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ECE 697B (667) Fall 2004 Synthesis and Verification of Digital Systems

ECE 697B (667) Fall 2004 Synthesis and Verification of Digital Systems. Boolean Functions SOP Representation. Slides adopted (with permission) from A. Kuehlmann, UC Berkeley 2003. Representation of Boolean Functions. Sum of Products:

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ECE 697B (667) Fall 2004 Synthesis and Verification of Digital Systems

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  1. ECE 697B (667)Fall 2004Synthesis and Verificationof Digital Systems Boolean Functions SOP Representation Slides adopted (with permission) from A. Kuehlmann, UC Berkeley 2003

  2. Representation of Boolean Functions • Sum of Products: • A function can be represented by a sum of cubes (products): f = ab + ac + bc Since each cube is a product of literals, this is a “sum of products” (SOP) representation • A SOP can be thought of as a set of cubes F F = {ab, ac, bc} • A set of cubes that represents f is called a cover of f. F1={ab, ac, bc} and F2={abc,abc’,ab’c,ab’c’,bc} are covers of f = ab + ac + bc. ECE 667 - Synthesis & Verification - Lecture 9b

  3. SOP • Covers (SOP’s) can efficiently represent many practical logic functions (i.e. for many, there exist small covers). • Two-level minimization seeks the minimum size cover (least number of cubes) ac bc = onset minterm Note that each onset minterm is “covered” by at least one of the cubes! None of the offset minterms is covered ab c b a ECE 667 - Synthesis & Verification - Lecture 9b

  4. Irredundant Cubes • Let F = {c1, c2, …, ck} be a cover for f, i.e. f = ik=1ci A cube ci F is irredundant if F\{ci}  f Example: f = ab + ac + bc ac bc bc Not covered ab ac c b F\{ab}  f a ECE 667 - Synthesis & Verification - Lecture 9b

  5. Prime Cubes • A literal j of cube ci F ( =f ) is prime if (F \ {ci })  {c”i }  f where c”i is ci with literal j of ci deleted. • A cube of F is prime if all its literals are prime. Example f = ab + ac + bc ci = ab; c”i = a (literal b deleted) F \ {ci }  {c”i } = a + ac + bc F=ac + bc + a = F \{ci }  {c”i } bc a ac c b Not equal to f since offset vertex is covered a ECE 667 - Synthesis & Verification - Lecture 9b

  6. Definitions - Prime, Irredundant, Essential • Definition 1A cube is primeif it is not contained in any other cube. A cover is primeif all its cubes are prime. • Definition 2A cover is irredundant if all its cubes are irredundant. • Definition 3A prime of f is essential (essential prime) if there is a minterm (essential vertex) in that prime that is in no other prime. • Definition 4 Two cubes are orthogonal if they do not have any minterm in common • E.g. c1= ab c2 = b’c are orthogonal c1= ab’ c2 = b’c are not orthogonal ECE 667 - Synthesis & Verification - Lecture 9b

  7. abc bd c b d cd a Prime and Irredundant Covers Example: f = abc + b’d + c’d is prime and irredundant. abc is essential since abcd’abc, but not in b’d or c’d or ad Why is abcd not an essential vertex of abc? What is an essential vertex of abc? What other cube is essential? What prime is not essential? abcd abcd’ ECE 667 - Synthesis & Verification - Lecture 9b

  8. n=3, m=3 a a b b c c f1 f2 f3 PLAs - Multiple Output Functions • A PLA is a function f : Bn Bm represented in SOP form: Personality Matrix abc f1f2f3 10- 1 - - -11 1 - - 0-0 - 1 - 111 - 1 1 00- - - 1 ECE 667 - Synthesis & Verification - Lecture 9b

  9. PLAs (cont.) • Each distinct cube appears just once in the AND-plane, and can be shared by (multiple) outputs in the OR-plane, e.g., cube (abc). • Extensions from single output to multiple output minimization theory are straightforward. • Multi-level logic can be viewed mathematically as a collection of single output functions. ECE 667 - Synthesis & Verification - Lecture 9b

  10. f = abc + abc fa = bc c c b b a a Shannon (Boole) Cofactors Let f : Bn B be a Boolean function, and x= (x1, x2, …, xn) the variables in the support of f. The cofactor fa of f w.r.t literal a=xi or a=x’i is: fxi (x1, x2, …, xn) = f (x1, …, xi-1, 1, xi+1,…, xn) fx’i (x1, x2, …, xn) = f (x1, …, xi-1, 0, xi+1,…, xn) The computation of the cofactor is a fundamental operation in Boolean reasoning ! Example: ECE 667 - Synthesis & Verification - Lecture 9b

