ICS 252 Introduction to Computer Design

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1 ICS 252 Introduction to Computer Design Logic Optimization Eli Bozorgzadeh Computer Science Department-UCI

2 Hardware compilation flow HDL RTL Synthesis netlist Logic synthesis library netlist Physical design layout [ Keutzer] ICS 252-Intro to Computer Design 3

3 Optimization of modules A B B C arrival time 15 ns required time 15 ns [ Keutzer] 4

4 Reduce to combinational optimization Flip-flops Combinational logic Input arrival time Output required time [ Keutzer] 5

5 Logic optimization netlist 2-level logic opt library Logic optimization netlist Tech independent Tech dependent library Multi-level logic opt [ Keutzer] 6

6 Logic Synthesis Target: Determine microscopic structure of a circuit (gate level representation) Two classes: combinatorial circuits (by Boolean functions) sequential circuits (using finite state machine diagram) 7

7 Logic Synthesis Two-level combinatorial logic optimization Modeled by sum-of-products or product-ofsums expressions forms Direct impact on macro-cell design like PLA Benefits the overall multi-level optimization Multi-level combinatorial logic optimization Synthesis of sequential logic 8

8 Two-level Logic Synthesis Input Boolean network, either sum-of-products (SOP) or product-of-sum (POS) Timing characterization for the module (input arrival time) Target library Objective Minimize the size of Boolean function either on SOP or POS [ Keutzer] 9

9 Basic Definitions B={0,1} Y={0,1,2} 2:don t care Input variables: X1,X2,,Xn Output Variables: Y1,Y2,,Ym Logic function f: F: B n Y m ON-SET i : the set of all input values for which f i (x)=1 (x i ON ) OFF-SET i : the set of all input values for which f i (x)=0 (x i OFF ) DC-SET i : the set of all input values for which f i (x)=2 (x i DC ) [ Gupta] 10

10 Boolean function f(x) x1 x2 x3 f1 f x x x ON 1 OFF 1 DC 1 = {[0,0,0],[0,0,1],[1,0,0],[1,0,1],[1,1,0]} = {[0,1,0],[0,1,1]} = {[1,1,1]} [ Gupta] 11

11 The Boolean n-cube x3 x1 x1 DC-SET OFF-SET ON-SET 12

12 Boolean functions f : complement of f ON-SET of fi is OOF-SET of fi and vice versa. Intersection (or and ) of f and g (f.g or f g) Intersection of ON_SET of f and g Union (or or) of f and g (fug, f+g) Union of ON-SET of f and g If ONSET of f is B n (i.e., OFF-SET for f is empty) then f is the tautology If ONSET of f is empty f is not satisfiable If f(x)=g(x) for all x, f and g are equivalent x1,x2, xn: variables x1,x1,x2,,xn,xn: literals [ Keutzer] 13

13 Don t care condition Value of the function is not immaterial. Related to its environment: Input value assignments that never occur Input assignments such that some output is never observed Important for logic optimization [ Gupta] 14

14 Prime Implicants Cubical representation: boolean variable Boolean literals: variable or complement Product of cube: product of literals Implicant: product that does not intersect with OFF-SET of function f Prime implicant: am implicant that is not contained by any implicant of the function Minterm: product of all input variables which does not intersect with OFF-SET of function f [ Gupta] 15

15 Cover A cover is a set of cubes C such that ON Why cover? f C and C Minimize number of implicants Same cost per implicant Minimize number of literals Literals define programming transistors Based on finding a cover f ON f DC [ Gupta] 16

16 Covers A set of implicants Cardinality: number of implicants Min cover: minimum cardinality An irredundant cover is a cover such that removing any cube from cover results in a set of cubes that is not a cover Prime cover is a cover whose cubes are all prime implicants [ Gupta] 17

17 Covers An essential prime of f is a prime that contains some minterm not contained in any other prime, A cover is said o be single cube containment minimal if no cube if the cover is contained in another cube of cover. A cover of f is said to be minimum if there exists no other cover with fewer cubes. [ Gupta] 18

18 Minimum covers A Minimum cover can always be found by restricting the search to prime and irredundant covers Exact methods Based on Quine-McClusky method, compute prime implicants and determine min cover Heuristic methods Find minimal cover [ Keutzer, Gupta] 19

19 Combinational logic design basics Two-level representation Cubes and covers Cover C of f if f ON C f ON Uf DC Minimum cover No cover with fewer cubes Irredundant cover If no subset of C is a cover of f No implicant can be dropped Minimal cover w.r.t. single-cube containment (SCC) No cube of C is contained in another cube of C Irredundant SCC (is it true for converse? 20

20 Cont Prime implicants Prime implicant not contained by any other implicant A product of literals where no literal can be dropped Geometrically : largest size cube without intersecting the OFF-SET Essential prime implicant Must be contained in any cover of the function 21

21 Exact optimization for two-level logic Goals: Reduce number of implicants Reduce number of literals Determine minimum cover of f Quine: There exists a minimum cover that is prime Look just for prime implicants Quine-McCluskey method: Compute prime implicants Determine minimum cover 22

22 Prime implicant table A binary-valued matrix A Columns are prime implicants Rows are minterms in the ON-SET a ij =1 iff prime cover j covers minterm I Example: f=a b c +a b c+ab c+abc+abc Primes? Implicant table? 23

23 Cover A Cover can be represented by a (hyper)graph with minterms as vertices and primes as edges (hyper) c b a 24

24 Covering problem Given A, cover all the rows with least number of columns A minimum cover is a minimum set of columns which covers all rows. Determine x such that Ax 1 Minimize cardinality of x A can be seen as incidence matrix of a hypergraph Column covering as an edge covering problem 25

25 Covering Brute force : consider all possible values Complexity? Use pruning: Petrick s method 26

26 Petrick s method Write covering clauses of the reduced table in POS form Multiply out POS form into SOP form Select cube of minimum size 27

27 Example POS form p 1 (p 1 +p 2 )(p 2 +p 3 )(p 3 +p 4 )p 4 =1 SOP form p 1 p 2 p 4 +p 1 p 3 p 4 =1 Two minimum covers of cardinality 3 Solution {p 1,p 2,p 4 }, {p 1,p 3,p 4 } 28

28 Heuristic Method Local minimum cover Given an initial cover Make it prime Make it irredundant Iterative improvement Reduce cover cardinality by modifying implicants. 29

29 Summary Logic synthesis basics Combinations logic synthesis Two-level logic optimization Two-level forms the theoretical foundation for multi-level logic synthesis Two-level optimization directly used for PLA/PLD design Two-level optimization is used as a subroutine in multi-level logic synthesis [ Keutzer] 30

Giovanni De Micheli. Integrated Systems Centre EPF Lausanne

Giovanni De Micheli. Integrated Systems Centre EPF Lausanne Two-level Logic Synthesis and Optimization Giovanni De Micheli Integrated Systems Centre EPF Lausanne This presentation can be used for non-commercial purposes as long as this note and the copyright footers