ECE 667 Synthesis & Verification - Boolean Functions 1 ECE 667 Spring 2013 ECE 667 Spring 2013...
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Transcript of ECE 667 Synthesis & Verification - Boolean Functions 1 ECE 667 Spring 2013 ECE 667 Spring 2013...
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ECE 667 Synthesis & Verification - Boolean Functions
ECE 667ECE 667Spring 2013Spring 2013
Synthesis and Verificationof Digital Circuits
Boolean Functions
and their Representations
Slides adopted (with permission) from A. Kuehlmann, UC Berkeley 2003
ECE 667 Synthesis & Verification - Boolean Functions 2
The Boolean Space The Boolean Space BBnn
B = { 0,1}, B2 = {0,1} X {0,1} = {00, 01, 10, 11}
BB00
BB11
BB22
Karnaugh Maps: Boolean Cubes:
BB33
BB44
ECE 667 Synthesis & Verification - Boolean Functions 3
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
ECE 667 Synthesis & Verification - Boolean Functions 4
Boolean FunctionsBoolean Functions
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
ECE 667 Synthesis & Verification - Boolean Functions 5
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
x1x2x3
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
x1
x2
x3
• Truth Table or Function Table:
ECE 667 Synthesis & Verification - Boolean Functions 6
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}
ECE 667 Synthesis & Verification - Boolean Functions 7
Cofactor and QuantificationCofactor and Quantification
Given 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 = {x | f(x1,x2,…,1,…,xn)=1}
• Negative Cofactor of f w.r.t variable xi
fxi’ = {x | f(x1,x2,…,0,…,xn)=1}
• Existential Quantification of function f w.r.t variable xi,
xi f = {x | f(x1,x2,…,0,…,xn)=1 f(x1,x2,…,1,…,xn)=1}
• Universal Quantification of function f w.r.t variable xi,
xi f = {x | f(x1,x2,…,0,…,xn)=1 f(x1,x2,…,1,…,xn)=1}
ECE 667 Synthesis & Verification - Boolean Functions 8
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)– Boolean formula– Binary Decision Tree, Binary Decision Diagram– Circuit (network of Boolean primitives)
• Canonicity – which forms are canonical?
ECE 667 Synthesis & Verification - Boolean Functions 9
Truth TableTruth Table
• Truth table (Function Table):The truth table of a function f : Bn B is a tabulation of
its values at each of the 2n vertices of Bn. (all mintems)
Example: f = a’b’c’d + a’b’cd + a’bc’d +
ab’c’d + ab’cd + abc’d +
abcd’ + abcd(Notation for complement: a’ = a )
The truth table representation is
- intractable for large n
- canonical
Canonical means that if two functions are the
same, then the canonical representations of each
are isomorphic (identical).
abcd f0 0000 01 0001 12 0010 03 0011 14 0100 05 0101 16 0110 07 0111 08 1000 0 9 1001 110 1010 011 1011 112 1100 013 1101 114 1110 115 1111 1
ECE 667 Synthesis & Verification - 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
ECE 667 Synthesis & Verification - Boolean Functions 11
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 function
f = (x1=1)(x2=0)(x3=1)
x1
x2
x3
c = x1
x1
x2
x3
f = x1x2
x1
x2
x3
f = x1x2x3
ECE 667 Synthesis & Verification - Boolean Functions 12
CubesCubes
• If C f, C a cube, then C is an implicant of f.
• If C Bn, and C has k literals, then |C| covers 2n-k vertices.
Example:C = xy B3
k = 2 , n = 3 => |C| = 2 = 23-2.
C = {100, 101}
• In an n-dimensional Boolean space Bn, an implicant with n literals is a minterm.
ECE 667 Synthesis & Verification - Boolean Functions 13
List of Cubes - CoverList of Cubes - Cover
Sum of Products (SOP)Sum of Products (SOP)
• A function can be represented by a sum of cubes (products):
f = ab + ac + bcSince 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
F = {ab, ac, bc}
• A set of cubes that represents f is called a cover of f.
Sets: F1={ab, ac, bc} and F2={abc, a’bc, ab’c, abc’}are some of possible covers of Boolean function
f = ab + ac + bc.
