ECE 667 Synthesis and Verification of Digital Circuits Boolean Functions Basics

1Boolean Functions

ECE 667ECE 667Synthesis and Verification

of Digital Circuits

Boolean Functions Basics

Maciej Ciesielski

[email protected]

Univ. of Massachusetts, Amherst

Slides adopted (with permission) from A. Kuehlmann, UC Berkeley 2003Slides adopted (with permission) from A. Kuehlmann, UC Berkeley 2003

Boolean Functions 2

OutlineOutline• Boolean function representation

– Boolean space, cubes, cover– SoP, truth tables– Canonicity

• Operations on Boolean functions– Cofactoring, Shannon expansion– Quantification

• Basic theorems– Quine’s theorem– Fundamental theorem, tautology

• Properties, special functions– Symmetry, unateness– Incompletely specified Boolean functions

• Two-level logic minimization– Recursive factorization (heuristic, espresso)– Exact method (Quine McCluskey)

Boolean Functions 3

The Boolean Space The Boolean Space BBnn

B = { 0,1}, B2 = {0,1} X {0,1} = {00, 01, 10, 11} , etc.

BB00

BB11

BB22

Karnaugh Maps: Boolean Spaces:

BB33

BB44

Boolean Functions 4

Boolean FunctionsBoolean Functions

1 2

1 2

1 1 2 2

1 2 1 2

Boolean Function: ( ) :

{0,1}

( , ,..., ) ;

- , ,... are

- , , , ,... are

- essentially: maps each vertex of to 0 or 1

Exampl

vari

e:

{(( 0, 0),0),(( 0,

ables

literals

1

n

nn i

n

f x B B

B

x x x x B x B

x x

x x x x

f B

f x x x x

1 2 1 2

),1),

(( 1, 0), 1),(( 1, 1),0)}x x x x

1x

2x

001

1

x2

x1

= on-set minterm (f = 1)= off-set minterm (f = 0)

Boolean Functions 5

On-set, Off-set, TautologyOn-set, Off-set, Tautology

1 1

1 0

1

0 1

- The of is { | ( ) 1} (1)

- The of is { | ( ) 0} (0)

- if , is the i.e. 1

- if ( ), is

Onset

Offset

tautology.

not satisfyabl , i.e. 0

- if ( ) ( ) , t

e

hen a

n

n

n

f x f x f f

f x f x f f

f B f f

f B f f f

f x g x for all x B f1

nd

- we say

are equiva

instea

le

nt

d of

g

f f

- literals: A is a variable or its negation , and represents a logic l functioniteral x x

Literal x1 represents the logic function f, where f = {x| x1 = 1}

Literal x1 represents the logic function g where g = {x| x1 = 0}

x1

x3

x2

f = x1

x1

x2

x3

f = x1

Notation: x’ = x


Boolean Functions 6

Set of Boolean FunctionsSet of Boolean Functions

• There are 2n vertices in input space Bn

• There are 22n distinct logic functions.

– Each subset of vertices is a distinct logic function: f Bn

x1

x2

x3

• Truth Table or Function Table:

X1x2x3 F

0 0 0 10 0 1 00 1 0 10 1 1 01 0 0 11 0 1 01 1 0 11 1 1 0

Boolean Functions 7

Boolean OperationsBoolean Operations - - AND, OR, COMPLEMENTAND, OR, COMPLEMENT

Given two Boolean functions:

f : Bn B

g : Bn B

• AND operation

f g = {x | f(x)=1 g(x)=1}

• The OR operation

f g = {x | f(x)=1 g(x)=1}

• The COMPLEMENT operation (^f or f’ )

f’ = {x | f(x) = 0}

Boolean Functions 8

Cofactor and QuantificationCofactor and QuantificationGiven a Boolean function:

f : Bn B, with the input variables (x1,x2,…,xi,…,xn)

• Positive Cofactor of function f w.r.t variable xi

fxi = f(xi=1)

• Negative Cofactor of f w.r.t variable xi

fxi’ = f(xi=0)

• Existential Quantification of function f w.r.t variable xi,

xi f = fxi fxi’

• Universal Quantification of function f w.r.t variable xi,

xi f = fxi fxi’

Boolean Functions 9

Representations of Boolean FunctionsRepresentations of Boolean Functions

• We need representations for Boolean Functions for two reasons:– to represent and manipulate the actual circuit we are “synthesizing”– as mechanism to do efficient Boolean reasoning

• Forms to represent Boolean Functions– Truth table– List of cubes: Sum of Products, Disjunctive Normal Form (DNF) – List of conjuncts: Product of Sums, Conjunctive Normal Form (CNF)– Binary Decision Tree, Binary Decision Diagram– Boolean formula– Boolean network

• Canonicity – which forms are canonical?

