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Example: Verification

A. B. Example: Verification. e.g., input/output specification of multiplier. e.g., multi-level logic representation. 1. 0. 0. 1. Binary Decision Diagrams. Graph-based Representation of Boolean Functions. Introduced by Lee ( 1959 ). Popularized by Bryant ( 1986 ).

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Example: Verification

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  1. A B Example: Verification e.g., input/output specification of multiplier e.g., multi-level logic representation

  2. 1 0 0 1 Binary Decision Diagrams Graph-based Representation of Boolean Functions • Introduced by Lee (1959). • Popularized by Bryant (1986). • compact (functions of 50 variables) • efficient (linear time manipluation) Widely used; has had a significant impact on the CAD industry.

  3. x x x f 1 2 3 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 Binary Decision Diagrams Graph-based Representation of Boolean Functions 1 0 BDD is defined asDirectedAcyclicGraph 0 0 1 1

  4. output Digital Circuit Analysis of Digital Circuits Large domain, small range. inputs

  5. output Digital Circuit Analysis of Digital Circuits Large domain, small range. inputs 2mpossibilities 2possibilities

  6. x1 x2 x3 f mvariables 2mrows 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 Data Structures Truth Tables Example 2 variables 4 rows 3variables 8 rows 64variables 264rows

  7. S x1 0 1 x2 x2 0 1 0 1 x3 x3 x3 x3 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 Data Structures Decision Diagrams Example x1 x2 x3 f 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1

  8. S x1 0 1 x2 x2 0 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 0 1 Data Structures Decision Diagrams Optimize by merging nodes: x3 x3 x3 x3

  9. x2 0 0 1 0 x3 1 0 0 1 Data Structures Decision Diagrams S Optimize by merging nodes: x1 0 1 x2 1 x3 x3 x3 0 1 0 1 1

  10. S x1 0 1 0 1 1 0 1 1 0 0 x3 x3 x3 x3 0 0 1 1 0 1 1 0 1 Data Structures Decision Diagrams Optimize by merging nodes: x2 x2

  11. 1 1 x2 0 1 0 1 Data Structures Decision Diagrams S Optimize by merging nodes: x1 0 x2 1 0 x3 0 1

  12. S T U x1 x1 x1 0 1 0 1 0 1 x2 x2 x2 1 0 1 1 0 0 x3 x3 x3 0 0 0 1 1 1 0 1 0 1 1 0 Data Structures Logic Operations = AND

  13. Decision Diagrams Properties: • Canonical: uniqueup to variable ordering • Compact: represent functions of up to1000 variables • Efficient:performlogic operations inlinear-time

  14. Variable Ordering

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