Boolean Satisfiability (SAT) Algorithms

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Indirect Implication (AND gate) Select 2 pins (fanins or the gate itself) in a gate to watch • Almost the same as CNF SAT, except that the “watched value” depends on the gate type and I/O • For each gate, two lists of watching gates • When a gate gets a value, perform direct implication and/or update watch for the gates on the watching list watching-0: { a } watching-1: { } a c watching-0: { } watching-1: { } watched pins b watching-0: { } watching-1: { a, b } Boolean SAT Algorithms / FLOLAC 2009 Prof. Chung-Yang (Ric) Huang http://dvlab.ee.ntu.edu.tw 98
Watch Scheme for XOR Gate n-input XOR gate • 2 n (n+1)-literal clauses • e.g. (a + b + f) (a + b + f) (a + b + f) (a + b + f) Implication occurs only when n variables become “known” 2-watch; watch-known watching-0: { } watching-1: { } watching-known: { a } a c watching-0: { } watching-1: { } watching-known: { } Boolean SAT Algorithms / FLOLAC 2009 Prof. Chung-Yang (Ric) Huang http://dvlab.ee.ntu.edu.tw b watching-0: { } watching-1: { b } watching-known: { a } 99
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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
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
Page 79 and 80: What’s the improvement Worse case
Page 81 and 82: Sounds good for all-1’s forward i
Page 83 and 84: 2-Watched-Literal Algorithm H. Zhan
Page 85 and 86: 2-Watched-Literal Algorithm Example
Page 87 and 88: Caching Effect: Reducing from O(n)
Page 89 and 90: Difference between circuit and CNF
Page 91 and 92: Any problem Remember there is only
Page 93 and 94: A Closer Look 0 1 1 0 0 0 11 1 11 1
Page 95 and 96: Direct Implication 1. Single source
Page 97: Direct vs. Indirect Implications 0
Page 101 and 102: Applications of Logic Implication W
Page 103 and 104: Redundancy in a Combinational Circu
Page 105 and 106: Mandatory Assignment Example (1) Fa
Page 107 and 108: 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
Page 113 and 114: Is (g5 g9) redundant c 0 b 0 g1 1
Page 115 and 116: Filtering Out Impossible Candidate
Page 117 and 118: How do we add “something” to a
Page 119 and 120: A Two-Way Redundancy Addition and R
Page 121 and 122: RAR Example (w t : g6 g7) c g4 0 0
Page 123 and 124: 2-Way RAR Algorithm 1. Given a targ
Page 125 and 126: Creating a Redundant Gate (2) g s1
Page 127 and 128: A closer look at the previous examp
Page 129 and 130: Single Wire Replacement Theorem in
Page 131 and 132: 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
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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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More decision heuristics... Variabl
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Remember when we talked about confl
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Various Learning Techniques Other
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Learned by Signal Correlations A p
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Conflict vs. Success-Driven Learnin
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Success-Driven Learning Ref: Shuo,
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What can we do to make the learning
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Recalled, the SAT algorithms we hav
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SAT-Based Verification We know tha
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However, in the above approach, we
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Bounded Model Checking (BMC) Algori
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Application of BMC If the property
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K-induction Induction: SSS2000 •
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Induction SAT for (k = 0 to infinit
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Does “Induction SAT” guarantee
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Induction over simple paths Let sim
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Is the recurrence diameter the same
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Outline Overview of Hardware Verif
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Symbolic model checking without BDD
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Limitation of Formal Engine Still
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Localization abstraction Property:
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Image and over-approximated image
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Remember: Resolution a ∨ b ∨ ¬
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Generating refutations Refutation
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Extraction of Unsatisfiability Core
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Some Definitions for Unsatisfiabili
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Interpolants from Proofs Deriving
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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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