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.