Boolean Satisfiability (SAT) Algorithms

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Static Learning Learn by contrapositive (a b Ξ !b !a) e.g. b a = 1 b = 1 Learned b = 0 a = 0 The question is: which gate to learn a Ref: “SOCRATES: A Highly Efficient Automatic Test Pattern Generation System”, Schulz et.al, TCAD 1988 Boolean SAT Algorithms / FLOLAC 2009 Prof. Chung-Yang (Ric) Huang http://dvlab.ee.ntu.edu.tw 184
Learned by Signal Correlations A proof-based approach • Since learned information is universally true, we can create some internal interesting properties, and use these properties to derive some interesting learning (by conflict analysis) e.g. By simulation, if we find a gate ‘g’ is very likely to stuck at some value ‘v’ Witness “g = ¬v” (should produce many conflicts) e.g. By simulation, if two signals respond almost the same Witness “p != q” No matter the proof is finished or not • We can always learn something Ref: Feng Lu, et. al, “A Circuit SAT Solver with Signal Correlation Guided Learning”, DATE 2003 Boolean SAT Algorithms / FLOLAC 2009 Prof. Chung-Yang (Ric) Huang http://dvlab.ee.ntu.edu.tw 185
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Boolean Satisfiability (SAT) Algori
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What is Verification What’s the p
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#Transistors in Intel’s CPUs Year
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What is Verification What’s the p
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And of course, don’t forget the i
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Improvement on Floating-Point Execu
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A 3-D plot of the ratio 4195835/314
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Aftermath... Intel hired 1000+ PhD
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How do you verify your design Bool
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Simulation-Based Verification Testb
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Like Finding a Bug in an Ocean… B
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Then why is simulation still the ma
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And of course, simulation approach
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Using hardware to speed up simulati
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Verification vs. Testing Objective
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Observability Problem --- Solved In
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The Fact More than 90% of properti
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ABV Example --- OVL module assert_n
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Any Problem Whose Problem “My big
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Think --- If you simulate your desi
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Simulation and Assertion-Based Veri
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Formal Verification Technologies De
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Proving “assert_always(p)” In
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Set of Reachable States t0 t1 t2 t3
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Combinational Invariance It is wor
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In this topic, we will use SAT engi
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Boolean Satisfiability (SAT) Fundam
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Complexity of SAT solver We have le
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Something like --- A Decision Tree
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Boolean Satisfiability Checking for
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A Typical Combinational SAT Algorit
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A Typical Combinational SAT Algorit
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a b c d combSat (g7, 0) 1 g1 g2 g3
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a b c d combSat (d, 1) 1 1 g1 g2 g3
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a b c d combSat (d, 1) 1 1 g1 g2 g3
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How many decisions did we make Ther
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Logic Implications Also called “
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How to schedule and evaluate these
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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
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Direct vs. Indirect Implications 0
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Watch Scheme for XOR Gate n-input
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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
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How to add an extra wire to make th
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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
Page 133 and 134: SRAR-Wire vs. Original RAMBO (FYI)
Page 135 and 136: Single Gate Replacement in SatRAR *
Page 137 and 138: Alternative Sub-circuit by SatRAR M
Page 139 and 140: 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
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
Page 169 and 170: 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
Page 175 and 176: Static Decision Ordering Decision
Page 177 and 178: zChaff’s Variable State Independe
Page 179 and 180: More decision heuristics... Variabl
Page 181 and 182: Remember when we talked about confl
Page 183: Various Learning Techniques Other
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
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Overapproximation An overapproxima
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Adequate image Img(P,C) P Img’(P,
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Huh A A' B P C C C C C C C F A ⇒
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Reachability algorithm let k = 0 re
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Interpolation-based MC Fully SAT-ba
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Conclusion on Boolean SAT Algorithm
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Some popular SAT engines SATO •
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Boolean Satisfiability (SAT) Algorithms

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