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EE219BEE219BLogic SynthesisLogic Synthesis

R. K. Brayton R. K. Brayton

University of CaliforniaUniversity of California

BerkeleyBerkeley

Spring 2000Spring 2000

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IntroductionIntroduction

MotivationMotivation• Commercial success - used almost Commercial success - used almost

everywhere VLSI is doneeverywhere VLSI is done• But more general – But more general –

– Generally - discrete functions of Generally - discrete functions of discrete valued variablesdiscrete valued variables

• Body of useful and general Body of useful and general techniques techniques – can be applied to other areascan be applied to other areas

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ApplicationsApplications

FoundationFoundation for for• Combinational and sequential logic synthesisCombinational and sequential logic synthesis• Automatic test vector generationAutomatic test vector generation• Timing and false paths analysisTiming and false paths analysis• Formal verificationFormal verification• Asynchronous synthesisAsynchronous synthesis• Automata theoryAutomata theory• Optimal clocking schemesOptimal clocking schemes• Hazard analysisHazard analysis• Power estimationPower estimation• General combinatorial problemsGeneral combinatorial problems

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Outline of ClassOutline of ClassClass notes: (see class web page)Class notes: (see class web page)• IntroductionIntroduction

– Logic functions and their representationLogic functions and their representation– Unate recursive paradigm,Unate recursive paradigm,– Two Level logic minimization (ESPRESSO) Two Level logic minimization (ESPRESSO) – Quine McCluskey methodQuine McCluskey methodMidtermMidterm

• Multi-level logic synthesisMulti-level logic synthesis– Introduction (Boolean networks, factored forms)Introduction (Boolean networks, factored forms)– DivisionDivision– SimplificationSimplification

– TestingTesting– Technology mappingTechnology mapping

• Advanced technology mapping Advanced technology mapping

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Outline of ClassOutline of Class

• Multi-level logic synthesis (contd.)Multi-level logic synthesis (contd.)– Delay analysis (true and false paths)Delay analysis (true and false paths)– Timing optimizationTiming optimization– Constant delay paradigm (sizing)Constant delay paradigm (sizing)

midtermmidterm

• Sequential logic synthesisSequential logic synthesis– Introduction (FSM networks) Introduction (FSM networks) – Retiming and resynthesisRetiming and resynthesis– Asynchronous synthesisAsynchronous synthesis– Node minimization?Node minimization?– ** State minimization **** State minimization **

• Final project presentations and reportFinal project presentations and report

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The Boolean n-cube BThe Boolean n-cube Bnn

• B = { 0,1}B = { 0,1}• BB22 = {0,1} X {0,1} = {00, 01, 10, 11} = {0,1} X {0,1} = {00, 01, 10, 11}

B0B1 B2

B3B4

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Boolean FunctionsBoolean Functions ff(x) : B(x) : Bnn B B B = {0, 1}, x = (xB = {0, 1}, x = (x11, x, x22, …, x, …, xnn))

• xx11, x, x22, … are , … are variablesvariables• xx11, x, x11, x, x22, x, x22, … are , … are literalsliterals• each vertex of Beach vertex of Bnn is mapped to 0 or 1 is mapped to 0 or 1• the the onset onset of of ff is is {x|{x|ff(x)=1} =(x)=1} =f f 1 1 = = f f -1-1(1) (1) • the the offset offset of of ff is is {x|{x|ff(x)=0} =(x)=0} =f f 0 0 = = f f -1-1(0)(0)• if if f f 11 = B = Bnn, , ff is the is the tautologytautology, i.e. , i.e. ff 1 1 • if if f f 00 = B = Bnn ( (f f 11 = = ), ), ff is is not satisfiablenot satisfiable• if f(x) = g(x) for all x if f(x) = g(x) for all x BBnn, then f and g are, then f and g are

equivalentequivalentWe write simply We write simply ff instead of instead of f f 11

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LiteralsLiteralsA literal is a variable or its negationA literal is a variable or its negation

y, yy, y

It represents a It represents a logic functionlogic function

Literal xLiteral x11 represents the logic function f, where represents the logic function f, where

ff = {x| x = {x| x11 = 1} = 1}

f = xf = x1 1 f = xf = x1 1

xx1 1 xx11

Literal xLiteral x11 represents logic function g where represents logic function g where

gg = {x| x = {x| x11 = 0} = 0}

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Boolean FormulasBoolean Formulas

