Geometric interpretation & Cubes Paolo PRINETTO Politecnico di Torino (Italy) University of Illinois...

Geometric Geometric interpretation interpretation

& Cubes& Cubes

Geometric Geometric interpretation interpretation

& Cubes& Cubes

Paolo PRINETTOPolitecnico di Torino (Italy)

University of Illinois at Chicago, IL (USA)

[email protected] [email protected]

www.testgroup.polito.it

Lecture

3.2

2 3.2

Goal

This lecture first introduces a geometric interpretation of Boolean Algebras, focusing, in particular, on the concept of K-cubes.

It then presents several ways of representing K-cubes.

Eventually some advanced operators on cubes are defined.

3 3.2

Prerequisites

Lecture 3.1

4 3.2

Homework

No particular homework is foreseen

5 3.2

Further readings

Students interested in a deeper knowledge of advanced operators on cubes can refer, for instance, to:

G.D. Hachtel, F. Somenzi: “Logic Synthesis and Verification Algorithms,” Kluwer Academic Publisher, Boston MA (USA), 1996

6 3.2

Outline

Geometric interpretation of Boolean Algebras

K-cubes representation

Advanced operations on K-cubes.

7 3.2

Geometric interpretation

The carrier Bn of a Boolean Algebra can be seen as an n-dimensional space, where each generic element v Bn (usually called a vertex), is represented by a vector of n coordinates, each B.

8 3.2

x

z

00 10

1101

B = { 0, 1}

B 2

9 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1}

B3

10 3.2

x

y

zB = { 0, 1 }

B4

w

11 3.2

K-cube

A k-dimensional sub-space (or k-cube)

Sk Bn

is a set of 2k vertices, in which n-k variables get the same constant value.

12 3.2

x

z

00 10

1101

0-cube = vertex

B = { 0, 1 }

B2

13 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3

0-cube = vertex

14 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3

1-cube = edge

15 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3

2-cube = face

16 3.2

B = { 0, 1 }

B4

2-cube = face

x

y

z

w

17 3.2

Remarks

Not all the sub-sets of Bn with cardinality 2k are k-cubes.

18 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3

An example of set of 22 vertices which is not a 2-cube

19 3.2

Remarks

Not all the sub-sets of Bn with cardinality 2k are k-cubes

The empty set is not a k-cube

20 3.2

Remarks

Not all the sub-sets of Bn with cardinality 2k are k-cubes

The empty set is not a k-cube

The total # of different k-cubes is:

N n! 2(n k)! k!

n-k

21 3.2

Examples

n = 3 k = 2 : 2-cubes : faces

N = 3! 2 / 1! 2! = 6

n = 3 k = 1 : 1-cubes : edges

N = 3! 4 / 2! 1! = 12

n = 3 k = 0 : 0-cubes : vertices

N = 3! 8 / 3! 0! = 8

N n! 2(n k)! k!

n-k

22 3.2

Outline




23 3.2


k-cubek-cube

24 3.2


AlgebraicAlgebraicnotationnotation

CubicCubicnotationnotation

k-cubek-cube

25 3.2




Product termProduct term Sum termSum term

k-cubek-cube

26 3.2



k-cubek-cube

list of n valueslist of n values { 0, 1, { 0, 1, –– }, },one for each one for each coordinatecoordinate

27 3.2

In such a list there will be:

n-k values set at 0, or at 1, and in particular:

at 0 if the corresponding coordinate gets the value 0 in all the vertices of the cube

at 1 if the corresponding coordinate gets the value 1 in all the vertices of the cube

k values set at a particular value, conventionally represented by “––” and read “don’t care” (they correspond to the k coordinates which assume all the 2k possible combinations of values).

Cubic notation (cont’d)

28 3.2

x

z

00 10

1101

0-cube in cubic notation

B = { 0, 1 }

B2

1010

29 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3


111111

30 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3


1100

31 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3


00

32 3.2

B = { 0, 1 }

B4


x

y

z

w

11 00

33 3.2

Cubes representation


k-cubek-cube

list of n-k literals, one for list of n-k literals, one for each variable that gets the each variable that gets the

same value in all the same value in all the vertices of the cubevertices of the cube

34 3.2



Product termProduct term

k-cubek-cube

35 3.2



Product termProduct term

k-cubek-cubeEach k-cube is represented by a Each k-cube is represented by a logiclogic product product of n-k variables where:of n-k variables where:• variables assuming constantly the variables assuming constantly the

1 value are asserted1 value are asserted• variables assuming constantly the variables assuming constantly the

0 value are complemented.0 value are complemented.

36 3.2

x

z

00 10

1101

0-cube by product terms

B = { 0, 1 }

B2

x zx z’’

37 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3


x y zx y z

38 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3


x zx z’’

39 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3


yy’’

40 3.2


B = { 0, 1 }

B4

x

y

z

w

x zx z’’

41 3.2

Remark

The universe set, i.e., the n-cube, which, according to the previous rules should be represented by “no” variable, is usually represented by the symbol 1.

