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.

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