Boolean formulasBoolean formulas can be represented by can be represented by formulas defined as catenations offormulas defined as catenations of

• parentheses ( , )parentheses ( , )• literals x, y, z, x, y, zliterals x, y, z, x, y, z• Boolean operators Boolean operators (OR), (OR), XX (AND) (AND)• complementation, e.g. x + ycomplementation, e.g. x + y

ExamplesExamples

f = xf = x11 XX x x22 + x + x11 XX x x22 = (x = (x11+x+x22) ) X X (x(x11+x+x22))

h = a + b X c = a h = a + b X c = a XX (b + c) (b + c)

We usually replace We usually replace XX by catenation, e.g. a by catenation, e.g. a X X bbabab

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Logic functions Logic functions

• There are 2There are 2nn vertices in input space B vertices in input space Bnn

• There are 2There are 222nn distinct logic distinct logic functionsfunctions. .

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

ff B Bnn

111111

000000

xx33

xx11

xx22

000 000 11001 001 00010 010 11011 011 00100 100 11101 101 00110 110 11111 111 00

““truth table”truth table”

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Logic FunctionsLogic Functions

• However, there are infinite However, there are infinite number of logic number of logic formulasformulas

f = x + yf = x + y

= xy + xy + xy= xy + xy + xy

= xx + xy + y= xx + xy + y

= (x + y)(x + y) + xy= (x + y)(x + y) + xy

• Synthesis = Find the best Synthesis = Find the best formula (or “representation”)formula (or “representation”)

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

f : Bf : Bnn B B

g : Bg : Bnn B B • AND - fg = h such thatAND - fg = h such that

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

• OR - f + g = h such thatOR - f + g = h such that

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

• COMPLEMENT - f = h such thatCOMPLEMENT - f = h such that

h = {x| f(x) = 0}h = {x| f(x) = 0}

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CubesCubes

• The AND of a set of literal functions The AND of a set of literal functions (“conjunction” of literals) is a (“conjunction” of literals) is a cubecube

C = xyC = xy

is a cubeis a cube

C = (x=1)(y=0)C = (x=1)(y=0)

x =1x =1 y =0y =0 xyxy

zz

yy

xx

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CubesCubes

• If C If C f, C a cube, then C is an f, C a cube, then C is an implicantimplicant of f.of f.

• If C If C B Bnn, and C has k literals, then |C| has, and C has k literals, then |C| has

22n-kn-k vertices. vertices.

Example 1Example 1 C = x y C = x y B B33..

k = 2 , n = 3.k = 2 , n = 3.

C = {100, 101}.C = {100, 101}.

|C| = 2 = 2|C| = 2 = 23-2.3-2.

• If k=n, the cube is a If k=n, the cube is a minterm.minterm.

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Representation of Representation of Boolean functionsBoolean functions

• The truth table of a function f : BThe truth table of a function f : Bnn B is a B is a tabulation of its value at each of the 2tabulation of its value at each of the 2n n

vertices of Bvertices of Bnn. . • For For

f = abcd + abcd + abcd + abcd + abcd + abcd + abcd + f = abcd + abcd + abcd + abcd + abcd + abcd + abcd + abcdabcd

the truth table is the truth table is

This is intractable for large nThis is intractable for large n

(but (but canonicalcanonical))

Canonical means that if two functions are the Canonical means that if two functions are the same, then the canonical representations of same, then the canonical representations of each are isomorphic.each are isomorphic.

abcd fabcd f0 0000 00 0000 01 0001 11 0001 12 0010 02 0010 03 0011 13 0011 14 0100 04 0100 05 0101 15 0101 16 0110 06 0110 07 0111 07 0111 0

abcd fabcd f 8 1000 08 1000 0 9 1001 19 1001 110 1010 010 1010 011 1011 111 1011 112 1100 012 1100 013 1101 113 1101 114 1110 114 1110 115 1111 115 1111 1

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Sum-of-Products Sum-of-Products Representation - SOPRepresentation - SOP

• A function can be represented by a sum of A function can be represented by a sum of cubes (products):cubes (products):

f = ab + ac + bcf = ab + ac + bc

Since each cube is a product of literals, this Since each cube is a product of literals, this is a “sum of products” representationis a “sum of products” representation

• A SOP can be thought of as a set of cubes FA SOP can be thought of as a set of cubes FF = {ab, ac, bc} = CF = {ab, ac, bc} = C

• A set of cubes that represents f is called a A set of cubes that represents f is called a covercover of f. F={ab, ac, bc} is a cover of of f. F={ab, ac, bc} is a cover of f = ab + ac + bc. f = ab + ac + bc.

