Logic Synthesis

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# Logic Synthesis - PowerPoint PPT Presentation

Logic Synthesis. Multi-Valued Logic. Multi-Valued Logic. 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)

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### Logic Synthesis

Multi-Valued Logic

Multi-Valued Logic

Up to now…two-valued synthesis

• Binary variables take only values {0, 1}

Multi-Valued logic

• Multi-valued variable Xi can take on values Pi = {0,…,|Pi|-1} (integers - but no ordering implied)
• Symbolic variables take values from symbolic set, e.g. state: {s0,s1,…,sn} or X: {a,b,c}.
• Enumeration types from RTL
• Set of values for each dimension is finite!!!!
Multi-Valued Logic
• Formally: (sometimes called anmv-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!!
Example “Truth Table”
• P1={0,1,2}, P2={0,1}
• Here “2” means the value 2 and not {0,1}

f(0,0) = 1 f(2,1) = 1

f(1,0) = 0 f(2,0) = *unspecified (don’t cares)

MV Function

on

off

Don’t care

Terminology
• Vertex:
• Cube:
• Containment:
• Implicant:
Terminology
• Onset minterm:
• Prime Implicant:
• Cover of F :
• Prime Cover of F:
Notation-MV Literals

Definition: A multi-valued literalis a binary logic function of the form

where

Definition: A cube can be written as the product of

MV-literals:

Notation-MV Literals
• If ci=Pi we may omit from the expression (since =1)
• Note analogy to two-valued case:
• Actually, multi-valued notation is superior to binary notation.
Example

Rows marked as a (b) form single mv-cube implicant

The following are cube covers of F. F2 is a prime cover

Positional Notation

Example: Cube1 P1={A,B,C,D}, P2={R,S} (Symbolic)

A B C D R SCube1: 1 1 0 0 1 0

Cube2: 1 1 1 1 0 1

• A cube does not depend on variable Xi if it has all 1’s in the set of columns associated with Xi (Cube2 does nor depend on X1).
• 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.
Positional Notation

Extension of Espresso notation

(value=0) (value=1)0 1  1 1 0  0

1 1  2

Example:

X1 X2 X3

C1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1

C2 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0

C3 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1

C4 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0

C5 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0

Positional Notation

X1 X2 X3

C1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1

C2 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0

C3 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1

C4 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0

C5 0 0 0 0 1 1 1 1 1 1 1 0 1 1 0

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)
Applications of Multi-Valued Logic

Theorem: minimizing a two-valued (n input) (m output) logic function gis equivalent to minimizing a single binary-output MV-logic function:

f : {0,1}  {0,1}  ...  {0,…,m-1}  {0,1}

Proof( sketch):Let g = {f0,…,fm-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} ):

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 fi(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.

Xf1 f2 f3 f4 f5 f6

g

x-cube 0 1 0 1 1 0

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
Example - after minimization

Prime and irredundant SOP of f:

(five cubes 1+2+3+4+5)

Equivalent to:

Example - after minimization

f0 f1 f2

Note: is not a prime of f0, but is a prime of f.

Similarly for .