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Tadeusz Łuba Institute of Telecommunications

Methods of Logic Synthesis and Their Application in Data Mining Prezentacja wygłaszana na KNU Daegu (Korea), 25.11.2012. Tadeusz Łuba Institute of Telecommunications. Faculty of Electronics and Information Technology . Warsaw University of Technology. Logic synthesis vs. Data Mining.

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Tadeusz Łuba Institute of Telecommunications

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  1. Methods of Logic Synthesis and Their Application in Data MiningPrezentacja wygłaszana na KNU Daegu (Korea), 25.11.2012 Tadeusz Łuba Institute of Telecommunications Faculty of Electronics and Information Technology Warsaw University of Technology

  2. Logic synthesis vs. Data Mining • applicability of the logic synthesis algorithmsin Data Mining • data mining extendstheapplication of LS: • medicine • pharmacology • banking • linguistics • telecommunication • environmentalengineering Data Mining is the process of automatic discovery of significant and previously unknown information from large databases

  3. It is able todiagnose the patient It is able tomake a survey It is able toclassify data It is able todecide of granting a loanto the bank customer Data mining is also called knowledge discovery in databases

  4. Gaining the knowledge from databases at the abstract level of data mining algorithms - itmeansusingtheprocedure of: • reduction of attributes, • generalization of decision rules, • making a hierarchical decision These algorithmsare similar to thoseused in logic synthesis!

  5. Data mining vs. logic synthesis • generalization of decision rules • reduction of attributes • hierarchical decision making • minimization of the Boolean function • reduction of arguments • functional decomposition (rule induction) (logic minimization)

  6. Data mining systems ROSETTA Rough Set Toolkit forAnalysis of Data: Biomedical Centre (BMC), Uppsala, Sweden. http://logic.mimuw.edu.pl/~rses/ http://www.lcb.uu.se/tools/rosetta/

  7. Breast Cancer Database: Diagnosis of breast cancer • Number of instances: 699 training cases • Number of attributes : 10 • Classification (2 classes) Clump Thickness Uniformity of Cell Size Uniformity of Cell Shape …. 9. Mitoses Sources: Dr. WIlliam H. Wolberg (physician); University of Wisconsin Hospital ;Madison; Wisconsin; USA

  8. Breast Cancer database These are the data after discretization

  9. RULE_SET breast_cancer RULES 35 (x9=1)&(x8=1)&(x2=1)&(x6=1)=>(x10=2) (x9=1)&(x2=1)&(x3=1)&(x6=1)=>(x10=2) (x9=1)&(x8=1)&(x4=1)&(x3=1)=>(x10=2) (x9=1)&(x4=1)&(x6=1)&(x5=2)=>(x10=2) ………………….. (x9=1)&(x6=10)&(x1=10)=>(x10=4) (x9=1)&(x6=10)&(x5=4)=>(x10=4) (x9=1)&(x6=10)&(x1=8)=>(x10=4) REDUCTS (27) { x1, x2, x3, x4, x6 } { x1, x2, x3, x5, x6 } { x2, x3, x4, x6, x7 } { x1, x3, x4, x6, x7 } { x1, x2, x4, x6, x7 } ……………. { x3, x4, x5, x6, x7, x8 } { x3, x4, x6, x7, x8, x9 } { x4, x5, x6, x7, x8, x9 }

  10. Increasingrequirements We areoverwhelmed with data! References: [1]

  11. UC Irvine Machine Learning Repository Are existing methods and algorithms for data mining sufficiently efficient? BreastCancerDatabase – 10 attr Audiology Database– 71 attr Dermatology Database– 34 attr Why does it take place? How these algorithms can be improved?

  12. Classic method Discernibility matrix (DM) Discernibility function (DF) Conjunction of clauses Clause is a disjunction of attributes References: [9] The key issue is to transform the DF: NP- HARD CNF DNF Every monomial corresponds to a reduct

  13. The method can be significantlyimproved ... by using the typical logic synthesis procedure for Boolean function complementation Instead of transforming CNF to DNF we represent CNF in binary matrix M BM is treated as Boolean functionF Complement of function F F is always unate function!

  14. Using Complement Theorem… M: fM = x1x2x4 + x3x4 + x1x2 + x1x4 = x1x4 + x3x4 + x1x2 .i 4 .o 1 .p 4 11-1 1 --11 1 11-- 1 1--1 1 .end The same result! Discernibility function F = (x1 + x2 + x4) (x3 + x4) (x1 + x2)(x1 + x4) = (x1 + x2)(x1 + x4) (x3 + x4) = (x1 + x2x4) (x3 + x4) = x1x3+ x2x4 + x1x4

  15. The key element Fast ComplementationAlgorithm F = RecursiveComplementationTheorem: whereiscalledthecofactor of F withrespect to variablexj The problem of complementingfunctionF istransformedintothe problem of findingthecomplements of twosimplercofactors

