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Logic Synthesis. Minimization of Boolean logic Technology-independent mapping Objective: minimize # of implicants, # of literals, etc. Not directly related to precise technology (# transistors), but correlated – consistent with objectives for any technology Technology-dependent mapping

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## Logic Synthesis

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**Logic Synthesis**• Minimization of Boolean logic • Technology-independent mapping • Objective: minimize # of implicants, # of literals, etc. • Not directly related to precise technology (# transistors), but correlated – consistent with objectives for any technology • Technology-dependent mapping • Linked to precise technology/library • Technology-independent mapping • Two-level minimization – sum of products (SOP)/product of sums (POS) • Karnaugh maps – “visual” technique • Quine-McCluskey method – algorithmic • Heuristic minimization – fast and “pretty good,” but not exact • Multi-level minimization**Basic Definitions**• Specification of a function f • On-set fon: set of input combinations for which f evaluates to 1 • Off-set foff: set of input combinations for which f evaluates to 0 • Don’t care set fdc: set of input combinations over which function is unspecified • Cubes • Can represent a function of k variables over a k-dimensional space • Example: f(x1,x2,x3) = m(0,3,5,6) + d(7) fon = {0,3,5,6}; fdc = {7}; foff = {1,2,4} • Graphically: 011 x3 111 101 001 x2 110 010 x1 000 100**k-cube: k-dim. subset of fon**0-cube = vertex in fon k-cube = a pair of (k-1) cubes with a Hamming distance of 1 Examples A 0-cube is a vertex A 1-cube is an edge A 2-cube is a face A 3-cube is a 3D cube A 4-cube is harder to visualize but can be shown as k-cubes**More defintions**• Implicant • A k-cube whose vertices all lie in the fon fdc and contains at least one element of fon • Prime implicant • A k-cube implicant such that no k+1-cube containing this cube is an implicant • Cover • A set of implicants whose union contains all elements of fon and no elements of foff (may contain some elements of fdc) • Minimum cover • A cover of minimum cost (e.g., cardinality) • A min cardinality cover composed only of prime implicants exists (if not, can combine some implicants into larger prime implicants)**Quine-McCluskey Method**• Illustration by example: f(x1,x2,x3,x4) = m(0,5,7,8,9,10,11,14,15) 0-cubes 1-cubes 2-cubes 0 (0000) x 0,8 (-000) A8,9,10,11 (10--) D 5 (0101) x 5,7 (01-1) B10,11,14,15 (1-1-) E 7 (0111) x 7,15 (-111) C 8 (1000) x 8,9 (100-) x 9 (1001) x 8,10 (10-0) x 10 (1010) x 9,11 (10-1) x 11 (1011) x 10,11 (101-) x 14 (1110) x 10,14 (1-10) x 15 (1111) x 11,15 (1-11) x 14,15 (111-) x • “x” implies that the cube has been combined into a larger cube Prime implicants**Prime implicant table**• Essential Prime Implicant (PIs): • The only PI that covers a minterm (encircled in the table) • Must be included in any cover • Here, essential PIs = A,B,D,E – form a cover! • WARNING: this was luck – in general, essential PIs will not form a cover!**Reducing the prime implicant table**• PI table reduction • In general, essential PIs will not form a cover • Reduce table by removing essential PIs, corresponding minterms • Further reduction: can remove Dominating rowsDominated columns Row m1 dominates row m2 Column J dominates column K**May still not have a cover**Example: example from previous slide after removing dominating row m1 and consequently empty column P Can enumerate possibilities using a search tree Binary search tree: include or exclude PI Branch-and-bound algorithm Q include exclude Done Cover = {R,S} R Reduced PI table exclude include Done Cover = {Q,R} Done Cover = {Q,S}**Branch-and-bound algorithm (contd.)