Transcript Boolean Functions I - The University of Texas at Austin

Logic Synthesis
Boolean Functions and Circuits
Courtesy RK Brayton (UCB)
and A Kuehlmann (Cadence)
1
Remember: What is Logic Synthesis?


X
D
Given: Finite-State Machine F(X,Y,Z, ,  ) where:
Y
X:
Y:
Z:
:
:
Input alphabet
Output alphabet
Set of internal states
XxZ
Z (next state function)
XxZ
Y (output function)
These are Boolean Functions!!
Target: Circuit C(G, W) where:
G: set of circuit components g  {Boolean gates,
flip-flops, etc}
W: set of wires connecting G
2
The Boolean Space B
•
•
n
B = { 0,1}
B2 = {0,1} X {0,1} = {00, 01, 10, 11}
Karnaugh Maps:
Boolean Cubes:
B0
B1
B2
B3
B4
3
Boolean Functions
Boolean Function: f ( x ) : B n  B
B  {0,1}
x  ( x1, x2 ,..., xn )  B n ; xi  B
- x1, x2 ,... are variables
- x1, x1, x2 , x2 ,... are literals
- essentially: f maps each vertex of B n to 0 or 1
x2
Example:
f  {(( x1  0, x2  0),0),(( x1  0, x2  1),1),
(( x1  1, x2  0), 1),(( x1  1, x2  1),0)}
x1
0 1
1 0
x2
x1
4
Boolean Functions
- The Onset of f is { x | f ( x )  1}  f 1(1)  f 1
- The Offset of f is { x | f ( x )  0}  f 1(0)  f 0
- if f 1  B n , f is the tautology. i.e. f  1
- if f 0  B n (f 1   ), f is not satisfyable, i.e. f  0
- if f ( x )  g ( x ) for all x  B n , then f and g are equivalent
- we say f instead of f 1
- literals: A literal is a variable or its negation x, x and represents a logic function
Literal x1 represents the logic function f, where f = {x| x1 = 1}
Literal x1 represents the logic function g where g = {x| x1 = 0}
f = x1
f = x1
x3
x3
x2
x1
x2
x1
5
Set of Boolean Functions
•
Truth Table or Function table:
x3
x2
x1
x1x2x3
000
1
001
0
010
1
011
0
100 1
101
0
110
1
111
•
•
0
There are 2n vertices in input space Bn
n
There are 22 distinct logic functions.
– Each subset of vertices is a distinct logic function:
f  Bn
6
Boolean Operations AND, OR, COMPLEMENT
Given two Boolean functions:
f : Bn  B
g : Bn  B
•
The AND operation h = f  g is defined as
h = {x | f(x)=1  g(x)=1}
•
The OR operation h = f + g is defined as
h = {x | f(x)=1  g(x)=1}
•
The COMPLEMENT operation h = ^f is defined as
h = {x | f(x) = 0}
7
Cofactor and Quantification
Given a Boolean function:
f : Bn  B, with the input variable (x1,x2,…,xi,…,xn)
•
The positive cofactor h = fxi is defined as
h = {x | f(x1,x2,…,1,…,xn)=1}
•
The negative cofactor h = fxi is defined as
h = {x | f(x1,x2,…,0,…,xn)=1}
•
The existential quantification of variable xi h = $ xi . f is defined as
h = {x | f(x1,x2,…,0,…,xn)=1 f(x1,x2,…,1,…,xn)=1}
•
The universal quantification of variable xi h = " xi . f is defined as
h = {x | f(x1,x2,…,0,…,xn)=1 f(x1,x2,…,1,…,xn)=1}
8
Representation 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)
9
Truth Table
•
Truth table (Function Table):
The truth table of a function f : Bn  B is a tabulation of its value at
each of the 2n vertices of Bn.
In other words the truth table lists all mintems
Example: f = abcd + abcd + abcd +
abcd + abcd + abcd +
abcd + abcd
The truth table representation is
- intractable for large n
- canonical
0
1
2
3
4
5
6
7
abcd
0000
0001
0010
0011
0100
0101
0110
0111
f
0
1
0
1
0
1
0
0
8
9
10
11
12
13
14
15
abcd
1000
1001
1010
1011
1100
1101
1110
1111
f
0
1
0
1
0
1
1
1
Canonical means that if two functions are the same, then the
canonical representations of each are isomorphic.
