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1ECE 667 - Synthesis & Verification - Lecture 11

ECE 697B 667ECE 697B 667Spring 2007Spring 2007

Synthesis and Verification

of Digital Systems

Unate Recursive Paradigm


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ECE 667 - Synthesis & Verification - Lecture 11 2

Shannon ExpansionShannon Expansion

f : BnBShannon Expansion:

Theorem: F is a cover of f. Then

We say that f(F) is expanded aboutxi.

xi is called the splitting variable.

i i

ii x x f x f x f

i iii x x

F x F x F %


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Example

Cube bcist split into two cubes

( ) ( )a a

F ab ac bc

F aF aF a b c bc a bc

ab ac abc abc

%

c

a

bc

a

bc

ac

ab

Shannon Expansion (cont.)Shannon Expansion (cont.)


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List of Cubes (Cover Matrix)List of Cubes (Cover Matrix)

We often use matrix notation to represent a cover:

Example: F = ac + cd + bcd

a b c d a b c d

a c 1 2 1 2 1 - 1 - cd 2 2 0 1 or - - 0 1

bcd 2 1 1 0 - 1 1 0

Each row represents a cube

1 means that the positive literal appears in the cube

0 means that the negative literal appears in the cube

The 2 (or -) represents that the variable doesnot appearin the cube.

Finding factors from matrix representation is easy.


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Definition: A function f : Bn

B issymmetricwith respect tovariablesxi andxjiff

f(x1,,xi,,xj,,xn ) = f(x1,,xj,,xi,,xn)

Definition: A function f : BnB istotally symmetric iffanypermutation of the variables in f does not change the function

Some Special FunctionsSome Special Functions

i j i j x x x xf f

Symmetry can be exploited in searching the binary decision tree

because:

- That means we can skip one of four sub-cases

- used in automatic variable ordering for BDDs


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Definition: A function f: Bn

B ispositive unateinvariablexiiff

This is equivalent tomonotone increasinginxi:

for all minterm pairs (m-, m+) where

For example, m-3=1001, m+3=1011 (where i =3)

ii xx f f

)()( + mfmf

,

0

1

j j

i

i

m m j i

m

m

Unate FunctionsUnate Functions


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Similarly fornegative unate

monotone decreasing:

A function isunateinxi if it is either positive unate or negative unate inxi.

Definition: A function isunateif it is unate in each variable.

Definition: A cover F ispositive unateinxi iffxicj for all cubes cjF

i ix x f f

)()( + mfmf

Unate FunctionsUnate Functions


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c

ba

mc+

mc-

fis positive unate in a,b :

f(ma+) f(ma-)

f(mb+) f(mb-)

and negative unate in c:

f(mc-) = 1 f(mc+) = 0

f ab bc ac

Example - unatenessExample - unateness

off

on

Minterms associated with c

mc- = (010) = 1mc+ = (011) = 0


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The Unate Recursive ParadigmThe Unate Recursive Paradigm

Key pruning technique based on exploiting the properties ofunatefunctions

based on the fact that unate leaf cases can be solved efficiently

New case splitting heuristic

splitting variable is chosen so that the functions at lower nodes

of the recursion tree become unate


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Unate covers Fhave many extraordinary properties: If a coverFis minimal with respect to single-cube

containment, all of its cubes are essential primes.

In this case F is the unique minimum cube representation ofits logic function.

A unate cover represents the tautology iff it contains a cubewith no literals (constant 1).

Positive unate: f = x fx+ fx

Negative unate: f = fx+ xfx

This type of implicit enumeration applies to many sub-problems(prime generation, reduction, complementation, etc.).

The Unate Recursive Paradigm


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Unate Recursive ParadigmUnate Recursive Paradigm

Create cofactoring tree stopping atunate covers choose, at each node, the most binate variable for splitting

recurse until no binate variable left (unate leaf)

Operateon the unate cover at each leaf to obtain the result for thatleaf. Return the result

At each non-leaf node,merge(appropriately) the results of the twochildren.

Main idea: Operationon unate leaf is computationally less complex Operations: complement, simplify, tautology, generate-primes,...etc.

a

cbmerge


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Two Useful Theorems - TautologyTwo Useful Theorems - Tautology

Theorem:

Checking for tatutology for is simplified for unate functions

Positive unate (f = x fx+ fx) f1 fx= 1

Negative unate (f = fx+ xfx)f1 fx= 1

Theorem: LetA be a unate cover matrix.

ThenA1 if and only ifA has a row of all -s.

Proof:If. A row of all -s is the tautology cube.

Only if. Assume no row of all -s. Without loss of generality, suppose function ispositive unate. Then each row has at least one 1 in it. Consider the point

(0,0,,0). This is not contained in any row of A. HenceA1.

1 ( 1) ( 1)j j

x x F F F


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Recursive Tautology termination rulesRecursive Tautology termination rules


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Recursive Tautology - exampleRecursive Tautology - example


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Recursive Complement OperationRecursive Complement Operation

x x f x f x f

0

1

x x

x x

g x f x f

f x f x f

f gg f

f g

Theorem:

Proof:


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COMPLEMENT OperationCOMPLEMENT Operation

Algorithm COMPLEMENT(List_of_Cubes C) {

if(C contains single cube c) {

Cres = complement_cube(c) // generate one cube per

return Cres // literal l in c with ^l

}else {

xi = SELECT_VARIABLE(C)

C0= COMPLEMENT(COFACTOR(C,^xi)) ^xiC

1= COMPLEMENT(COFACTOR(C,x

i )) x

i

return OR(C0,C1)

}

}


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Recursive Complement termination rulesRecursive Complement termination rules


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Recursive Complement example (split)Recursive Complement example (split)


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Recursive Complement example (merge)Recursive Complement example (merge)


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Incompletely Specified Boolean FunctionsIncompletely Specified Boolean Functions

F = (f, d, r) : Bn {0, 1, *}where * represents a dont care.

f= onset function - f(x)=1 F(x)=1

r= offset function - r(x)=1 F(x)=0 d= dont care function - d(x)=1 F(x)=*

(f,d,r) forms a partition ofBn, i.e. f + d + r = Bn

fd =fr = dr= (pairwise disjoint)


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A completely specified Boolean function gis a cover forF= (f,d,r) if

f g f+dNote:

g r =

ifxd, then g(x) = 0 or 1 (dont care) ifxf, then g(x)=1

ifxr, then g(x)=0.

Also: r = fd g f

Incompletely Specified Boolean FunctionsIncompletely Specified Boolean Functions

fd

r


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Example: Logic Minimization (single output)Example: Logic Minimization (single output)

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

abcis redundanta is primeF3= a+abc

Expandabc bc

Expandabca

F2= a+abc + abc

F4= a+bc

off

on

Dont care

F1= abc + abc+ abc


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Two-level minimization (multiple-outputs)Two-level minimization (multiple-outputs)

Initial representation: x y z0 0

0 1

1 1

1 1

f1 f20 1

0 1

1 0

1 0

x

yz

000 100

110010

111011

010

000 100

110010

111011

010

f1f2

101 101

f1 f2

000 100

110

010

111011

010

000 100

110

010

111011

010101 101

x y z

0 0

0 1 1

1 1

f1 f2

0 1

1 1

1 0

Minimized cover:

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