20130108 ast signature based boolean matching in the presence of don’t cares_mori
•Download as PPTX, PDF•
0 likes•229 views
The document outlines an approach for signature based Boolean matching in the presence of don't cares. It discusses previous work on computing canonical forms and signatures. Key aspects of the presented approach include determining the phase of functions, creating an input matching graph, applying graph reduction techniques, and using a branch and bound algorithm. Experimental results on MCNC benchmark circuits show the approach has significantly better runtimes than previous methods for Boolean matching with don't cares.
More Related Content
What's hot
What's hot (20)
Viewers also liked
Viewers also liked (12)
Similar to 20130108 ast signature based boolean matching in the presence of don’t cares_mori
Similar to 20130108 ast signature based boolean matching in the presence of don’t cares_mori (20)
Recently uploaded
Recently uploaded (20)
20130108 ast signature based boolean matching in the presence of don’t cares_mori
- 1. Signature Based Boolean Matching in the Presence of Don’t Cares Afshin Abdollahi Department of Electrical Engineering University of California Riverside, CA
- 2. Outline Introduction Previous work Preliminaries Workflow – – – – Determining phase of function Creating input match graph Graph reduction Branch and bound algorithm Experimental results Conclusions
- 3. Introduction Boolean Matching – is to determine whether two functions are equivalent under input permutation and input/output phase assignment NPN-equivalent – Boolean functions that are equivalent under input permutation and input/output phase assignment
- 4. Boolean Matching f (X ) x1 x2 x3 g (Y ) T ( x1 , x2 , x3 ) ( y2 , y3 , y1 ) f ' (Y ) f (T ( X )) g ( X ) y1 y2 y3
- 5. Previous Work Compute canonical forms of the Boolean functions Signatures – completely specified functions (CSF) – incompletely specified functions (ISF) Algebraic framework Multiple-value functions Matrix
- 7. Compatibility x 1 x 2 x3 f y1 y2 y3 g 0 0 0 0 1 1 1 1 0 0 0 d d d 1 1 0 0 0 0 1 1 1 1 0 0 d d d 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 f off f dc f on f f off g on = 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 g f on g off = g off g dc g on
- 8. Transformation A Transformation, T, is an onto-mapping from X = (x1, x2, …, xn) to Y = (y1, y2, …, yn) by a phase assignment T (xi) = yj or yj Example : – For vectors X = (x1, x2, x3) and Y =(y1, y2, y3) – an example of a transformation is T (x1, x2, x3) = (y2, y3, y1 )
- 10. Equivalency x1 x2 x3 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 y2 y3 y1 0 d 0 d 1 d 1 1 f g g 0 0 0 0 1 1 1 1 f 0 d 0 1 d d d 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 f off g on = y2 y3 y1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 1 1 1 1 0 0 0 0 f on g y1 y2 y3 g d 1 d 1 0 0 d d 0 0 0 0 1 1 1 1 d 1 d 1 0 0 d d g off = 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1
- 11. Signature
- 12. Satisfy Count Range x 1 x 2 x3 2 |f | 5 [f ] = [2, 5] = {2, 3, 4, 5} f 0 0 0 0 1 1 1 1 0 0 0 d d d 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 f off f dc f on