10
Boolean Formula
•
A Boolean formula is defined as an expression with the following
syntax:
formula ::=
|
|
|
|
‘(‘ formula ‘)’
<variable>
formula “+” formula
formula “” formula
^ formula
(OR operator)
(AND operator)
(complement)
Example:
f = (x1x2) + (x3) + ^^(x4  (^x1))
typically the “” is omitted and the ‘(‘ and ‘^’ are simply reduced by priority,
e.g.
f = x1x2 + x3 + x4^x1
11
Cubes
•
A cube is defined as the AND of a set of literal functions (“conjunction”
of literals).
Example:
C = x1x2x3
represents the following function
f = (x1=1)(x2=0)(x3=1)
c = x1
f = x1x2
x3
f = x1x2x3
x3
x3
x2
x1
x2
x1
x2
x1
12
Cubes
•
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}
•
An implicant with n literals is a minterm.
13
List of Cubes
• Sum of Products:
•
A function can be represented by a sum of cubes (products):
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
F = {ab, ac, bc}
•
A set of cubes that represents f is called a cover of f.
F1={ab, ac, bc} and F2={abc,abc,abc,abc}
are covers of
f = ab + ac + bc.
14
Binary Decision Diagram (BDD)
Graph representation of a Boolean function f
- 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
f = ab+a’c+a’bd
root
node
c+bd
a
b
b
b
c
c+d
c
c
1
d
d
0
0
1
15
Boolean Circuits
•
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 we have in CAD
– memory complexity is same as the size of circuits we are actually
building
•
Close to input representation and output representation in logic
synthesis
16
Definitions
Definition:
A Boolean circuit is a directed graph C(G,N) where G are the gates and
N GG 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 gate g is assigned a Boolean function fg which computes the
output of the gate in terms of its inputs.
17
Definitions
The fanin FI(g) of a gate g are all predecessor vertices of g:
FI(g) = {g’ | (g’,g) N}
The fanout FO(g) of a gate g are all successor vertices of g:
FO(g) = {g’ | (g,g’) N}
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
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Example
8
7
1
4
6
2
9
5
3
I
FI(6) = {2,4}
FO(6) = {7,9}
CONE(6) = {1,2,4,6}
SUPPORT(6) = {1,2}
O
19
Circuit Function
•
Circuit functions are defined recursively:
if gi  I
 xi
hgi  
 f gi (hg j ,..., hgk ), g j ,..., g k  FI ( gi ) otherwise
•
If G is implemented using physical gates that have positive (bounded)
delays for their evaluation, the computation of hg depends in general on
those delays.
Definition:
A circuit C is called combinational if for each input assignment of C for
t the evaluation of hg for all outputs is independent of the internal
state of C.
Proposition:
A circuit C is combinational if it is acyclic.
20
Cyclic Circuits
Definition:
A circuit C is called cyclic if it contains at least one loop of the form:
((g,g1),(g1,g2),…,(gn-1,gn),(gn,g)).
...
g
...
In order to check whether the loop is combinational or sequential, we
need to derive the loop function hloop by cutting the loop at gate g
21
Cyclic Circuits
hloop
...
...
g
g
...
v
...
hloop  h( x, v)  f g (h 'g j ,..., h ' gk ), g j ,..., g k  FI ( g )
 xi
if gi  I

h ' gi   v
if gi  g
 f (h ' ,..., h ' ), g ,..., g  FI ( g ) otherwise
gk
j
k
i
 gi g j
22
Cyclic Circuits
xi
hloop
v
The following equation computes the sensitivity of h
with respect to v:
hloop ( x, v)v  hloop ( x, v)v  1
For any solution x, hloop will toggle if v toggles.
