This document discusses using rough set theory to improve circuit miniaturization and efficiency. It presents an example of applying rough set concepts like indiscernibility relations, lower and upper approximations, and decision rules to reduce the number of logic gates in a circuit without changing the output. The example generates data from each gate, analyzes it using rough set approximations and rules to identify redundant gates, allowing the circuit to be minimized. This technique could help reduce chip size, switching power usage, and increase the number of transistors implemented on a chip.

1. Improving circuit miniaturization
and its efficiency using Rough Set
Theory
Presented by :Sarvesh Singh Rawat

2. Introduction
One goal of the Knowledge Discovery is extract meaningful
knowledge.
Rough Sets theory was introduced by Z. Pawlak (1982) as
a mathematical tool for data analysis.
Rough sets have many applications in the field of
Knowledge Discovery: feature selection, discretization
process, data imputations and create decision Rules.
Rough set have been introduced as a tool to deal with,
uncertain Knowledge in Artificial Intelligence Application.

3. Equivalence Relation
Let X be a set and let x, y, and z be elements of X.
An equivalence relation R on X is a Relation on X
such that:
Reflexive Property: xRx for all x in X.
Symmetric Property: if xRy, then yRx.
Transitive Property: if xRy and yRz, then xRz.

4. Rough Sets Theory
Let T (U , A, C, D), be a Decision system data,
Where: U is a non-empty, finite set called the universe ,
A is a non-empty finite set of attributes, C and D are subsets
of A, Conditional and Decision attributes subsets
respectively.
is called the value set of a ,
The elements of U are objects, cases, states, observations.
The Attributes are interpreted as features, variables,
characteristics conditions, etc.
a :U
Va for a
A, V a

5. Indiscernibility Relation
The Indecernibility relation IND(P) is an
equivalence relation.
Let a A , P , the indiscernibility
A
relation IND(P), is defined as follows: IND( P) {(x, y) U U :
for all a
P,
a ( x)
a( y)}

6. Indiscernibility Relation
The indiscernibility relation defines a partition in U.
Let P A, U/IND(P) denotes a family of all equivalence
classes of the relation IND(P), called elementary sets.
Two other equivalence classes U/IND(C) and
U/IND(D), called condition and decision equivalence
classes respectively, can also be defined.

7. R-lower approximation
Let X U and R C , R is a subset of conditional
features, then the R-lower approximation
set of X, is the set of all elements of U which
can be with certainty classified as elements of X.
RX
{Y
U / R :Y
X}
R-lower approximation set of X is a subset of X

8. R-upper approximation
the R-upper approximation set of X, is the
set of all elements of U such that:
RX
{Y
U / R :Y
X
}
X is a subset of R-upper approximation set of X.
R-upper approximation contains all data which can possibly
be classified as belonging to the set X
the R-Boundary set of X is defined as:
BN ( X )
RX
RX

9. Representation of the approximation sets
If
If
RX
RX
RX
RX
then, X is R-definible (the boundary set is empty)
then X is Rough with respect to R.
ACCURACY := Card(Lower)/ Card (Upper)

10. Decision Class
The decision d determines the partition
CLASS T (d ) { X 1 ,..., X r ( d ) } of the universe U.
Where X k {x U : d ( x) k} for 1 k r (d )
CLASS T (d )
will be called the classification of objects
in T
determined by the decision d.
The set Xk is called the k-th decision class of T

11. Decision Class
This system data information has 3 classes, We represent the
partition: lower approximation, upper approximation and boundary
set.

12. Dispensable feature
Let R a family of equivalence relations and let P R,
P is dispensable in R if IND(R) = IND(R-{P}),
otherwise P is indispensable in R.
CORE
The set of all indispensable relation in C will be called the
core of C.
CORE(C)= ∩RED(C), where RED(C) is the family of all
reducts of C.

13. CASE STUDY
Circuit - miniaturization
● In this section, a simple structure using logic gates
is shown which is the magnified view of a portion of
complicated circuit and it is further reduced using
Rough Set based on logical classifier and rules.

14. Information –Table (I)
,
,
{
,
{{
, ,
● A set of data is generated by each gate in binary form
(either 0 or 1), and the wires represents that attributes.
● The basic idea behind circuit miniaturization is to mine te
data that is obtained as a result of each gate in a logical
manner using algebraic developments so that the final
result is not altered.
● The example that is shown here is a small circuit but the
same technique can be implemented in bigger circuits
using the same procedure.

15. Information Table (II)

16. Approximations
● Let X U and R C R is a subset of conditional
,
features, then the R-lower approximation
RX
{Y
U / R :Y
X}
● The R-upper approximation set of X, is the
set of all elements of U such that:
RX
{Y
U / R :Y
X
● The accuracy of approximation is given by
| ( PX ) |
p(X )
| ( PX ) |
}

17. Approximation
● From our information system, we have two classes of
decision set as 0 and 1. As the data value is discrete
(either 0 or 1) so the total number of lower
approximations is equal to that of the upper
approximations.

18. Decision rules
The algorithm should minimize the number of features
included in decision rules.

19. Conclusions and outcomes
 We have reduced the number of gates without affecting
the output of the given circuit using the mathematical
model of Rough Set Theory.
 It saves a lot of time and power that is wasted in
switching of gates , the wiring-crises is reduced, crosssectional area of chip is reduced, the number of
transistors that can implemented in chip is multiplied
many folds.

20. References
 Pawlak, Z. (1997). Rough set approach to knowledge-based
decision support. European journal of operational
research, 99(1), 48-57.
 Pawlak, Z. (1998). Rough set theory and its applications to
data analysis. Cybernetics & Systems, 29(7), 661-688.
 Roy, S, S. Viswanatham, V, M. Krishna, P, V. Saraf, N, Gupta.
A, and Mishra, R. (2013). Applicability of Rough Set Technique
for Data Investigation and Optimization of Intrusion Detection
System. 9th International Conference, QShine 2013, India,
January 11-12, 2013,(pp.479-484).
 Roy, S, S. Viswanatham, V, M. Rawat, S, S. Shah, H. (2013).
Multicriteria decision examination for electrical power grid
monitoring system. Intelligent Systems and Control (ISCO),
2013 7th International Conference on pp. 26-30.

21. THANK YOU!