Specialized Applications of Decision Diagrams Alan Mishchenko Electrical and Computer Engineering

1. Specialized Applications of Decision Diagrams Alan Mishchenko Electrical and Computer Engineering Portland State University October 22, 2001

2. Overview • Applications of Decision Diagrams (DDs) • Typical, not so typical, and specialized • Fast checks • Generic DD traversal procedure • Case study: Checking for redundant variables • Other specialized operators • Decomposability checks • Encoding • Two-level SOP minimization • Computing of Walsh and Haar spectra • Conclusions Electronic Systems Design Seminar, UC Berkeley

3. Applications of DD operators • Standard • Apply (Boolean AND,OR,XOR), If-Then-Else (ITE), quantifications, cofactoring • Non-standard • ISOP, compatible projection, disjoint cover • Specialized • Useful for one application only • Performance improvements are possible by tailoring DD operators to the application • Significant improvements are possible if processing can be reduced to checking conditions on the DD structure, without building new DD nodes ( example: Cudd_bddIteConstant() ) Electronic Systems Design Seminar, UC Berkeley

4. Generic DD Traversal Procedure dd Traversal( dd A, dd B ) { (1) terminal cases (2) cache lookup (3) cofactoring w.r.t. the top variable in A and B (4) recursively solving subproblems (5) deriving the solution from the partial solutions (6) cache insert (7) returning the result } Electronic Systems Design Seminar, UC Berkeley

5. Traversal Procedure: Boolean AND bdd AND( bdd A, bdd B ) { (1) if ( A == 0 ) return 0; if ( B == 0 ) return 0; if ( A == 1 ) return B; if ( B == 1 ) return A; if ( A == B ) return A; if ( A == B’ ) return 0; (2) cache lookup (3) (A0,A1)=Cofactors(A,x); (B0,B1)=Cofactors(F,x); (4) R0 = AND( A0, A1 ); R1 = AND( B0, B1 ); (5) R = ITE( x, R1, R0 ); (6) cache insert (7) return R; } Electronic Systems Design Seminar, UC Berkeley

6. Checking Variable Redundancy c c ab ab Theorem. Variablec is redundant if and only if and Electronic Systems Design Seminar, UC Berkeley

7. Ellimination of Redundant Variables c c c ab ab ab Electronic Systems Design Seminar, UC Berkeley

8. Redundancy Checking Procedure A procedure to perform the redundancy check for a set of variables without building new BDD nodes boolCheckRedundant(bdd F, bdd G, bdd Vars ); • Arguments • F is the on-set; G is the off-set; Vars is the variable set • Return value • TRUE, if variables in Vars are redundant in the incompletely specification function and can be removed from the support • FALSE, otherwise Electronic Systems Design Seminar, UC Berkeley

9. Redundancy Checking Procedure • Terminal cases • If one of the arguments is 0, Vars are redundant • If one of the arguments is 1, Vars are not redundant • Recursive step • If the topmost variable does not belong to Vars, call the procedure for both cofactors • If the topmost variable belongs to Vars, check that the redundancy conditions are true Electronic Systems Design Seminar, UC Berkeley

10. Redundancy Checking Pseudocode boolCheckRedundant(bdd F, bdd G, bdd Vars ) { if ( F == 0 || G == 0 ) return TRUE; if ( F == 1 || G == 1 ) return FALSE; x = TopVar( F, G, Vars ); (F0, F1) = Cofactors( F, x ); (G0, G1) = Cofactors( G, x ); if ( x  Vars ) { /* the topmost variable belongs to Vars */ Res = CheckRedundant( F0, G1, Vars – x ); if ( Res == TRUE ) Res = CheckRedundant( F1, G0, Vars – x ); } else { /* the topmost variable does not belong to Vars */ Res = CheckRedundant( F0, G0, Vars ); if ( Res == TRUE ) Res = CheckRedundant( F1, G1, Vars ); } return Res; } Electronic Systems Design Seminar, UC Berkeley

11. Resubstitution • Algebraic resubstitution exists if the result of division of F by G is not an empty cover • Boolean resubstitution exists if adding the output of G to the support of F leads to the simplification of F F F  G G Electronic Systems Design Seminar, UC Berkeley

12. Boolean Resubstitution • Force variable g = G(x) into the support of F • Minimize the support of FR(x,g) • Accept the transformation, if the resulting function is simpler than F Electronic Systems Design Seminar, UC Berkeley

13. Impact on Boolean Resubstitution A - algebraic resubstitution A+B - algebraic resubstitution followed by boolean resubstitution BF - brute force approach S - smart approach Electronic Systems Design Seminar, UC Berkeley

14. Disjoint-Support Decomposition (DSD) • DSD is the decomposition into logic blocks with disjoint support • Theorem. For a completely specified function, the structure of DSD of the finest granularity is canonical (unique up to the complementation of inputs and outputs of the blocks) F d e G H a b c f g Electronic Systems Design Seminar, UC Berkeley

15. Checking Existence of DSD • The algorithm [Bertacco, Damiani, ICCAD’97] computes the DSD structure from the shared BDD of a set of completely specified Boolean functions • The condition that should be verified many times in the process of DSD: Is it true that two Boolean functions, F and G, are equal when restricted to the domains DF and DG? • A specialized checking procedure has been designed to perform this check Electronic Systems Design Seminar, UC Berkeley