Boolean Functions 10

Boolean FormulaBoolean Formula

• A Boolean formula is defined as an expression with the following syntax:

formula ::= ‘(‘ formula ‘)’

| <variable>

| formula “+” formula (OR operator)

| formula “” formula (AND operator)

| ^ formula (complement)

Example:

f = (x1x2) + (x3) + ^^(x4 (^x1))

typically the “” is omitted and the ‘(‘ and ‘^’ are simply reduced by priority, e.g.

f = x1x2 + x3 + x4^x1


CubesCubes

• A cube is defined as the product (AND) of a set of literal functions (“conjunction” of literals).Example:

C = x1x’2x3

represents the following logic function

f = (x1=1)(x2=0)(x3=1)

Other examples of cubes:

x1

x2

x3

c = x1

x1

x2

x3

f = x1x2

x1

x2

x3

f = x1x2x3



Set of Cubes – SoP vs CoverSet of Cubes – SoP vs Cover

• A function can be represented by a sum of cubes:f = ab + ac + bc

Since each cube is a product of literals, this is a “sum of products” (SOP) representation

• A SOP can be thought of as a set of cubes F (a cover of f ). F = {ab, ac, bc}

• Definition: cover F of function f = set of implicants that cover all minterms of function f

• Cover is non-unique, e.g., F1={ab, ac, bc} and F2={abc, a’bc, ab’c, abc’}are some of possible covers of Boolean function

f = ab + ac + bc.


Cover minimizationCover minimization

c

a

bc ac

ab

b

= on-set minterm (f = 1)= off-set minterm (f = 0)= don’t care-set minterm (f = x)

• Two-level minimization seeks a minimum size cover (least number of cubes). Reason: minimize number of product terms in PLA

Note: each onset minterm is “covered” by at least one of the cubes and none of the offset minterms is covered.


PLAs - Multiple Output FunctionsPLAs - Multiple Output Functions

• A PLA is a function f : Bn Bm represented in SOP form• Each distinct cube appears only once in the AND-plane, and can

be shared by (multiple) outputs in the OR-plane, e.g., cube abc.

f2 f3f1

n=3, m=3

a a b b c c

abc f1f2f3

10- 1 - -

-11 1 - -

0-0 - 1 -

111 - 1 1

00- - - 1

Personality Matrix

AND plane

OR plane


Irredundant CubesIrredundant Cubes

bc ac

abc

a

b

bc

ac

not covered,so ab is irredundant

F \ {ab} f

• Definition: Let F = {c1, c2, …, ck} be a cover for f, i.e. f = ik=1 ci

A cube ci F is irredundant if F \ {ci} f

A cover is irredundant if all its cubes are irredundant.

Example: f = ab + ac + bc


Prime, Essential, IrredundantPrime, Essential, Irredundant

• Definition: A cube is prime if it is not contained in any other cube. A cover is prime if all its cubes are prime.

• Definition: A prime of f is essential if it contains a minterm that is not contained by any other prime.

Example: f = abc + b’d + c’d

is prime and irredundant.

abc is essential since abcd’ abc but not in the other cubes

abc

bd

cdda

cb

abcdabcd’


Prime and Irredundant CoversPrime and Irredundant Covers

Example: f = abc + b’d + c’d is prime and irredundant.

abc

bd

cdda

cb

abcdabcd’

• Why is abcd not an essential vertex of abc?

• What is an essential vertex of abc?

• What other cube is essential? What prime is not essential?


ImplicantsImplicants

• Formally: cube C f is an implicant of f– It is called implicant since C = 1 f = 1

– In an n-dimensional Boolean space Bn, an implicant with n literals is a minterm.

• Prime implicants• Essential implicants• Irredundant implicants

are defined similarly as for cubes.


Quine’s TheoremQuine’s Theorem

Importance of having prime cubes: • minimum SOP solution can be composed of prime

cubes only

Theorem (Quine): There exists a minimum cover that is prime

Given initial cover for F = (f,d,r), find a minimum cover G of primes where: f G f+d

G is a prime cover of F ; f = on-set, r = off-set, d = don’t care set


Shannon (Boole) CofactorsShannon (Boole) Cofactors

Let f : Bn B be a Boolean function, and x= (x1, x2, …, xn) the variables in the support of f.

The cofactor fa of f w.r.t literal a=xi or a=x’i is:

fxi (x1, x2, …, xn) = f (x1, …, xi-1, 1, xi+1,…, xn)

fx’i (x1, x2, …, xn) = f (x1, …, xi-1, 0, xi+1,…, xn

Computation of cofactors is a fundamental operation in Boolean reasoning

Shannon (Boole) expansion

f = x fx + x’ fx’


Generalized CofactorGeneralized Cofactor

• The generalized cofactor fC of f by a cube C is f with the fixed values indicated by the literals of C,

• Eg. if C= x1 x’4 x6

fC is just the function f restricted to the subspace

where x1 =x6 =1 and x4 =0.