  11. Generalized Cofactor • The generalized cofactor fC of f by a cube C is f with the fixed values indicated by the literals of C, e.g. if C=xi x’j, then xi =1, and xj =0. • if C= x1 x’4 x6 fC is just the function f restricted to the subspace where x1 =x6 =1 and x4 =0. • As a function, fC does not depend on x1,x4 or x6 anymore (However, we still consider fC as a function of all n variables, it just happens to be independent of x1,x4 and x6). • x1f  fx1 Example: f= ac + a’c , af = ac, fa=c ECE 667 - Synthesis & Verification - Lecture 9b

  12. Fundamental Theorem Theorem:Let c be a cube and f a function. Then c  f  fc  1. Proof.We use the fact that xfx = xf, and fx is independent of x. If : Suppose fc  1. Then cf=fcc=c. Thus, c  f. f c ECE 667 - Synthesis & Verification - Lecture 9b

  13. C=xz m= 000 m’= 101 m’ Proof (cont.) Only if. Assume c  f Then c  cf = cfc. But fc is independent of literals i  c. If fc 1, then  m  Bn, fc(m)=0. We will construct a m’ from m and c in the following manner: mi’=mi, if xic and xic, mi’=1, if xi  c, mi’=0, if xi  c. i.e. we made the literals of m’ agree with c, i.e. m’  c Þ c(m’)=1 Also, fc is independent of literals xi,xi  c Þ fc(m’) =fc(m) = 0 Þ fc(m’) c(m’)= 0 contradicting c  cfc. m ECE 667 - Synthesis & Verification - Lecture 9b

  14. Cofactor of Covers Definition: The cofactor of a cover F is the sum of the cofactors of each of the cubes of F. Note: If F={c1, c2,…, ck} is a cover of f, then Fc= {(c1)c, (c2)c,…, (ck)c} is a cover of fc. Suppose F(x) is a cover of f(x), i.e. Then for 1jn, is a cover of fxj(x) ECE 667 - Synthesis & Verification - Lecture 9b

  15. Cofactor of Cubes Definition: The cofactor Cxj of a cube C with respect to a literal xj is • C if xj and x’j do not appear in C • C\{xj} if xj appears positively in C, i.e. xjC •  if xj appears negatively in C, i.e. xjC Example 1: C = x1 x’4 x6, Cx2 = C (x2 and x2 do not appear in C ) Cx1 = x’4 x6 (x1 appears positively in C) Cx4 =  (x4 appears negatively in C) Example 2: F = abc + b’d + c’d Fb = ac + c’d (Just drop b everywhere and throw away cubes containing literalb) ECE 667 - Synthesis & Verification - Lecture 9b

  16. Shannon Expansion f : Bn B Shannon Expansion: Theorem: F is a cover of f. Then We say that f (F) is expanded about xi. xi is called the splitting variable. ECE 667 - Synthesis & Verification - Lecture 9b

  17. ac bc ab c c b b a a Shannon Expansion (cont.) Example Cube bc got split into two cubes ECE 667 - Synthesis & Verification - Lecture 9b

  18. List of Cubes (Cover Matrix) We often use a matrix notation to represent a cover: Example:F = ac + c’d = a b c d a b c d a c  1 2 1 2 or 1 - 1 - c’ d  2 2 0 1 - - 0 1 • Each row represents a cube • 1 means that the positive literal appears in the cube • 0 means that the negative literal appears in the cube • The 2 (or -) represents that the variable doesnot appearin the cube. It implicitly represents both 0 and 1 values. ECE 667 - Synthesis & Verification - Lecture 9b

  19. Some Special Functions Definition: A function f : Bn B issymmetricwith respect tovariables xi and xjiff f(x1,…,xi,…,xj,…,xn) = f(x1,…,xj,…,xi,…,xn) Definition: A function f : Bn B istotally symmetric iff any permutation of the variables in f does not change the function • Symmetry can be exploited in searching the binary decision tree • because: • - That means we can skip one of four sub-cases • - used in automatic variable ordering for BDDs ECE 667 - Synthesis & Verification - Lecture 9b

  20. Unate Functions Definition:A function f : Bn B ispositive unateinvariable xiiff This is equivalent tomonotone increasingin xi: for all min-term pairs (m-, m+) where For example, m-3=1001, m+3=1011(where i=3) ECE 667 - Synthesis & Verification - Lecture 9b

