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

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The Fanout Structure of Switching Functions

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The fanout structure of switching functions

The Fanout Structure of Switching Functions

Author: John P. Hayes

Speaker: Johnny Lee


Outline

Outline

  • Introduction

  • Notation and Background Material

  • Fanout-Free functions

  • Concluding Remarks


Introduction

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


Outline1

Outline

  • Introduction

  • Notation and Background Material

  • Fanout-Free functions

  • Concluding Remarks


Notation and background material

Notation and Background Material


Notation and background material cont d

Notation and Background Material (cont’d)


Notation and background material cont d1

Notation and Background Material (cont’d)


Notation and background material cont d2

Notation and Background Material (cont’d)


Theorem 1

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


Outline2

Outline

  • Introduction

  • Notation and Background Material

  • Fanout-Free functions

  • Concluding Remarks


Fanout free functions

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

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 d1

Fanout-Free Functions (cont’d)

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

x1

x2

f

x3


Fanout free functions cont d2

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

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) )


Theorem 4 proof

Theorem 4 Proof


Procedure 1

Procedure 1

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


Procedure 1 cont d

Procedure 1 (cont’d)


Procedure 1 cont d1

Procedure 1 (cont’d)


Procedure 1 cont d2

Procedure 1 (cont’d)

x1

φ11

x2

x3

φ21

φ13

x6

x4

φ12

φ22

f

x5


Review of theorem 4

Review of Theorem 4

φ1(X1)

X1

G1

φ2(X2)

N

X2

G2

f(X)

φm(Xm)

Xm

Gm


Characterization of fanout free functions

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

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


Outline3

Outline

  • Introduction

  • Notation and Background Material

  • Fanout-Free functions

  • Concluding Remarks


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


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