**• ESPRESSO-EXACT • Implementation of branching algorithm from previous slide • Traversal to a leaf node of the tree yields a cover (though possibly not a minimum cost cover) • ESPRESSO-EXACT adds bounding at any node: • If Costnode + LBsubtree > Best_cost_so_far, do not search the subtree • Costnode = cost (e.g., number of implicants) chosen so far • LBsubtree = a lower bound on the cost of a subtree (can be determined by solving a maximal independent set problem) • Best_cost_so_far = cost of best cover found so far through the traversal of the search tree; initialized to **Heuristic Logic Minimization**• Apply a sequence of logic transformations to reduce a cost function • Transformations • Expand: • Input expansion • Enlarge cube by combining smaller cubes • Reduces total number of cubes • Output expansion • Use cube for one output to cover another • Reduce: break up cube into sub-cubes • Increases total number of cubes • Hope to allow overall cost reduction in a later expand operation • Irredundant • Remove redundant cubes from a cover**Example**• Expand: input expansion z Redundant! y x On-set member Off-set member Examples from G. Hachtel and F. Somenzi, “Logic Synthesis and Verification Algorithms,” Kluwer Academic Publishers, Boston, MA, 1996.**Example**• Expand: output expansion • Two output functions of three variables each with initial covers shown below z z y y x x f1 f2 f1 f2 Examples from G. Hachtel and F. Somenzi, “Logic Synthesis and Verification Algorithms,” Kluwer Academic Publishers, Boston, MA, 1996.**Other operators**• Reduce • Irredundant Future Expand operation Reduce Identified as redundant; removed Irredundant**Example of an application of operators**(12 cubes) (10 cubes) (9 cubes) reduce expand expand reduce expand Example from S. Devadas, A. Ghosh and K. Keutzer, “Logic Synthesis,” McGraw-Hill, New York, NY, 1994.**F = Expand(F, D)**F = Irredundant(F,D) do { Cost = |F| F = Reduce(F,D) F = Expand(F,D) F = Irredundant(F,d) } while (|F| < Cost) F= Make_sparse(F,D) Make_sparse reduces output parts of a cube (e.g., from 11 to 10) to remove redundant connections) Example: Example of a minimization loop**Implementation of operators**• Uses “unate recursive paradigm” • Definition: Shannon expansion • F(x1,x2, …, xi, …, xn) = xi F(x1,x2, …, xi, …, xn) + xi’ F(x1,x2, …, xi, …, xn) = xi Fxi + xi’ Fxi’ (notationally) • Unate function • Positive unate in variable xi: Fxi Fxi’ F = xi Fxi + Fxi’ • Negative unate in variable xi: Fxi Fxi’ F = Fxi + xi’ Fxi’ • Unate function: positive unate or negative unate in each variable • Unate recursive paradigm • Recursively perform Shannon expansions about the variables until a unate function is obtained • Why unate functions? Various operations (tautology checking, complementation, etc. are “easy” for unate functions)**Unateness**• Example • Unate cover (not minimum) • Every column has only 1’s and –’s, or only 0’s and –’s • Nonunate cover: nonunate in y and z (both 1 and 0 appear in the columns) Note on notation: The table at left represents the on set**Unate recursive paradigm**Fc Fc’ Expand about binate variable (Unate function) Fe’ Fe (Unate function) (Unate function)**Complement**Complement Complement Complement Example: Unate complementation • (Example to show that unate operations are “easy”) Basic result: If F = x Fx + x’ Fx’ then F’ = x (Fx)’+ x’ (Fx’)’ Proof: Let G = x (Fx)’+ x’ (Fx’)’ Show that F+G = 1 and F.G = 0 x x’ y y’ Complement Empty set!**Can generalize the Shannon expansion to**F = c Fc + c’ Fc’ where c is an set of cubes Result: c F Fc is a tautology Example of finding a cofactor with respect to a cube: Cofactor of F = with respect to c = [1 1 – – ] Fc contains elements of Fon that agree with c at all non-don’t care positions (in this example, in variables p and q) If so: replace non-don’t cares by “–” and copy the rest of the cube Following this prescription, Fc is Cofactors with respect to sets of cubes**Checking for Tautology**• Checking for tautology: • F is a tautology Fxj is a tautology and Fxj’ is a tautology • Let C be a unate cover of F. Then “F is a tautology” “C has a row of all “-’s • Example Binate variable r’ r Tautology! Tautology! Since all leaf nodes are