- 13. Satisfy Count Range x1 x2 x3 f 0 0 0 0 1 1 1 1 0 0 0 d d d 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 f off f dc f on
- 14. Satisfy Count Range x 1 x 2 x3 f y1 y2 y3 g 0 0 0 0 1 1 1 1 1 d 1 d d 0 0 0 0 0 0 0 1 1 1 1 0 0 d d d 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 [f ] = [2, 5] [f ] = [3, 6] 0 0 1 1 0 0 1 1 [g] = [f ] [g ] [f ] [g ] 0 1 0 1 0 1 0 1 [3, 6]
- 15. Workflow Determining phase of function f Creating input match graph Graph reduction Branch and bound algorithm
- 16. Determining Phase of Function f Check two conditions : 1. [f] [g]≠ 2. [f] [g]≠
- 17. Workflow Determining phase of function f Creating input match graph Graph reduction Branch and bound algorithm
- 18. Input Matching Graph f x1 x2 ... xi ... ... xn yj ... yn + g y1 [ f xi ] [ g y j ] [ f xi ] [ g y j ] y2 ... ... +
- 19. Input Matching Graph f on x1 x2 f off x1 x3 x4 f x1 x2 + + g y1 g on y1 y2 x1 ( x2 x3 + y2 x4 + + y3 g off x3 ) x2 ( x3 x4 ) + y4 y2 y1 ( y3 y4 )
- 20. Input Matching Graph f on x1 x2 f f off x1 x3 x4 x1 x2 + g g on y1 y1 y2 x1 ( x2 x3 x4 + + y2 y3 g off x3 ) x2 ( x3 x4 ) y4 y2 y1 ( y3 y4 )
- 21. Hall’s Matching Theorem In a bi-partite graph with parts X and Y a match exists if and only if any number of vertices in X (say m vetices) are connected to more than or equal to m vertices in Y. m=1 m=2 First step: Check Hall’s Theorem for m=1 and 2
- 22. Workflow Determining phase of function f Creating input match graph Graph reduction Branch and bound algorithm
- 23. Graph Reduction Identify the vertices from (X and Y ) that are connected to only one other vertex f x1 x2 x3 x4 g y1 y2 y3 y4
- 24. Graph Reduction Identify the vertices from (X and Y ) that are connected to only one other vertex f x1 x2 x3 x4 g y1 y2 y3 y4
- 27. Graph Reduction f x2 x1 x3 x4 y2 y3 y4 + g y1 [ f x1x2 ] [ g y1 y2 ]
- 28. Workflow Determining phase of function f Creating input match graph Graph reduction Branch and bound algorithm
- 29. Branch and Bound f on x1 x2 f off x1 x3 x4 f x1 x2 + + g g on y1 y1 y2 x1 ( x2 x3 + y2 x4 + + y3 g off x3 ) x2 ( x3 x4 ) + y4 y2 y1 ( y3 y4 )
- 30. Branch and Bound f x1 x2 + + g y1 y2 x3 + + y3 x4 + + y4
- 31. Branch and Bound f x1 x2 + g y1 y2 x3 + + y3 [ f x1x2 ] [ g y1 y2 ] x4 + + y4
- 32. Branch and Bound f x1 x2 + g y1 y2 x3 + + y3 [ f x1x3 ] [ g y1 y3 ] [ f x1x3 ] [ g y1 y3 ] x4 + + y4
- 33. Branch and Bound f x1 x2 + g y1 y2 x3 + + y3 [ f x1x3 ] [ g y1 y4 ] [ f x1x3 ] [ g y1 y4 ] x4 + + y4
- 34. Branch and Bound f x1 x2 + + g y1 y2 x3 + + y3 x4 + + y4
- 36. Branch and Bound f x1 x2 + g y1 x3 + y2 y3 x4 + y4 T ( x1 , x2 , x3 , x4 ) ( y2 , y1 , y3 , y4 ) f (T ( X )) g (Y )
- 37. Experimental Results MCNC benchmark a machine with an AMD Opteron processor running at 1.8 Ghz with 2GB RAM They generated a number of circuits of up to 10 inputs by extracting clusters (cones) from MCNC benchmarks
- 38. Results Runtimes (ms) Number of Inputs Our Approach Ref [11] Ref [10] Ref [9] 4 4.2 10-5 1.5 10-4 0.13 0.4 5 7.3 10-5 6 1.8 10-4 6 10-4 0.56 117 7 3.7 10-4 8 8.2 10-4 3.2 10-3 8 >1 hour 9 20.1 10-4 10 44.8 10-4
- 39. Conclusions They presented a signature and graph based approach that proved to be very effective in pruning the search space and resulted in reasonable runtimes