• negative feedback - oscillator (astable multivibrator)
• positive feedback - flip-flop (bistable multivibrator)
23
Cyclic Circuits
Theorem: A circuit loop involving gate g is combinational if:
hloop ( x, v)v  hloop ( x, v)v  0
Theorem: Output y of a circuit containing a loop with gate g is
combinational if:
(hloop ( x, v)v  hloop ( x, v)v )  ( y ( x, v) v  y ( x, v) v )  0
“Either the loop is combinational or the output does not depend on
the loop value.”
24
Circuit Representations
For general circuit manipulation (e.g. synthesis):
•
Vertices have an arbitrary number of inputs and outputs
•
Vertices can represent any Boolean function stored in different ways,
such as:
– other circuits (hierarchical representation)
– Truth tables or cube representation (e.g. SIS)
– Boolean expressions read from a library description
– BDDs
•
Data structure allow very general mechanisms for insertion and
deletion of vertices, pins (connections to vertices), and nets
– general but far too slow for Boolean reasoning
25
Circuit Representations
For efficient Boolean reasoning (e.g. a SAT engine):
•
Circuits are non-canonical
– computational effort is in the “checking part” of the reasoning
engine (in contrast to BDDs)
•
Vertices have fixed number of inputs (e.g. two)
•
Vertex function is stored as label, well defined set of possible function
labels (e.g. OR, AND,OR)
•
on-the-fly compaction of circuit structure
– allows incremental, subsequent reasoning on multiple problems
26
Boolean Reasoning Engine
Engine application:
- traverse problem data structure and build
Boolean problem using the interface
- call SAT to make decision
Engine Interface:
void INIT()
void QUIT()
Edge VAR()
Edge AND(Edge p1,
Edge p2)
Edge NOT(Edge p1)
Edge OR(Edge p1
Edge p2)
...
int SAT(Edge p1)
External reference pointers attached
to application data structures
Engine Implementation:
...
...
...
...
27
Basic Approaches
• Boolean reasoning engines need:
– a mechanism to build a data structure that represents the problem
– a decision procedure to decide about SAT or UNSAT
• Fundamental trade-off
– canonical data structure
• data structure uniquely represents function
• decision procedure is trivial (e.g., just pointer comparison)
• example: Reduced Ordered Binary Decision Diagrams
• Problem: Size of data structure is in general exponential
– non-canonical data structure
• systematic search for satisfying assignment
• size of data structure is linear
• Problem: decision may take an exponential amount of time
28
AND-INVERTER 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
f
g
g
29
Data 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
30
Data Representation
Hash Table
one
6423
….
8456
….
Constant
One Vertex
zero
1345
….
0456
0 left
1 right
next
fanout
...
0455
0456
0457
...
7463
….
hash value
complement bits
left pointer
right pointer
next in collision chain
array of fanout pointers
0456
0 left
0 right
next
fanout
31
Hash Table
Algorithm HASH_LOOKUP(Edge p1, Edge p2) {
index = HASH_FUNCTION(p1,p2)
p
= &hash_table[index]
while(p != NULL) {
if(p->left == p1 && p->right == p2) return p;
p = p->next;
}
return NULL;
}
Tricks:
- keep collision chain sorted by the address (or index) of p
- that reduces the search through the list by 1/2
- use memory locations (or array indices) in topological order of circuit
- that results in better cache performance
32
Basic Construction Operations
Algorithm AND(Edge p1,Edge p2){
if(p1 == const1) return p2
if(p2 == const1) return p1
if(p1 == p2)
return p1
if(p1 == ^p2)
return const0
if(p1 == const0 || p2 == const0) return const0
if(RANK(p1) > RANK(p2)) SWAP(p1,p2)
if((p = HASH_LOOKUP(p1,p2)) return p
return CREATE_AND_VERTEX(p1,p2)
}
33
Basic Construction Operations
Algorithm NOT(Edge p) {
return TOOGLE_COMPLEMENT_BIT(p)
}
Algorithm OR(Edge p1,Edge p2){
return (NOT(AND(NOT(p1),NOT(p2))))
}
34
Cofactor Operation
Algorithm POSITIVE_COFACTOR(Edge p,Edge v){
if(IS_VAR(p)) {
if(p == v) {
if(IS_INVERTED(v) == IS_INVERTED(p)) return const1
else
return const0
}
else
return p
}
if((c = GET_COFACTOR(p)) == NULL) {
left = POSITIVE_COFACTOR(p->left, v)
right = POSITIVE_COFACTOR(p->right, v)
c = AND(left,right)
SET_COFACTOR(p,c)
}
if(IS_INVERTED(p)) return NOT(c)
else
return c
}
35
Cofactor Operation
- similar algorithm for NEGATIVE_COFACTOR
- existential and universal quantification build from AND, OR and COFACTORS
Question: What is the complexity of the circuits resulting from quantification?
