Boolean Satisfiability (SAT) Algorithms

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Some Definitions for Unsatisfiability Proof Let (A,B) be a pair of clause sets and let Π be a proof of unsatisfiability of A ∪ B • Π is a DAG (V Π , E Π ) • Each vertex c ∈ Πin the graph corresponds to a clause and has exactly 2 predecessors, say c1, c2 • c is called the “resolvent” of c1 and c2 • The resolved variable v is called the “pivot” variable • Π has exactly 1 leaf vertex which is a False (null clause) • The roots are original clauses in A ∪ B Global/Local variable/literal • With respect to (A,B), a variable/literal is global if it appears in both A and B • It is called local to A if it appears only in A Boolean SAT Algorithms / FLOLAC 2009 Prof. Chung-Yang (Ric) Huang http://dvlab.ee.ntu.edu.tw 230
Interpolants from Proofs Deriving interpolant from Π Calling itp(leaf vertex) itp(c) { // c ∈ V Π let p(c) be a if c is a root, then if c ∈ A then itp(c) = the disjunction of the global literals in c else itp(c) = constant True else, let c1, c2 be the predecessors of c and let v be their pivot variable if v is local to A then itp(c) = itp(c1) ∨ itp(c2) else itp(c) = itp(c1) ∧ itp(c2) } Boolean SAT Algorithms / FLOLAC 2009 Prof. Chung-Yang (Ric) Huang http://dvlab.ee.ntu.edu.tw 231
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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
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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
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Davis Putnam Algorithm M .Davis, H.
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Basic DLL Procedure - DFS (a’ + b
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Potentially exponential complexity!
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Conflict-Driven Learning (a’ + b
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Non-Chronological Backtracking (a
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Deduced Implication from Learned Cl
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Deduced Implication from Learned Cl
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A Closer Look at the Implication Gr
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Conflict Analysis 2 a = 1 a = 1 a1
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Which constraint is the best to add
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Conflict-Driven Learning Decision l
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A closer look at binary decision tr
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Conflict-Driven Non-Chronological B
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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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Page 181 and 182: Remember when we talked about confl
Page 183 and 184: Various Learning Techniques Other
Page 185 and 186: Learned by Signal Correlations A p
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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
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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: Some Definitions for Unsatisfiabili
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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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