16. Speeding-up DSD BD – Bertacco/Damiani, ICCAD’97 150 MHz PC M – Matsunaga, SASHIMI’98 266 MHz PC New – our implementation 933 MHz PC Dash (-) means that the result is not available in the publications. Electronic Systems Design Seminar, UC Berkeley

17. Ashenhurst-Curtis Variable grouping Decomposition table Graph coloring Encoding Bi-Decomposition Variable grouping Checking conditions Deriving A and B using boolean formulas Non-Disjoint Decomposition These schemes work for binary and MV functions/relations Electronic Systems Design Seminar, UC Berkeley

18. DD Operators for MV Decomposition • Ashenhurst-Curtis • Checking column compatibility Problem: Given MV relation R(X,Y,V) and two columns C1(X) and C2(X), find out whether these columns are compatible bool CheckColumnCompatibility( bdd R, bdd C1, bdd C2 ); • Bi-Decomposition • Checking the existence of MAX/MIN-bi-decomposition Problem: Given MV relation R(X,V) and the partitioning of X into three sets Xa, Xb, and Xc, find out whether there exists bi-decomposition using MAX or MIN gate bool CheckMMDecomposability( bdd R, bdd Xa, bdd Xb ); Electronic Systems Design Seminar, UC Berkeley

19. Optimal Non-Strict Encoding • Definition. Encoding is strict, if a code is a minterm depending on the encoding variable. • Definition. Encoding is non-strict, if a code is a set of minterms depending on the encoding variable. (There should be no overlap between the sets.) • Problem: Given a set of N Boolean or MV functions, Fi(X), and an array of variables V, |V|  |log2N|, find a non-strict encoding of the set of function, which guarantees that the resulting code-bit functions are simple in some sense. bdd FindEncoding(bdd * F, int N, bdd * Vars, int nVars); Electronic Systems Design Seminar, UC Berkeley

20. Encoding: Good and Bad Before encoding Encoding   After encoding Electronic Systems Design Seminar, UC Berkeley

21. Two-Level SOP Minimization • The problem with long history • Explicit solutions • Quine (1952), Espresso (1984), Espresso-Signature (1993) • Implicit solutions • Swami et al (1992), Coudert/Madre (1993), Scherzo (1994) • Specialized DD operators in the implicit approach • Prime computation • Covering table reduction • Minimization of the number of literals in the SOP • SOP minimizer Rondo was designed after Scherzo http://www.ee.pdx.edu/~alanmi/research/min/minSop.htm Electronic Systems Design Seminar, UC Berkeley

22. Espresso vs. Rondo E – Espresso R – Rondo Electronic Systems Design Seminar, UC Berkeley

23. Walsh Spectrum • Applications: • Logic synthesis, technology mapping, NPN checking • Traditionally computed using the Walsh matrix [Clarke et al, 1993] • Walsh matrix can be defined recursively • Examples: Wn = W0 = W1 = W2 = Electronic Systems Design Seminar, UC Berkeley

24. Computing Walsh Spectrum • Brute-force approach • Deriving the matrix and multiplying it by the truth vector • Smart approach • Developing a specialized DD procedure Illustration of the brute force approach: F = a + b. • Truth vector is (0,1,1,1) • Encoded truth vector is (1,-1,-1,-1) • Spectrum is ( -2, 2, 2, 2 ) x = Electronic Systems Design Seminar, UC Berkeley

25. Computing Walsh Spectrum • Smart approach • Developing a specialized DD procedure based on a recursive definition of the Walsh matrix • Walsh “butterfly diagram” [Thornton et al, RM’01] • Illustration of the smart approach: F = a + b A A + B B A - B Electronic Systems Design Seminar, UC Berkeley

26. Walsh Spectrum Computation spectrumWalsh( bdd F, varset V ) { if ( F = 0 ) return +1; /* S-encoding */ if ( F = 1 ) return -1; x = TopVariable( V ); (F0, F1) = Cofactors( F, x ); W0 = Walsh( F0 , V- x); W1 = Walsh( F1 , V- x); R0 = W0 + W1; R1 = W0 - W1; return ITE( x, R1, R0 ); } Electronic Systems Design Seminar, UC Berkeley

27. Haar Spectrum Computation spectrumHaar(bdd F, varset V ) { if ( F = 0 ) return 0; /* R-encoding */ if ( F = 1 ) return 1; var x = TopVariable( V ); (F0, F1) = Cofactors( F, x ); H0 = Haar( F0 , V- x); H1 = Haar( F1 , V- x); C0 = NegPathCoef( H0 ); C1 = NegPathCoef( H1 ); R0 = Update(H0 , C0 +C1); R0 = Update(H0 , C0 - C1); return ITE( x, R1, R0 ); } Electronic Systems Design Seminar, UC Berkeley

28. Experimental Results Electronic Systems Design Seminar, UC Berkeley

29. Conclusions • Analyzed the generic DD traversal procedure • Discussed standard and specialized DD operators • Looked into several applications of specialized operators • Experimental results show that the performance may increase several orders of magnitude if the DD operators are skillfully tailored to the application • You are welcome to use the source code at http://www.ee.pdx.edu/~alanmi/research/extra.htm Electronic Systems Design Seminar, UC Berkeley