36
SAT and Tautology
•
•
Tautology:
– Find an assignment to the inputs that evaluate a given vertex to “0”.
SAT:
– Find an assignment to the inputs that evaluate a given vertex to “1”.
– Identical to Tautology on the inverted vertex
•
SAT on circuits is identical to the justification part in ATPG
– First half of ATPG: justify a particular circuit vertex to “1”
– Second half of ATPG (propagate a potential change to an output)
can be easily formulated as SAT (will be covered later)
•
Basic SAT algorithms:
– branch and bound algorithm as seen before
• branching on the assignments of primary inputs only (Podem
algorithm)
• branching on the assignments of all vertices (more efficient)
37
General Davis-Putnam Procedure
•
•
search for consistent assignment to entire cone of requested vertex by
systematically trying all combinations (may be partial!!!)
keep a queue of vertices that remain to be justified
– pick decision vertex from the queue and case split on possible
assignments
– for each case
• propagate as many implications as possible
– generate more vertices to be justified
– if conflicting assignment encountered
» undo all implications and backtrack
• recur to next vertex from queue
Algorithm SAT(Edge p) {
queue = INIT_QUEUE()
if(IMPLY(p) return TRUE
return JUSTIFY(queue)
}
38
General Davis-Putnam Procedure
Algorithm JUSTIFY(queue) {
if(QUEUE_EMPTY(queue)) return TRUE
mark = ASSIGNMENT_MARK()
v = QUEUE_NEXT(queue) // decision vertex
if(IMPLY(NOT(v->left)) {
if(JUSTIFY(queue)) return TRUE
}
// conflict
UNDO_ASSIGNMENTS(mark)
if(IMPLY(v->left) {
if(JUSTIFY(queue)) return TRUE
}
// conflict
UNDO_ASSIGNMENTS(mark)
return FALSE
}
39
Example
Queue
SAT(NOT(9))??
4
1
Assignments
9
9
9
9
7
4
5
1
2
7
2
9
5
0
8
6
3
First case for 9:
1
2
1
0
4
1
1
1
7
1
0
9
5
0
8
3
6
Conflict!!
- undo all assignments
- backtrack
40
Example
Queue
Second case for 9:
4
1
0
Note:
- vertex 7 is justified
by 8->5->7
2
7
0
9
5
8
3
6
1
4
5
6
1
1
0
0
1
0
0
Assignments
9
7
8
5
6
First case for 5:
0
2
0
7
9
5
8
3
0
6
0
1
0
0
9
7
8
5
6
2
3
Solution cube: 1 = x, 2 = 0, 3 = 0
41
Implication Procedure
•
•
Fast implication procedure is key for efficient SAT solver!!!
– don’t move into circuit parts that are not sensitized to current SAT
problem
– detect conflicts as early as possible
Table lookup implementation (27 cases):
– No implications:
x
x
x
x
x
x
1
0
0
1
1
x
x
0
0
1
0
0
x
0
0
1
0
0
0
1
1
– Implications:
0
x
x
x
0
x
0
0
x
x
x
1
x
1
1
1
1
x
42
Implication Procedure
– Implications:
x
0
1
1
0
x
1
x
1
1
x
0
0
x
1
– Conflicts:
0
1
x
0
0
1
0
1
1
x
0
1
1
0
1
1
0
1
– Case Split:
x
0
x
43
Ordering of Case Splits
•
•
•
•
•
various heuristics work differently well for particular problem classes
often depth-first heuristic good because it generates conflicts quickly
mixture of depth-first and breadth-first schedule
other heuristics:
– pick the vertex with the largest fanout
– count the polarities of the fanout separately and pick the vertex with
the highest count in either one
– run a full implication phase on all outstanding case splits and count
the number of implications one would get
• some cases may already generate conflicts, the other case is
immediately implied
pick vertices that are involved in small cut of the circuit
== 0?
