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# The Fanout Structure of Switching Functions PowerPoint PPT Presentation

The Fanout Structure of Switching Functions. Author: John P. Hayes Speaker: Johnny Lee. Outline. Introduction Notation and Background Material Fanout-Free functions Concluding Remarks. Introduction. Fanout-free circuits are easy to test and require very few test patterns

The Fanout Structure of Switching Functions

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## The Fanout Structure of Switching Functions

Author: John P. Hayes

Speaker: Johnny Lee

### Outline

• Introduction

• Notation and Background Material

• Fanout-Free functions

• Concluding Remarks

### Introduction

• Fanout-free circuits are easy to test and require very few test patterns

• Properties of functions that can be implemented without using fanout are investigated

### Outline

• Introduction

• Notation and Background Material

• Fanout-Free functions

• Concluding Remarks

### Theorem 1

• f(X) is unate

•  f(X) can be written as a SOPs expression in which no variable appears both complemented and uncomplemented

•  If some prime implicant of f(X) contains xe, then no prime implicant contains xe’

•  f’(X) is unate

### Outline

• Introduction

• Notation and Background Material

• Fanout-Free functions

• Concluding Remarks

### Fanout-Free Functions

• Definition 1. A single-output network N is a fanout-free network if every line in N is connected to an input line of at most one gate

• Definition 2. The constant functions 0 and 1 and the 1-variable functions x and x’ are fanout-free functions.

### Fanout-Free Functions (cont’d)

• Theorem 2. A function is fanout-free  it can be realized by a fanout-free network

• Theorem 3. Every fanout-free function is unate (proved by induction on |X|)

• Unateness is not a sufficient condition for a function to be fanout-free

### Fanout-Free Functions (cont’d)

• The unate function f=x1x2+x2x3+x1x3 is not fanout-free

x1

x2

f

x3

### Fanout-Free Functions (cont’d)

• Definition 3. xi≠ xj. xi is adjacent to xj if f(ai)=f(aj) for some pair of constants ai and aj. It is denoted by =a

• Adjacency is a reflexive and symmetric relation

• Lemma 1. Adjacency is transitive

• Adjacency is an equivalence relation

### Theorem 4

Let the variables of f(X) be partitioned into blocks X1,X2,…,Xm by the adjacency relation.

There exists a set of m elementary functions φ1(X1), φ2(X2), …, φm(Xm) and an m-variable function F such that f(X)=F(φ1(X1), φ2(X2), …, φm(Xm) )

### Procedure 1

• To determine if f(X) is fanout-free, and to find a fanout-free realization if one exists

x1

φ11

x2

x3

φ21

φ13

x6

x4

φ12

φ22

f

x5

φ1(X1)

X1

G1

φ2(X2)

N

X2

G2

f(X)

φm(Xm)

Xm

Gm

### Characterization of Fanout-Free Functions

• Definition 4. xi≠xj. xi masks xj if f(ai,0j)=f(ai,1j) for some constant ai. (denoted by xi→xj)

• Lemma 2. The masking operation is transitive

• Lemma 3. xi≠xj. xi =axj  xi→xj and xj→xi

### Characterization of Fanout-Free Functions (cont’d)

• Definition 5. Let Xi and Xj be disjoint subsets of the variables of f(X). Xi→Xj if there is an assignment Ai to Xi and a variable xj∈Xj such that f(Ai)=f(Ai,0j)=f(Ai,1j)

• Definition 6. f(X) has the masking property if for every proper subset Xi, Xi →(X-Xj)

• Lemma 4. If f(X) has the masking property and |X|≥2, then at least two distinct variables of X are adjacent

• Theorem 5. f(X) is fanout-free f(X) has the masking property

### Outline

• Introduction

• Notation and Background Material

• Fanout-Free functions

• Concluding Remarks

### Concluding Remarks

• A procedure is proposed to determine if a function is fanout-free and realize a fanout-free function if one exists

• Two relations, adjacency and masking, are used to characterize fanout-free functions