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SOPSOP

• Covers (SOP’s) can efficiently represent Covers (SOP’s) can efficiently represent many logic functions (i.e. for many, there many logic functions (i.e. for many, there exist small covers).exist small covers).

• Two-level minimization seeks the minimum Two-level minimization seeks the minimum size cover (least number of cubes)size cover (least number of cubes)

bcbc

acacabab

cc

aa

bb

= onset minterm= onset mintermNote that each onset Note that each onset minterm is “covered” minterm is “covered” by at least one by at least one of the cubes, andof the cubes, and

covers no offset mintermcovers no offset minterm..

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IrredundantIrredundant

• Let F = {cLet F = {c11, c, c22, …, c, …, ckk} be a cover for f.} be a cover for f.

f = f = iikk=1=1 c cii

A cube cA cube cii F is F is irredundantirredundant if F\{c if F\{cii} } f f

Example 2:Example 2: f = ab + ac + bc f = ab + ac + bcbcbc

acacabab

cc

aa

bb

bcbc

acacNot coveredNot covered

F\{ab} F\{ab} f

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PrimePrime

• A A literalliteral jj of cube of cube ccii FF ( = ( =f f ) is ) is primeprime if if

((F F \\ {{ccii }) }) { {c’c’ii } } ff

where where c’c’ii is is ccii with literal with literal jj of of ccii deleted. deleted.

• A A cubecube of F is prime if all its literals are of F is prime if all its literals are primeprime..

Example 3Example 3

f = ab + ac + bcf = ab + ac + bc

cci i = ab; c’= ab; c’i i = a (literal b deleted)= a (literal b deleted)

F \ {cF \ {ci i } } {c’ {c’i i } = a + ac + bc } = a + ac + bc

bcbc

acacaa

cc

aa

bb

Not equal to f sinceNot equal to f sinceoffset vertex is coveredoffset vertex is covered

F=ac + bc + a =F=ac + bc + a =F \{ci } {c’i }

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Prime and Irredundant Prime and Irredundant CoversCovers

• Definition 1 Definition 1 A cover is prime A cover is prime (irredundant) if all its cubes are prime (irredundant) if all its cubes are prime (irredundant). (irredundant).

• Definition 2 Definition 2 A prime of f is A prime of f is essentialessential (essential prime) if there is a minterm (essential prime) if there is a minterm (essential vertex) in that prime but in no (essential vertex) in that prime but in no other prime. other prime.

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Prime and Irredundant Prime and Irredundant CoversCovers

Example 4Example 4f = abc + bd + cd is prime and irredundant.f = abc + bd + cd is prime and irredundant.

abc is abc is essentialessential since abcd since abcdabc, but not in bd abc, but not in bd or cd or ador cd or ad

Why is abcd not an essential vertex of abc?Why is abcd not an essential vertex of abc?

What is an essential vertex of abc?What is an essential vertex of abc?

What other cube is essential? What What other cube is essential? What primeprime is not is not essential?essential?

abcabc

bdbd

cdcdddaa

ccbb

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PLA’s - Multiple Output PLA’s - Multiple Output Boolean FunctionsBoolean Functions

• A PLA is a function f : BA PLA is a function f : Bnn B Bmm

represented in SOP form:represented in SOP form:

aa

aa

aa

aa

aa

~b~b cc dd

bb

~b~b

~b~b

~b~b

~c~c

~c~c

cc

cc

~d~d

~d~d

dd

~d~d

f2f2 f3f3f1f1

n=4, m=3n=4, m=3

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PLA’s (contd)PLA’s (contd)

• Each distinct cube appears just Each distinct cube appears just once in the AND-plane, and can be once in the AND-plane, and can be shared by (multiple) outputs in the shared by (multiple) outputs in the OR-plane, e.g., cube a~bc~d.OR-plane, e.g., cube a~bc~d.