“small cut”
44
Learning
•
•
•
Learning is the process of adding “shortcuts” to the circuit structure that
avoids case splits
– static learning:
• global implications are learned
– dynamic learning:
• learned implications only hold in current part of the search tree
Learned implications are stores as additional network
Back to example:
– First case for vertex 9 lead to conflict
– If we were to try the same assignment again (e.g. for the next SAT call), we
would get the same conflict => merge vertex 7 with Zero-vertex
Zero Vertex
1
2
1
0
4 1
1
1
7
0
9
5
8
3
1
0
- if rehashing is invoked
vertex 9 is simplified and
and merged with vertex 8
6
45
Static Learning
•
Implications that can be learned structurally from the circuit
– Example:
(( x  y )  0)  ( x  y )  0)  ( x  0)
– Add learned structure as circuit
Use hash table to find structure in circuit:
Algorithm CREATE_AND(p1,p2) {
. . . // create new vertex p
if((p’=HASH_LOOKUP(p1,NOT(p2))) {
LEARN((p=0)&(p’=0)(p1=0))
}
if((p’=HASH_LOOKUP(NOT(p1),p2)) {
LEARN((p=0)&(p’=0)(p2=0))
}
Zero Vertex
46
Back to Example
Queue
Original second case for 9:
1
4
7
0
2
0
9
5
8
3
6
5
6
1
1
0
0
1
0
0
Assignments
9
7
8
5
6
Second case for 9 with static learning:
1
4
7
0
2
9
5
8
3
0
6 0
a
Solution cube: 1 = x, 2 = x, 3 = 0
1
1
b
0
0
Zero Vertex
9
7
8
5
6
a
3
47
Static Learning
•
Socrates algorithm: based on contra-positive:
( x  y)  ( y  x )
foreach vertex v {
mark = ASSIGNMENT_MARK()
IMPLY(v)
LEARN_IMPLICATIONS(v)
UNDO_ASSIGNMENTS(mark)
IMPLY(NOT(v))
LEARN_IMPLICATIONS(NOT(v))
UNDO_ASSIGNMENTS(mark)
}
0
0
y
x
0
(( x  0)  ( y  1))  (( y  0)  ( x  1))
1
•
1
Problem: learned implications are far too many
– solution: restrict learning to non-trivial implications
– mask redundant implications
x
y
0
0
0
Zero Vertex
48
Recursive Learning
•
Compute set of all implications for both cases on level i
– static implications (y=0/1)
– equivalence relations (y=z)
Intersection of both sets can be learned for level i-1
•
(( x  1)  ( y  1)  ( x  0)  ( y  1))  ( y  1)
1
1
x
1
y
x
0
x
y
y
0
y
1
x
1
x
0
0
0
x
1
•
Apply learning recursively until all case splits exhausted
– recursive learning is complete but very expensive in practice for
levels > 2,3
– restrict learning level to fixed number -> becomes incomplete
49
Recursive Learning
Algorithm RECURSIVE_LEARN(int level) {
if(v = PICK_SPLITTING_VERTEX()) {
mark = ASSIGNMENT_MARK()
IMPLY(v)
IMPL1 = RECURSIVE_LEARN(level+1)
UNDO_ASSIGNMENTS(mark)
IMPLY(NOT(v))
IMPL0 = RECURSIVE_LEARN(level+1)
UNDO_ASSIGNMENTS(mark)
return IMPL1  IMPL0
}
else {
// completely justified
return IMPLICATIONS
}
}
50
Dynamic Learning
Learn implications in sub-tree of search
• cannot simply add permanent structure because not globally valid
– add and remove learned structure (expensive)
– add the branching condition to the learned implication
• of no use unless we prune the condition (conflict learning)
– use implication and assignment mechanism to assign and undo
assigns
• e.g. dynamic recursive learning with fixed recursion level
•
Dynamic learning of equivalence relations (Stalmarck procedure)
– learn equivalence relations by dynamically rewriting the formula
51
Dynamic Learning
•
Efficient implementation of dynamic recursive learning with level 1:
– consider both sub-cases in parallel
– use 27-valued logic in the IMPLY routine
{level0-value ´ level1-choice1 ´ level1-choice2}
{{0,1,x} ´ {0,1,x} ´ {0,1,x}}
– automatically set learned values for level0 if both level1 choices
agree
x
{x,1,0}
{1,1,1}
1
0
0
0
{x,x,1}
x
52
Conflict-based Learning
•
Idea: Learn the situation under which a particular conflict
occurred and assert it to 0
• imply will use this “shortcut” to detect similar conflict earlier
Definition:
An implication graph is a directed Graph I(G’,D), G’ G are the gates
of C with assigned values vgx, D  G G are the edges, where each
edge (gi,gj) D reflects an implication for which an assignment of gate
gi lead to the assignment of gate gj.
