1. Courtesy RK Brayton (UCB) and A Kuehlmann (Cadence) 1 Logic Synthesis Multi-Valued Logic

2. 2 Up to now…two-valued synthesis –Binary variables take only values {0, 1} Multi-Valued logic –Multi-valued variable X i can take on values P i = {0,…,|P i |-1} (integers - but no ordering implied) –Symbolic variables take values from symbolic set, e.g. state: {s 0,s 1,…,s n } or X: {a,b,c}. –Enumeration types from RTL –Set of values for each dimension is finite!!!!

3. 3 Multi-Valued Logic Formally: (sometimes called an mv-function ). Problem: find the minimum SOP form for an incompletely-specified function of this kind Big News: Nothing really changes because solution space is still finite!!

4. 4 Example “Truth Table” P 1 ={0,1,2}, P 2 ={0,1} Here “2” means the value 2 and not {0,1} f(0,0) = 1f(2,1) = 1 f(1,0) = 0f(2,0) = * unspecified (don’t cares)

5. 5 MV Function off on Don’t care

6. 6 Terminology Vertex: Cube: Containment: Implicant:

7. 7 Terminology Onset minterm: Prime Implicant: Cover of F : Prime Cover of F:

8. 8 Notation-MV Literals Definition: A multi-valued literal is a binary logic function of the form where Definition: A cube can be written as the product of MV-literals:

9. 9 Notation-MV Literals If c i =P i we may omit from the expression (since =1) Note analogy to two-valued case: Actually, multi-valued notation is superior to binary notation.

10. 10 Example Rows marked as a (b) form single mv-cube implicant The following are cube covers of F. F 2 is a prime cover

11. 11 Positional Notation Example: Cube1 P 1 ={A,B,C,D}, P 2 ={R,S} (Symbolic) A B C D R S Cube1:1 1 0 0 1 0 Cube2:1 1 1 1 0 1 A cube does not depend on variable X i if it has all 1’s in the set of columns associated with X i (Cube2 does nor depend on X 1 ). Each of the columns of a variable is called a part of that variable. There is one part for each value a variable can take.

12. 12 Positional Notation (value=0) (value=1) 0 1  1 1 0  0 1 1  2 Extension of Espresso notation Example: X 1 X 2 X 3 C 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 C 2 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 C 3 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 C 4 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 C 5 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0

13. 13 Positional Notation X 1 X 2 X 3 C 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 C 2 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 C 3 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 C 4 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 C 5 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0

14. 14 Minimization for Multi-Valued Logic Given: Cover F of  and a cover D of the don’t-care set d, Find: A minimum sum-of-products form for  Same problem as for two-valued Generate primes of (f+d) Generate covering table Solve the covering table (unate covering problem)

15. 15 Applications of Multi-Valued Logic Theorem: minimizing a two-valued (n input)  (m output) logic function g is equivalent to minimizing a single binary-output MV-logic function: f : {0,1}  {0,1} ...  {0,…,m-1}  {0,1} Proof( sketch): Let g = {f 0,…,f m-1 } be the multiple output function. Consider the characteristic function f of the multiple output function, (defined on (n+1) variables with the last one, y, being multi-valued on {0,1,…,m-1} ) :

16. 16 Applications of Multi-Valued Logic Note: An implicant of g (the multi-output function) is a cube c in the x-space where each output is turned on only if f i (c)=1. Any output not turned on means no information (not offset), since the each output is the OR of all of its input cubes. X f 1 f 2 f 3 f 4 f 5 f 6 x-cube 0 1 0 1 1 0 g

17. 17 Other Applications: Encoding Input Encoding problem –bit-grouped PLA structure (decoded PLA) Output encoding problem –output phase optimization State encoding problem –Minimize symbolically to get constraints on a possible binary encoding –solve constraints to derive binary code –Re-minimize binary problem –Implement in binary

18. 18 Multi-Valued Minimization Example

19. 19 Prime and irredundant SOP of f: (five cubes 1+2+3+4+5) Equivalent to: Example - after minimization

20. 20 Note: is not a prime of f 0, but is a prime of f. Similarly for. Example - after minimization f 0 f 1 f 2