  21. Unate Functions Similarly fornegative unate monotone decreasing: A function isunatein xi if it is either positive unate or negative unate in xi. Definition:A function isunateif it is unate in each variable. Definition:A cover F ispositive unatein xi iff xi  cj for all cubes cjF ECE 667 - Synthesis & Verification - Lecture 9b

  22. Example positive unate in a,b negative unate in c m+ c f(m-)=1  f(m+)=0 b m- a ECE 667 - Synthesis & Verification - Lecture 9b

  23. The Unate Recursive Paradigm • Key pruning technique based on exploiting the properties ofunatefunctions • based on the fact that unate leaf cases can be solved efficiently • New case splitting heuristic • splitting variable is chosen so that the functions at lower nodes of the recursion tree become unate ECE 667 - Synthesis & Verification - Lecture 9b

  24. The Unate Recursive Paradigm Unate covers F have many extraordinary properties: • If a cover F is minimal with respect to single-cube containment, all of its cubes are essential primes. • In this case F is the unique minimum cube representation of its logic function. • A unate cover represents the tautology iff it contains a cube with no literals, i.e. a single tautologous cube. This type of implicit enumeration applies to many sub-problems (prime generation, reduction, complementation, etc.). Hence, we refer to it as the Unate Recursive Paradigm. ECE 667 - Synthesis & Verification - Lecture 9b

  25. a merge b c Unate Recursive Paradigm • Create cofactoring tree stopping atunate covers • choose, at each node, the “most binate” variable for splitting • recurse till no binate variable left (unate leaf) • “Operate” on the unate cover at each leaf to obtain the result for that leaf. Return the result • At each non-leaf node,merge(appropriately) the results of the two children. • Main idea: “Operation” on unate leaf is computationally less complex • Operations: complement, simplify,tautology,generate-primes,... ECE 667 - Synthesis & Verification - Lecture 9b

  26. Recursive Complement Operation • We have shown how tautology check (SAT check) can be implemented recursively using the Binary Decision Tree • Similarly, we can implement Boolean operations recursively • Example COMPLEMENT operation: Theorem: Proof: ECE 667 - Synthesis & Verification - Lecture 9b

  27. COMPLEMENT Operation Algorithm COMPLEMENT(List_of_Cubes C) { if(C contains single cube c) { Cres = complement_cube(c) // generate one cube per return Cres // literal l in c with ^l } else { xi = SELECT_VARIABLE(C) C0 = COMPLEMENT(COFACTOR(C,^xi)) Ù ^xi C1 = COMPLEMENT(COFACTOR(C,xi)) Ù xi return OR(C0,C1) } } ECE 667 - Synthesis & Verification - Lecture 9b

  28. Complement of a Unate Cover • Complement of a unate cover can be computed more efficiently • Idea: • variables appear only in one polarity on the original cover (ab + bc + ac) = (a+b)(b+c)(a+c) • when multiplied out, a number of products are redundant aba + abc + aca + acc + bba + bbc + bca + bcc = ab + ac + bc • we just need to look at the combinations for which the variables cover all original minterms • this works independent of the polarity of the variables because of symmetry to the (1,1,1,…,1) case ECE 667 - Synthesis & Verification - Lecture 9b

  29. Incompletely Specified Functions F = (f, d, r) : Bn {0, 1, *} where * represents a don’t care. • f = onset function - f(x)=1  F(x)=1 • r = offset function - r(x)=1  F(x)=0 • d = don’t care function - d(x)=1  F(x)=* (f,d,r) forms a partition of Bn. i.e. • f + d + r = Bn • fd =fr = dr =  (pairwise disjoint) ECE 667 - Synthesis & Verification - Lecture 9b

  30. Incompletely Specified Functions A completely specified function g is a cover for F=(f,d,r) if f g  f+d (Thus gr = ). Thus, if xd (i.e. d(x)=1), then g(x) can be 0 or 1, but if xf, then g(x)=1 and if xr, then g(x)=0. (We “don’t care” which value g has at xd) ECE 667 - Synthesis & Verification - Lecture 9b

  31. F1= abc + abc+ abc Example:Logic Minimization Consider F(a,b,c)=(f,d,r), where f={abc, abc, abc} and d ={abc, abc}, and the sequence of covers illustrated below: Expand abca on off F2= a+abc + abc Don’t care abc is redundant a is prime F3= a+abc Expand abc bc F4= a+bc ECE 667 - Synthesis & Verification - Lecture 9b

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