Circuit:
Implication graph:
0 (decision vertex)
1
0
1’
2
3
4
1
0 (decision vertex)
2’
3’
4’
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Conflict-based Learning
•
The roots of the implication graph correspond to the decision vertices,
the leafs corresponds to the implication frontier
Cut assignment (Ci)
•
There is a strict implication order in the graph from the roots to the leafs
– We can completely cut the graph at any point and identical value
assignments to the cut vertices, we result in identical implications toward
the leafs
C1 C2 Cn-1 Cn
(C1: Decision vertices)
54
Conflict-based Learning
•
If an implication leads to a conflict, any cut assignment in the
implication graph between the decision vertices and the conflict will
result in the same conflict!
(Ci Conflict) (NOT(Conflict) NOT(Ci))
•
We can learn the complement of the cut assignment as circuit
– find minimal cut in I (costs less to learn)
– find dominator vertex if exists
– restrict size of cuts to be learned, otherwise exponential blow-up
55
Non-chronological Backtracking
•
If we learned only cuts on decision vertices, only the decision vertices
that are in the support of the conflict are needed
Decision levels: 5
Decision Tree:
1
2
4
3
1
4
3
6
5
6
2
•
The conflict is fully symmetric with respect to the unrelated decision
vertices!!
– Learning the conflict would prevent checking the symmetric parts again
BUT: It is too expensive to learn all conflicts
56
Non-chronological Backtracking
•
We can still avoid exploring symmetric parts of the decision tree by
tracking the supporting decision vertices for all conflicts.
If no conflict of the first choice on a decision vertex depends on that vertex,
the other choice(s) will result in symmetric conflicts and their evaluation can
be skipped!!
•
By tracking the implications of the decision vertices we can skip
decision levels during backtrack
0
1
2
{2,0}
3
4
{2,3}
{2,4}
{2,4,0}
{4,3}
{4,0}
57
Modified Justify Algorithm
Algorithm JUSTIFY(queue) {
...
if((decision_levels0 = IMPLY(NOT(v->left))) == ) {
if((decision_levels0 = JUSTIFY(queue)) == ) return 
}
// conflict
UNDO_ASSIGNMENTS(mark)
if((decision_levels1 = IMPLY(v->left)) == ) {
if((decision_levels1 = JUSTIFY(queue)) == ) return 
}
// conflict
UNDO_ASSIGNMENTS(mark)
decision_levels = decision_levels0  decision_levels1;
return decision_levels;
}
58
D-Algorithm
• In addition to controllability, we need to check observability of a
possible signal change at a vertex:
vertex need to be justified to 1 (0)
value change must be observable
at the output
• Two implementations:
– D-algorithm using five-valued logic {0,1,x,D,D}
• inject D at internal vertex
• expect D or D at one of the circuit outputs
– regular SAT problem based on replicated fanout structure
59
Five-Valued Implication Rules
B=~A
A B
0 1
1 0
X X
D ~D
~D D
C=A&B
A\B 0 1
0 0 0
1 0 1
X 0 X
D 0 D
~D 0 ~D
X
0
X
X
X
X
D ~D
0 0
D ~D
X X
D 0
0 ~D
60