tautologies, the function is a tautology**The Expand Operator and Tautology**• Consider the function f with a cover G and fdc specified in the table • Objective: to expand 000 | 1 to 0 – 0 | 1 • Need to check if the expansion is valid, i.e., it does not overlap with foff • Define ci = 0 0 0 | 1 • di = difference between ci and 0 – 0 | 1 = 0 1 0 | 1 here • Need to know if di Q = (G \ ci) fdc : if so, can expand • In other words, check if Qdi is a tautology • Qdi can easily be verified here to be – – –| 1 here, which is a tautology**The Irredundant Operator and Tautology**• Objective: to check if a cube ci in a cover G of function F is redundant • In other words, check if ci Q = (G \ ci) Fdc • In other words, check if Qci is a tautology**Multilevel logic optimization**• Motivation • Two-level optimization (SOP, POS) is too limiting • Useful for structures like PLA’s, but most circuits are not designed in that way • May require gates with a large number of inputs • Restricts “sharing” of logic gates between outputs • Multilevel optimization permits more than two levels of gates between the inputs and the outputs • Necessarily heuristic Reference for this part: G. De Micheli, “Synthesis and Optimization of Digital Circuits,” McGraw-Hill, New York, NY, 1994.**Basic Transformations**• Elimination r = p + a’; s = r + b’ s = p + a’ + b’ • Decomposition v = a’d + bd + cd + a’e j = a’ + b + c; v = j d + a’e • Extraction p = ce+de; t = ac+ad+bc+bd+e k = c+d; p = ke; t = ka+kb+e • Simplification u = q’c+qc’+qc u = q+c • Substitution t = ka+kb+e; q = a+b t = kq+e • (Others exist; these are the most common)**Transformations**• Apply the transformations heuristically • Two methods: • Algorithmic: algorithm for each transformation type • Rule-based: according to a set of rules injected into the system by a human designer**A typical synthesis script**• script.rugged in the SIS synthesis system from Berkeley sweep; eliminate –1 simplify –m nocomp eliminate –1 sweep; eliminate 5 simplify –m nocomp resub –a fx resub –a; sweep eliminate –1; sweep full_simplify –m nocomp Explanation sweep: eliminates single-input vertices (w = x; y = w+z becomes y = x+z) eliminate k: eliminate defined earlier; Eliminates vertices so that area estimate increases by no more than k simplify –m nocomp: simplify defined earlier Invokes ESPRESSO to minimize without computing full off-set (“nocomp”) full_simplify –m nocomp: as above, but uses a larger don’t care set resub –a: algebraic substitute for vertex pairs fx: extracts double cube and single cube expressions**Algebraic model**• Also known as weak division • Manipulation according to rules of polynomial algebra • Support of a function • Sup(f) = set of all variables v that occur as v or v’ in a minimal representation of f • Sup(ab+c) = {a,b,c}; Sup(ab+a’b) = {b} • f is orthogonal to g (or f g) if Sup(f) Sup(g) = • g is an algebraic (or weak) divisor of f when • f= g h + r, provided h and g h • g divides f evenly if r = • Example • If f = ab+ac+d; g = b+c, then f = ag + d (here h = a, r = d) • The quotient, loosely referred to as f/g, is the largest cube h such that f = gh + r**Computing the quotient f/g**• Given f = {set of cubes ci}, g = {set of cubes ai} • Define hi = {bj | ai bj f} for all cubes ai g (all multipliers of a cube ai that produce elements in f) • f/g = i=1 to |g|hi • Example • f = abc + abde + abh + bcd, or f = {abc,abde,abh,bcd} • g = c + de + h, or g = {c,de,h} • h1 = f/c = ab + bd, or {ab,bd}; h2 = f/de = ab, or {ab}; h3 = f/h = ab, or {ab} • f/g =h1 h2 h3 = {ab} • (Confirmation: f = ab(c+de+h) + bcd = (f/g) g + r) • Complexity of this method = |f|.