• Extensions from single output to Extensions from single output to multiple output minimization multiple output minimization theory are straightforward.theory are straightforward.

• Multi-level logic can be viewed Multi-level logic can be viewed mathematically as a connection of mathematically as a connection of single output functions.single output functions.

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Shannon (Boole) cofactorsShannon (Boole) cofactors

Let Let f : Bf : Bnn B be a Boolean function, B be a Boolean function, and x= and x= (x(x11, x, x22, …, x, …, xnn) the variables in the ) the variables in the support of f. The cofactor fsupport of f. The cofactor faa of f by a of f by a

literal a=xliteral a=xii or a=x or a=xii is is

ffxxii (x (x11, x, x22, …, x, …, xnn) = f (x) = f (x11, …, x, …, xi-1i-1, 1, x, 1, xi+1i+1,…, x,…, xnn))

ffxxii (x (x11, x, x22, …, x, …, xnn) = f (x) = f (x11, …, x, …, xi-1i-1, 0, x, 0, xi+1i+1,…, x,…, xnn) )

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Shannon (Boole) CofactorShannon (Boole) Cofactor• The The cofactorcofactor f fCC of f by a of f by a cubecube C is just f with C is just f with

the fixed values indicated by the literals of C, the fixed values indicated by the literals of C, e.g. if C=xe.g. if C=xi i xxjj, then x, then xi i =1, and x=1, and xj j =0.=0.

• If C= xIf C= x1 1 xx4 4 xx66, f, fCC is just the function f restricted is just the function f restricted to the subspace where xto the subspace where x1 1 =x=x6 6 =1 and x=1 and x4 4 =0.=0.

• As a function, fAs a function, fCC does not depend on x does not depend on x11,x,x4 4 or or xx66

(However, we still consider f(However, we still consider fCC as a function of as a function of all all nn variables, it just happens to be variables, it just happens to be independent of xindependent of x11,x,x4 4 and xand x66).).

• xx11f f f fxx11

Example:Example: f= ac + a c , af = ac, f f= ac + a c , af = ac, faa=c=c

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Fundamental TheoremFundamental Theorem

Theorem 1Theorem 1 Let c be a cube and f a Let c be a cube and f a function. Then c function. Then c f f f fcc 1. 1.

ProofProof. We use the fact that xf. We use the fact that xfxx = xf, and f = xf, and fx x is is independent of x.independent of x.

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

c c f. f.

ffcc

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Proof (contd)Proof (contd)Only if.Only if. Assume c Assume c f f

Then c Then c cf = cf cf = cfcc. But f. But fcc is independent of is independent of literals l literals l c. If f c. If fcc 1, then 1, then m m B Bnn, f, fcc(m)=0. (m)=0.

Let mLet mii’=m’=mii, if x, if xiic and xc and xiic. c.

or if mor if mii=0, x=0, xi i c c

or mor mii=1, x=1, xi i c. c.

mmii’=m’=mii otherwise. otherwise.

i.e. we make the literals of m’ agree with c, i.e. i.e. we make the literals of m’ agree with c, i.e. m’ m’ c. But then f c. But then fcc(m’) = f(m’) = fcc(m) = 0,(m) = 0,

Hence, c(m’)=1 Hence, c(m’)=1

and fand fcc(m’) c(m’)= 0,(m’) c(m’)= 0,

contradicting c contradicting c cf cfcc..

m= 000m= 000m’= 101m’= 101

mm

m’m’

C=xzC=xz

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End of Lecture 1End of Lecture 1

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Cofactor of CoversCofactor of CoversDefinition 3Definition 3 The The cofactor of a covercofactor of a cover F is the sum F is the sum

of the cofactors of each of the cubes of F.of the cofactors of each of the cubes of F.

Note:Note: If F={c If F={c11, c, c22,…, c,…, ckk} is a cover of f, then F} is a cover of f, then Fcc= = {(c{(c11))cc, (c, (c22))cc,…, (c,…, (ckk))cc} is a cover of f} is a cover of fcc..

Suppose F(x) is a cover of f(x), i.e.Suppose F(x) is a cover of f(x), i.e.