• As a function, fC does not depend on x1,x4 or x6 anymore

(However, we still consider fC as a function of all n variables, it just happens to be independent of x1,x4 and x6).

• x1f fx1

Example: f = ac + a’c , af = ac, fa=c


Cofactor of CoverCofactor of Cover

Definition: The cofactor of a cover F is the sum of the cofactors of each of the cubes of F.

Given the cover The cofactor of the cover is simply

Note: If F={c1, c2,…, ck} is a cover of f, then Fc= {(c1)c, (c2)c,…, (ck)c}

is a cover of fc.

( ) ( )j jx i x

i

F x c

( ) { }ii j i

i i j

F x c c { c i }


Definition: The cofactor Cxj of a cube C with respect to a literal xj is

• C if xj and x’j do not appear in C

• C\{xj} if xj appears positively in C, i.e. xjC

• if xj appears negatively in C, i.e. xj’C

Example 1: C = x1 x’4 x6

Cx2 = C (x2 and x’2 do not appear in C )

Cx1 = x’4 x6 (x1 appears positively in C)

Cx4 = (x4 appears negatively in C)

Example 2: F = abc + b’d + c’d

Fb = ac + c’d

(Just drop b everywhere and delete cubes containing literal b)

Cofactor of CubesCofactor of Cubes


Fundamental TheoremFundamental Theorem

Theorem: Let c be a cube and f a function.

Then c f fc 1.

Proof. We use the fact that x fx = x f, and fx is independent of x.

If : Suppose fc 1. Then cf =fcc = c. Thus, c f.

ff

cc


Proof (cont.)Proof (cont.)

Only if. Assume c f

Then c cf = cfc. But fc is independent of literals i c.

If fc 1, then m Bn, fc(m)=0.

We will construct a m’ from m and c in the following manner:

mi’=mi, if xic and xic,

mi’=1, if xi c,

mi’=0, if xi c.

i.e., we made the literals of m’ agree with c, i.e. m’ c c(m’)=1

Also, fc is independent of literals xi,xi c fc(m’) = fc(m) = 0

fc(m’) c(m’)= 0

contradicting c cfc.

m

m= 000m’= 101

m’

C=xz


Application of Containment Test: Application of Containment Test: c F


Extra slidesExtra slides


Prime and Essential ImplicantsPrime and Essential Implicants

• Quine’s Theorem (Willard Quine, 1908 – 2000, Harvard Univ.):

– Boolean function can be implemented with only prime implicants (but other solutions exist)

– The number of such implicants is minimum

• Essential prime implicant– Prime implicant that covers a minterm that is not covered by any

other prime implicant


Essential Prime ImplicantsEssential Prime Implicants

• Example: F= (0,1,2,6,8,9,10,14)

• Essential implicants : – Implicant 1 = (0,1,8,9)

– Implicant 2 = (2,6,14,10)

• Non-essential implicant: (0,2,8,10)

1

1

1

1

1

1

1

1


Don’t CaresDon’t Cares: Incompletely Specified Functions: Incompletely Specified Functions

• Output of function is irrelevant for some inputs– Input might never occur– Input is invalid

• Function is incompletely specified– Multiple completely specified functions can implement it

• Karnaugh map is marked with ‘x’ for don’t cares– Output can be set to 1 or 0 (hence “don’t care”)– Choose most convenient output – Maximize block size

• Specification of don’t care conditions:– Function: F(w,x,y,z) = (1,3,7,11,15)– Don’t cares: d(w,x,y,z) = ∑(0,2,5)


Incompletely Specified FunctionsIncompletely Specified Functions

• Design a logic function that determines if a BCD digit is divisible by 3– Find minimum SOP logic implementation

• What are the possible inputs for this function?– Only (0000)2 (1001)2 are used– Other inputs are not valid

• What should we do with invalid inputs?– “Don’t care”– Output can take any value

• F(A,B,C,D) = AD + B’CD + BCD’ + A’B’C’D’

1

X

1

X

1

X

X

1

X

XA

B

C

D


Example with Don’t CaresExample with Don’t Cares

• Minimization example: – F(w,x,y,z) = (1,3,7,11,15) and d(w,x,y,z) = (0,2,5)

• Solution is non-unique

Date post:	31-Dec-2015
Category:	Documents
Upload:	gillian-thomas
View:	38 times
Download:	0 times

ECE 667 Synthesis and Verification of Digital Circuits Boolean Functions Basics

Documents