisfiability Checking for
Page 61 and 62: A Typical Combinational SAT Algorit
Page 63 and 64: A Typical Combinational SAT Algorit
Page 65 and 66: a b c d combSat (g7, 0) 1 g1 g2 g3
Page 67 and 68: a b c d combSat (d, 1) 1 1 g1 g2 g3
Page 69 and 70: a b c d combSat (d, 1) 1 1 g1 g2 g3
Page 71 and 72:
How many decisions did we make Ther
Page 73 and 74:
Logic Implications Also called “
Page 75 and 76:
How to schedule and evaluate these
Page 77 and 78:
Trying to avoid fruitless implicati
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What’s the improvement Worse case
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Sounds good for all-1’s forward i
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2-Watched-Literal Algorithm H. Zhan
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2-Watched-Literal Algorithm Example
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Caching Effect: Reducing from O(n)
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Difference between circuit and CNF
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Any problem Remember there is only
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A Closer Look 0 1 1 0 0 0 11 1 11 1
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Direct Implication 1. Single source
Page 97 and 98:
Direct vs. Indirect Implications 0
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Watch Scheme for XOR Gate n-input
Page 101 and 102:
Applications of Logic Implication W
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Redundancy in a Combinational Circu
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Mandatory Assignment Example (1) Fa
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1. How do we know a wire in a combi
Page 109 and 110:
How to add an extra wire to make th
Page 111 and 112:
RAMBO: Redundancy Addition and remo
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Is (g5 g9) redundant c 0 b 0 g1 1
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Filtering Out Impossible Candidate
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How do we add “something” to a
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A Two-Way Redundancy Addition and R
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RAR Example (w t : g6 g7) c g4 0 0
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2-Way RAR Algorithm 1. Given a targ
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Creating a Redundant Gate (2) g s1
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A closer look at the previous examp
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Single Wire Replacement Theorem in
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A closer look... MA(w t ) a MA(g d
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SRAR-Wire vs. Original RAMBO (FYI)
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Single Gate Replacement in SatRAR *
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Alternative Sub-circuit by SatRAR M
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CNF-Based SAT Algorithm 1. Davis, P
Page 141 and 142:
Davis Putnam Algorithm M .Davis, H.
Page 143 and 144:
Basic DLL Procedure - DFS (a’ + b
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Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Potentially exponential complexity!
Page 153 and 154:
Conflict-Driven Learning (a’ + b
Page 155 and 156:
Non-Chronological Backtracking (a
Page 157 and 158:
Deduced Implication from Learned Cl
Page 159 and 160:
Deduced Implication from Learned Cl
Page 161 and 162:
A Closer Look at the Implication Gr
Page 163 and 164:
Conflict Analysis 2 a = 1 a = 1 a1
Page 165 and 166:
Which constraint is the best to add
Page 167 and 168:
Conflict-Driven Learning Decision l
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A closer look at binary decision tr
Page 171 and 172:
Conflict-Driven Non-Chronological B
Page 173 and 174:
What we have learned on SAT... 1. E
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Static Decision Ordering Decision
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zChaff’s Variable State Independe
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More decision heuristics... Variabl
Page 181 and 182:
Remember when we talked about confl
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Various Learning Techniques Other
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Learned by Signal Correlations A p
Page 187 and 188:
Conflict vs. Success-Driven Learnin
Page 189 and 190:
Success-Driven Learning Ref: Shuo,
Page 191 and 192:
What can we do to make the learning
Page 193 and 194:
Recalled, the SAT algorithms we hav
Page 195 and 196:
SAT-Based Verification We know tha
Page 197 and 198:
However, in the above approach, we
Page 199 and 200:
Bounded Model Checking (BMC) Algori
Page 201 and 202:
Application of BMC If the property
Page 203 and 204:
K-induction Induction: SSS2000 •
Page 205 and 206:
Induction SAT for (k = 0 to infinit
Page 207 and 208:
Does “Induction SAT” guarantee
Page 209 and 210:
Induction over simple paths Let sim
Page 211 and 212:
Is the recurrence diameter the same
Page 213 and 214:
Outline Overview of Hardware Verif
Page 215 and 216:
Symbolic model checking without BDD
Page 217 and 218:
Limitation of Formal Engine Still
Page 219 and 220:
Localization abstraction Property:
Page 221 and 222:
Image and over-approximated image
Page 223 and 224:
Remember: Resolution a ∨ b ∨ ¬
Page 225 and 226:
Generating refutations Refutation
Page 227 and 228:
Extraction of Unsatisfiability Core
Page 229 and 230:
Some Definitions for Unsatisfiabili
Page 231 and 232:
Interpolants from Proofs Deriving
Page 233 and 234:
SAT to compute set of reachable sta
Page 235 and 236:
Overapproximation An overapproxima
Page 237 and 238:
Adequate image Img(P,C) P Img’(P,
Page 239 and 240:
Huh A A' B P C C C C C C C F A ⇒
Page 241 and 242:
Reachability algorithm let k = 0 re
Page 243 and 244:
Interpolation-based MC Fully SAT-ba
Page 245 and 246:
Conclusion on Boolean SAT Algorithm
Page 247 and 248:
Some popular SAT engines SATO •
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Boolean Satisfiability (SAT) Algorithms

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