Then for 1Then for 1jjn,n,

is a cover of fis a cover of fxxjj(x)(x)

( ) { }ii j i

i i j

F x c c

( ) ( )j jx i x

i

F x c

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Definition 4Definition 4 The cofactor C The cofactor Cxxjj of a cube C of a cube C

with respect to a literal xwith respect to a literal xjj is is

• C if xC if xjj and x and xjj do not appear in C do not appear in C

• C\{xC\{xjj} if x} if xjj appears positively in C, i.e. appears positively in C, i.e. xxjjCC

• if xif xjj appears negatively in C, i.e. x appears negatively in C, i.e. xjjCC

Example 5Example 5

If C = xIf C = x1 1 xx44 x x66,,

CCxx22 = C (x = C (x22 and x and x22 do not appear in C ) do not appear in C )

CCxx11 = x = x44 x x6 6 (x(x11 appears positively in C) appears positively in C)

CCxx4 4 = = (x (x44 appears negatively in C) appears negatively in C)

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Example 6Example 6F = abc + bd + cdF = abc + bd + cd

FFbb = ac + cd = ac + cd

(Just drop b everywhere and throw away (Just drop b everywhere and throw away cubes containing literal b)cubes containing literal b)

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Shannon ExpansionShannon Expansion

f : Bf : Bnn B BTheorem 2Theorem 2

Theorem 3Theorem 3 F is a cover of f. Then F is a cover of f. Then

We say that f (F) is We say that f (F) is expanded aboutexpanded about x xii. x. xii is is called the called the splitting variablesplitting variable..

i i

ii x xf x f x f

i iii x x

F x F x F

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Example 7Example 7

=>=>

Cube Cube bcbc got split into two cubes got split into two cubes

( ) ( )a a

F ab ac bc

F aF aF a b c bc a bc

ab ac abc abc

cc

aa

bbcc

aa

bb

bcbc

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Cover MatrixCover Matrix

We sometimes use matrix notation to We sometimes use matrix notation to represent a cover:represent a cover:

Example 8Example 8 F = ac + cd =F = ac + cd =

a b c da b c d a b c da b c dacac 1 2 1 2 or 1 - 1 - 1 2 1 2 or 1 - 1 -

cdcd 2 2 0 1 - - 0 1 2 2 0 1 - - 0 1

Each row represents a cube. 1 means that the Each row represents a cube. 1 means that the positive literal appears in the cube, 0 the positive literal appears in the cube, 0 the negative. The 2 (or -) here represents that negative. The 2 (or -) here represents that the variable does the variable does not appearnot appear in the cube. It in the cube. It also represents both 0 and 1 values.also represents both 0 and 1 values.

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Example 9Example 9ac = 1 2 0 2 = {1, {0,1}, 0, {0,1}}ac = 1 2 0 2 = {1, {0,1}, 0, {0,1}}

= {1000, 1100, 1001, 1101}= {1000, 1100, 1001, 1101}

2 is sometimes called “input don’t care”, 2 is sometimes called “input don’t care”, but this is confusing so we won’t use the but this is confusing so we won’t use the term. term.

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Incompletely Specified Incompletely Specified FunctionsFunctions

FF = (f, d, r) : B = (f, d, r) : Bnn {0, 1, *} {0, 1, *}where * represents “don’t care”. where * represents “don’t care”.

(Sometimes we use 2 in place of *)(Sometimes we use 2 in place of *)

• f = onset function - f = onset function - f(x)=1 f(x)=1 FF(x)=1 (x)=1 • r = offset function - r = offset function - r(x)=1 r(x)=1 FF(x)=0 (x)=0 • d = don’t care function - d = don’t care function - d(x)=1 d(x)=1 FF(x)=*(x)=*

(f,d,r) forms a (f,d,r) forms a partitionpartition of B of Bnn. i.e.. i.e.• f + d + r = Bf + d + r = Bnn

• fd =fd = fr = dr = fr = dr = (pairwise disjoint) (pairwise disjoint)

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A completely specified function g is a A completely specified function g is a covercover for for FF=(f,d,r) if=(f,d,r) if

ff g g f+d f+d

((Thus gr = Thus gr = ). Thus, if x). Thus, if xd (i.e. d(x)=1), d (i.e. d(x)=1), then g(x) can be 0 or 1, but if xthen g(x) can be 0 or 1, but if xf, then f, then g(x)=1 and if xg(x)=1 and if xr, then g(x)=0.r, then g(x)=0.