|g|**Doing this more efficiently**• Encode ai g with integer codes with a unique bit position for each literal in sup(g) • g = {c,de,h}; sup(g) = {c,d,e,h}; encoding = {1000,0110,0001} • Encode ci f with the same encoding • f = {abc,abde,abh,bcd}; encoding = {1000,0110,0001,1100} • Sort {ai, cj} by their encodings to get • 1100: bcd • 1000:c, abc h1 = ab • 0110:de, abde h2 = ab • 0001:h, abh h3 = ab • (Not the same hi’s, but the intersection is the same) • Complexity = O(n log n) where n = |f| + |g|**Finding good divisors**• Now that we know how to divide – how do we find good divisors? • Primary divisors • P(f) = {f/c | f is a cube} • Example: f = abc + abde • f/a = bc + bde is a primary divisor • f/ab = c + de is a primary divisor • g is cube free if the only cube dividing g evenly (i.e., with remainder zero) is 1. Example: c+de • Kernels • K(f) = set of primary divisors that are cube-free • f/ab belongs to the set of kernels; f/a does not. • Kernels are good candidates for divisors**Kernels and co-kernels**• For f = abc + abde, f/ab = c + de • c+de is a kernel • ab is a co-kernel • Co-kernel of a kernel is not unique • f = acd + bcd + ae + be • f/a = f/b = cd+e Kernel = cd+e Co-kernels = {a,b} • f/cd = f/e = a+b Kernel = a+b Co-kernels = {cd,e}**Kernel (f)**Find cf so that f/cf is cube-free and cf has the largest number of literals K = Kernel1(0,f/cf) if (f is cube-free) return(f K) return(K) Kernel1(j,g) R = {g} for (i = j+1; i n; i++) if ( ith literal li has 0 or 1 terms) continue ce = cube that evenly divides g/li and has the max number of literals /* kernel already identified */ if (lk is not in ce for all k i) R = R Kernel1(i,(g/li)/ce) return(R) Finding all kernels**Example**F = abc(d+e)(k+l) + agh + m a F/a = bc(d+e)(k+l) + gh c b F/ab = c(d+e)(k+l) F/ac = b(d+e)(k+l) Triggers “if” condition that finds that this kernel was found earlier and prunes the search tree here [Leads to kernels (d+e) and (k+l)]**Example: extraction and resubstitution**F1 = ab(c(d+e)+f+g)+h F2 = ai(c(d+e)+f+j)+k • Generate kernels for F1, F2 • Select K1 K(F1) and K2 K(F2) such that K1 K2 is not a cube • Set the new variable v to K1 K2 • Rewrite Fi = v (Fi/v) + ri For the example: v1 = d+e F1 = ab(cv1+f+g)+h; F2 = ai(cv1+f+j)+k v2 = cv1+f F1 = ab(v2+g)+h; F2 = ai(v2+j)+k**Generic factorization algorithm**Factor(F) If (F has no factor) return(F); D = Divisor(F); (Q,R) = Divide(F,D); /* F = QD + R*/ return(Factor(Q),Factor(D),Factor(R)); • “Divisor” function identifies divisors, for example, based on a kernel-based algorithm • “Divide” function may be algebraic (weak) division**Don’t care based optimization: an outline**• Two types of don’t cares considered here • Satisfiability don’t cares • Observability don’t cares • Others: SPFD’s (sets of pairs of functions to be differentiated) • Satisfiability don’t cares (SDC’s) • Example: Consider • Y1 = a’b’ Y2 = c’d’ Y3 = Y1’Y2’ • Since Y1 = a’b’ is enforced by one equation, the minterms of “Y1 (a’b’)” can be considered to be don’t cares • In other words, Y1a’b’ + Y1(a+b) corresponds to a don’t care • Similarly, “Y2 (c’d’)” is also a don’t care**Don’t care based optimization (contd.)**• Observability don’t cares (ODC’s) • For r = p+q, if p = 1, then q is an observability don’t care • Similarly, can define ODC’s for AND operations, etc. • Example of don’t care based optimization • y1 = xw, y2 = x’+y, f = y1+y2 • Cost = 1 AND + 2 OR’s + 1 NOT • Minimize function for y1 • SDC(y1) = y2 (x’+y) = y2xy’+y2’x’+y2’y • ODC(y1) = y2 ODC SDC’s y1 = w y2 = x’+y, f = w + y2 y1 = w, y2 = x’+y, f = y1 + y2 (eliminate) (Cost: 2 OR’s + 1 NOT)**Acknowledgements**• Hardly anything in these notes is original, and they borrow heavily from sources such as • G. De Micheli, “Synthesis and Optimization of Digital Circuits,” McGraw-Hill, New York, NY, 1994. • S. Devadas, A. Ghosh and K. Keutzer, “Logic Synthesis,” McGraw-Hill, New York, NY, 1994 • G. Hachtel and F. Somenzi, “Logic Synthesis and Verification Algorithms,” Kluwer Academic Publishers, Boston, MA, 1996. • Notes from Prof. Brayton's synthesis class at UC Berkeley (http://www-cad.eecs.berkeley.edu/~brayton/courses/219b/219b.html) • Notes from Prof. Devadas's CAD class at MIT (http://glenfiddich.lcs.mit.edu/~devadas/6.373/lectures) • Possibly other sources that I may have omitted to acknowledge (my apologies)

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