1Section 4.2Switching Algebra

Unate & Threshold Functions

Alfredo BensoPolitecnico di Torino, Italy

Alfredo.benso@polito.it

Unate Functions

Functions that can be implemented using a Single-Rail Logic with AND and OR gates.

Single-Rail Logic to minimize pins and interconnect.

Usually it is possible to use a NOT gate to realize inversion in Single-Rail logic, but thereare situations in which inversion is very expensive to implement (RAM address decoding circuits)

2Unate Functions: definitions

f is positive unate in a dependent variable xi if xidoes not appear in the sum-of-products representation

f is negative unate in a dependent variable xi if xidoes not appear in the sum-of-products representation

f is vacuous in a dependent variable xi if neither xinor xi appears in the sum-of-products representation (otherwise it is essential)

f is mixed or binate in variable xi if it is not possible to write a sum-of-products representation in which x i and xi do not both appear

Unate functions

Example F(w, x, y, z)= wx+wz

Essential: w, x, z Vacuous: y Positive: x, y Negative: z, y Mixed: w

3Problem

Example F(w, x, y, z)= x + yz + xyw+wz

Essential: Vacuous: Positive: Negative: Mixed:

Problem

Example F(w, x, y, z)= x + yz + xyw+wz

Essential: w, x, y, z Vacuous: - Positive: x Negative: y Mixed: z

4Classification of functions

Monotone increasing (or Frontal or Positive) function: it is positive in all its variables it can be represented with AND and OR gates ONLY - (no inverters)

Monotone decreasing (or Backal or Negative)function: it is negative in each of its variables it can be represented in SOP form with ALL complemented literals

Unate function: it is positive or negative in each of its variables

Examples

f(x,y,z)=x y + y z Monotone Increasing (Frontal)

g(a,b,c)= ac +b c Monotone Decreasing (Backal)

k(A, B, C)= A B + A C Unate Function

h(X, Y, Z)= X Y + Y Z Unate in variables X and Z Binate in variable Y

5Threshold Functions

Definition Let (w1, w2, , wn) be an n-tuple of real-

numbered weights and t be a real number called the threshold. Then a threshold function, f, is defined as:

1 1 2 21, ...0,

n nw x w x w x tfotherwise

+ + + =

x1x2 w1

xn

w2

wn

ft

Threshold Functions

Example (2-valued logic) A 3-input majority function has a value of 1 iff 2 or

more variables are 1 Of the 16 Switching Functions of 2 variables, 14 are

threshold functions (but not necessarily majority functions)

w1 = w2 = -1 t = -0.5 f = x1 x2

All threshold functions are unate Majority functions are threshold functions where

n = 2m+1 t = m+1 w1 = w2 == wn = 1, majority functions equal 1 iff

more variables are 1 than 0 Majority functions are totally symmetric and monotone

increasing

6Problem

A threshold gate realization is to be found for the function f(x,y,z)=x(y+z)

Solution

From 5 and 4 wz>0 From 5 and 1 wx>0 From 6 and 4 wy>0 wx+ wy t > wy+ wz wx+ wz t > wx

wx= 2 wy = wz = 1 t=2.5

x y z f ai (0) 0 0 0 0 0 (1) 0 0 1 0 wz < t (2) 0 1 0 0 wy < t (3) 0 1 1 0 wy+ wz < t (4) 1 0 0 0 wx< t (5) 1 0 1 1 wx + wz t (6) 1 1 0 0 wx + wy < t (7) 1 1 1 1 wx + wy + wy t

7Problem

Is f(x,y) = xy + xy a threshold function?

No, since there is no solution to this set of inequalities.

However, two threshold functions could be used to achieve the same result.

x y f a i 0 0 0 0 0 1 1 wy t 1 0 1 wx t 1 1 0 wx + wy < t

Relations Among FunctionsAll Functions

Unate

Monotone

Threshold

Majority