(We “don’t care” which value g has at x(We “don’t care” which value g has at xd) d)

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Primes of Incompletely Primes of Incompletely Specified FunctionsSpecified Functions

Definition 5Definition 5 A cube c is A cube c is primeprime of of FF=(f,d,r) if =(f,d,r) if ccf+d (an implicant of f+d), and no other f+d (an implicant of f+d), and no other implicant (of f+d) contains c, i.e.implicant (of f+d) contains c, i.e.

(i.e. it is simply a prime of f+d)(i.e. it is simply a prime of f+d)

Definition 6Definition 6 Cube c Cube cjj of cover F={c of cover F={cii} is } is redundantredundant if f if f F\{c F\{cjj}. Otherwise it is }. Otherwise it is irredundantirredundant..

NoteNote that c that cf+d f+d cr = cr =

ccdfcc ~,~,~

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Example:Logic MinimizationExample:Logic Minimization

ConsiderConsider FF(a,b,c)=(f,d,r), where f={abc, (a,b,c)=(f,d,r), where f={abc, abc, abc} and d ={abc, abc}, and the abc, abc} and d ={abc, abc}, and the sequence of covers illustrated below:sequence of covers illustrated below:

FF11= = abc + abc+ abc

abc is redundanta is prime

FF33= a+= a+abc Expand Expand abc bc

Expand abcExpand abcaa

FF22= a+= a+abc + abc

FF44= a= a+bc

offoff

onon

Don’t Don’t carecare

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Checking for Prime and Checking for Prime and IrredundantIrredundant

Let G={cLet G={cii} be a cover of } be a cover of FF=(f,d,r). Let D be a =(f,d,r). Let D be a cover for d.cover for d.

• cciiG is G is redundantredundant iff iff

ccii (G\{c(G\{cii})})D D G Gi i (1)(1)

(Since c(Since cii G Gi i and fand fGGf+d then cf+d then cii c ciif+cf+ciidd

and cand ciiff G\{c G\{cii}. Thus f}. Thus f G\{c G\{cii}.)}.)

• A literal l A literal l c cii is is primeprime if (c if (cii\{ l }) \{ l }) ( = (c( = (cii))ll ) ) is is not an implicant of not an implicant of FF..

• A cube cA cube cii is a prime of is a prime of FF iff all literals l iff all literals l c cii are prime.are prime.

Literal l Literal l c cii is not prime is not prime (c (cii))ll f+d f+d (2)(2)

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Note:Note: Both tests Both tests (1)(1) and and (2)(2) can be can be checked by checked by tautologytautology::

1)1) (G(Gii))ccii 1 (implies c 1 (implies cii redundant) redundant)

2)2) (F(FD)D)((cci)i)ll 1 (implies l not prime) 1 (implies l not prime)

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ExampleExampleF={abc, abc, abc}F={abc, abc, abc}

D={ac}D={ac}

R={ab,ac}R={ab,ac}

Expand abc Expand abc ab= (c ab= (cii))ll = (abc) = (abc)cc

Check ab Check ab f+d f+d (f+d) (f+d)abab = 1 = 1

• (f+d)(f+d)abab = c +c = c +c 1 1 OKOK

Expand abExpand ab a. Check a a. Check a f+d f+d

• (f+d)(f+d)aa = bc + bc +c = bc + bc +c 1 1 OKOK

F= a + abc + abcF= a + abc + abc

Check if abc is redundantCheck if abc is redundant• abc abc a+ abc + ac = F\{abc} a+ abc + ac = F\{abc} D D

• (a+ abc + ac )(a+ abc + ac )abcabc = 1 = 1 OKOK

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Cover is now F = a + abc and D = ac Cover is now F = a + abc and D = ac

Check if abc is redundantCheck if abc is redundant

abc abc a+ ac = F\{abc } a+ ac = F\{abc } D D

• (a+ ac )(a+ ac )abcabc = = NOT REDUNDANTNOT REDUNDANT