42 3.2

f ( x1, x2, x3 )

10 -

- 1 -

110

From cubic notation to algebraic notation by product

terms

43 3.2

f ( x1, x2, x3 )

10 - x1 x2’

- 1 - x2

110 x1 x2 x3’

From cubic notation to algebraic notation by product

terms

44 3.2

f ( x1, x2, x3 )

x1 x2’

x2

x1 x2 x3’

From algebraic notation by product terms to cubic notation

45 3.2

f ( x1, x2, x3 )

x1 x2’ 10 -

x2 - 1 -

x1 x2 x3’ 110

From algebraic notation by product terms to cubic notation

46 3.2

Minterm

A minterm (or fundamental product) is the representation of a 0-cube by a product term.

k-cube representation

1 0 1 x y ’ z minterm

0 1 1 x’ y z minterm

0 0 x’ z’ not a minterm

1 y not a minterm

47 3.2



Sum termSum term

k-cubek-cube

48 3.2



Sum termSum term

k-cubek-cube

Each k-cube is represented by a Each k-cube is represented by a logiclogic sum sum of n-k variables where:of n-k variables where:• variables assuming constantly variables assuming constantly

the 1 value are complementedthe 1 value are complemented• variables assuming constantly variables assuming constantly

the 0 value are asserted.the 0 value are asserted.

49 3.2

x

z

00 10

1101

0-cube by sum terms

B = { 0, 1 }

B2

xx’ + ’ + yy

50 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3

0-cube by sum terms

xx’ + ’ + yy ’ + ’ + zz’’

51 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3

1-cube by sum terms

xx’ + ’ + zz

52 3.2

x

y

z

000

111

100

101001

010

011

110

B = { 0, 1 }

B3

2-cube by sum terms

yy

53 3.2

2-cube by sum terms

B = { 0, 1 }

B4

x

y

z

w

xx’ + ’ + zz

54 3.2

Remark

The universe set is usually represented, in this notation, by the symbol 0.

55 3.2

Maxterm

A maxterm (or fundamental sum) is the representation of a 0-cube by a sum term.

k-cube representation

1 0 1 x’ + y + z’ maxterm

0 1 1 x + y ’ + z’ maxterm

0 0 x + z not a maxterm

1 y ’ not a maxterm

56 3.2

Outline




57 3.2

Several advanced operators have been defined on k-cubes, including:

Splitting Coverage Intersection Distance Union Complement Cofactor Consensus …

Advanced operations on K-cubes

58 3.2

Note

Since most of them are of interest for peculiar applications, only (e.g., advanced techniques for logic minimization), they are not fatherly dealt with in the present course.

Just a couple of them are going to be presented here.

Interested students can refer to the book suggested in slide 5 for a deeper analysis.

59 3.2

Splitting

Splitting allows us to find all the vertices of a given k-cubes.

Algorithm

Assign each “–” all the possible combinations of the related input variable, until no “–” are present.

60 3.2

Example

–11–0

61 3.2

Example

–11–0111–0

011–0

62 3.2

Example

–11–0

11110

11100

01110

01100

111–0

011–0

63 3.2

Coverage

We have to distinguish between:

Coverage among cubes

Coverage among sets of cubes

64 3.2

Coverage among cubes

A cube a covers (or contains) a cube b, and we denote it as:

a b

Iff all the vertices of b are vertices of a as well.

In such a case we say that b implies a or that a is implied by b.

65 3.2

Example

a = - 0 -

b = - 0 0

a b

000

111

100

101001

010 110

011

b

aa

66 3.2

“Working” Definition

A cube a covers a cube b, iff the cube b can be derived from a by replacing, in a, one (or more) “–” by 0 or 1.

Examples

a = – 0 –

b = – 0 0 a b

a = 1 0 – 1

b = – 0 1 1 a b

67 3.2

Coverage among sets of cubes

A set of cubes C covers a cube a iff each vertex of a is a vertex of at least one of the cubes of C.

Example

C = { 0–1, 10– } a = –01 ?

68 3.2

Solution

We first split the vertices of a :

001, covered by C[1] = 0–1

101, covered by C[2] = 10–

As a consequence

C = { 0–1, 10– } a = –01

Even if, when considered individually

C[1] a and C[2] a

69 3.2

000

111

100

101001

010110

011

C[1]

C[2]

aC[1] = 0–1

C[2] = 10–

a = – 0 1

70 3.2

Hamming distance between cubes

We define Hamming distance between 2 cubes a and b, and we denote it as

D (a, b)

the # of coordinates of the 2 cubes such that:

a[i] = 0 and b[i] = 1

or

a[i] = 1 and b[i] = 0

71 3.2

Properties

Two cubes have a null distance iff they share one or more vertices

Two cubes having unit distance are usually referred to as being logically adjacent.

72 3.2

Example

y

z

000

111

100

101001

010

011

x

110

73 3.2

Example

y

z

000

111

100

101001

010

011

x

110aa

a = - 0 -

74 3.2

Example

y

z

000

111

100

101001

010

011

x

110aa

a = - 0 -

b

b = 1 - 0

75 3.2

Example

y

z

000

111

100

101001

010

011

x

110aa

a = - 0 -

b

b = 1 - 0 D(a, b) = 0

76 3.2

Example

y

z

000

111

100

101001

010

011

x

110aa

a = - 0 -

b

b = 1 - 0 D(a, b) = 0

c

c = 1 1 1

77 3.2

Example

y

z

000

111

100

101001

010

011

x

110aa

a = - 0 -

b

b = 1 - 0 D(a, b) = 0

c

c = 1 1 1 D(a, c) = 1

78 3.2

Example

y

z

000

111

100

101001

010

011

x

110aa

a = - 0 -

b

b = 1 - 0 D(a, b) = 0

c

c = 1 1 1 D(a, c) = 1

xyz 00 01 11 10

0

1

Geometric interpretation & Cubes Paolo PRINETTO Politecnico di Torino (Italy) University of Illinois...

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