ECE 667 Synthesis & Verification - Boolean Functions 14
Binary Decision Diagram (BDD)Binary Decision Diagram (BDD)
f = ab+a’c+a’bd
1
0
c
a
b b
c c
d
0 1
c+bd b
root node
c+d
d
Graph representation of a Boolean function - vertices represent decision nodes for variables
- two children represent the two subfunctions
f(x = 0) and f(x = 1) (cofactors)
- restrictions on ordering and reduction rules
can make a BDD representation canonical
- function is evaluated as sum of paths from
root to constant node 1
- paths form disjoint cubes.
ECE 667 Synthesis & Verification - Boolean Functions 15
Boolean NetworksBoolean Networks
• Used for two main purposes
– as representation for Boolean reasoning engine
– as target structure for logic implementation which gets restructured in a series of logic synthesis steps until result is acceptable
• Efficient representation for most Boolean problems
– memory complexity is same as the size of circuits we are actually building
• Close to input representation and output representation in logic synthesis
ECE 667 Synthesis & Verification - Boolean Functions 16
Definitions Definitions –– Boolean NetworkBoolean Network
Definition:
A Boolean network is a directed graph C(G,N) where G are the gates and N GG is the set of directed edges (nets) connecting the gates.
Some of the vertices are designated:
Inputs: I G
Outputs: O G, I O =
Each node g is assigned a Boolean function fg which computes the output of the gate in terms of its inputs.
ECE 667 Synthesis & Verification - Boolean Functions 17
Definitions Definitions –– fanin, fanout, supportfanin, fanout, support
• The (immediate) fanin FI(g) of a gate g are all predecessor vertices of g:
FI(g) = {g’ | (g’,g) N}• The transitive fanin FIT (g) is defined as follows:
if a FI(b) and b FI(c) then a FIT (c)
• The fanout FO(g) of a gate g are all successor vertices of g:
FO(g) = {g’ | (g,g’) N} The transitive fanout is defined similarly
The cone CONE(g) of a gate g is the transitive fanin of g and g itself.
The support SUPPORT(g) of a gate g are all inputs in its cone:
SUPPORT(g) = CONE(g) I
ECE 667 Synthesis & Verification - Boolean Functions 18
Example Example –– Boolean NetworkBoolean Network
I
O
FI(6) = {2,4}
FO(6) = {7,9}
CONE(6) = {1,2,4,6}
SUPPORT(6) = {1,2}
6
1
5
3
4
78
9
2
Nodes = logic functions of arbitrary complexity
ECE 667 Synthesis & Verification - Boolean Functions 19
Boolean Network RepresentationsBoolean Network Representations
• Vertices can have arbitrary number of inputs and outputs– typically single-output functions are used
• Vertices can represent any Boolean function stored in different
ways, such as:– other circuits (hierarchical representation)– truth tables or cube representation – Boolean expressions read from a library description– BDDs, AIGs, etc.
• Data structure allow very general mechanisms for insertion and
deletion of vertices, pins (connections to vertices), and nets
ECE 667 Synthesis & Verification - Boolean Functions 20
AND-INVERTERAND-INVERTER Circuits Circuits
• Base data structure uses two-input AND function for vertices and INVERTER attributes at the edges (individual bit)– use De’Morgan’s law to convert OR operation etc.
• Hash table to identify and reuse structurally isomorphic circuits
f
g g
f
Means complement
ECE 667 Synthesis & Verification - Boolean Functions 21
Data RepresentationData Representation
• Vertex:– pointers (integer indices) to left and right child and fanout
vertices– collision chain pointer– other data
• Edge:– pointer or index into array– one bit to represent inversion
• Global hash table holds each vertex to identify isomorphic structures
• Garbage collection to regularly free un-referenced vertices
ECE 667 Synthesis & Verification - Boolean Functions 22
Data RepresentationData Representation
0456left
rightnext
fanout1345….
8456….
6423….
7463….
01
hash value
left pointer
right pointer
next in collision chain
array of fanout pointers
complement bits
Constant
One Vertex
zero
one
04560455
0457
...
...
Hash Table
0456left
rightnext
fanout
00