  16. UnateComplementation The entire process reduces to three simple calculations: • The choice of the splitting variable • Calculation of Cofactors • Testing rules for termination F = xjF0 + F1 Matrix M Cofactor 1 Cofactor 0 …. …. Cofactor Cofactor Complement • Merging Complement Complement

  17. An Example F = x4 x6(x1 + x2) (x3 + x5 + x7)(x2 + x3)(x2 + x7) .i 7 .o 1 .p 6 11----- 1 --1-1-1 1 -11---- 1 -1----1 1 ---1--- 1 -----1- 1 .end

  18. An Example… x2, x3, x2, x5, x2, x7, x1, x3, x7 Reducts: {x2,x3,x4,x6} {x2,x4,x5,x6} {x2,x4,x6,x7} {x1,x3,x4,x6,x7}

  19. Verification Calculating reducts using the standard method: (x1 + x2) (x3 + x5 + x7) (x2 + x3) (x2 + x7) = = (x2 +x1)(x2 + x3)(x2 + x7)(x3 + x5 + x7) = (x3 + x5 + x7) = =(x2 +x1x3x7) = x2x3+ x2x5 +x2x7 + x1x3x7  {x4,x6} {x2,x4,x6,x7} {x2,x4,x5,x6} {x2,x3,x4,x6} The same set of reducts! {x1,x3,x4,x6,x7}

  20. Boolean function KAZ .type fr .i 21 .o 1 .p 31 100110010110011111101 1 111011111011110111100 1 001010101000111100000 1 001001101100110110001 1 100110010011011001101 1 100101100100110110011 1 001100100111010011011 1 001101100011011011001 1 110110010011001001101 1 100110110011010010011 1 110011011011010001100 1 010001010000001100111 0 100110101011111110100 0 111001111011110011000 0 101101011100010111100 0 110110000001010100000 0 110110110111100010111 0 110000100011110010001 0 001001000101111101101 0 100100011111100110110 0 100011000110011011110 0 110101000110101100001 0 110110001101101100111 0 010000111001000000001 0 001001100101111110000 0 100100111111001110010 0 000010001110001101101 0 101000010100001110000 0 101000110101010011111 0 101010000001100011001 0 011100111110111101111 0 .end All solutions : Withthesmallestnumber of arguments: 35, With the minimum number of arguments: 5574 01010 1 10110 1 00100 1 01001 1 01000 1 11010 1 10011 0 01110 0 10100 0 11000 0 11011 0 10000 0 00010 0 01111 0 00011 0 11111 0 00000 0 01101 0 00110 0 Computation time: RSES = 70 min. Proposedmethod= 234 ms. 18000 timesfaster!

  21. Conclusion The new method reduces the computation time a couple of orders of magnitude RSES How this acceleration will affect the speed of calculations for typical databases?

  22. Experimental results The absolute triumphof the complementation method!

  23. Further possibilities… of application of logic synthesis methods in issues of Data Mining • generalization of decision rules • hierarchical decision making • minimization of the Boolean function • Functional decomposition

  24. RSES vs Espresso ESPRESSO RSES .i 7 .o 1 .type fr .p 9 1000101 0 1011110 0 1101110 0 1110111 0 0100101 1 1000110 1 1010000 1 1010110 1 1110101 1 .e TABLE extlbis ATTRIBUTES 8 x1 numeric 0 x2 numeric 0 x3 numeric 0 x4 numeric 0 x5 numeric 0 x6 numeric 0 x7 numeric 0 x8 numeric 0 OBJECTS 9 1 0 0 0 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 1 (x1=1)&(x5=1)&(x6=1)&(x2=1)=>(x8=0) (x1=1)&(x2=0)&(x5=1)&(x3=0)&(x4=0)&(x6=0)=>(x8=0) (x4=0)&(x1=1)&(x2=0)&(x7=0)=>(x8=1) (x2=1)&(x4=0)&(x5=1)&(x6=0)=>(x8=1)

  25. Hierarchical decision making Is it possible to use the decomposition to solve difficult tasks of data mining? attributes attributes A B DT (G) Decomposition decisiontable F = H(A,G(B)) intermediatedecision DT (H) G  P(B) P(A)  G  PD final decision decision

  26. Data compression democrat n y y n y y n n n n n n y y y y republican n y n y y y n n n n n y y y n y democrat y y y n n n y y y n y n n n y y democrat y y y n n n y y y n n n n n y y democrat y n y n n n y y y y n n n n y y democrat y n y n n n y y y n y n n n y y democrat y y y n n n y y y n y n n n y y republican y n n y y n y y y n n y y y n y ……………………….......... democrat y y y n n n y y y n y n n n y y republican y y n y y y n n n n n n y y n y republican n y n y y y n n n y n y y y n n democrat y n y n n n y y y y y n y n y y democrat y n y n n n y y y n n n n n n y democrat y n y n n n y y y n n n n n y y G The data set HOUSE (1984 United States Congressional Voting Records Database) Decomposition Decomposition H 68% space reduction

  27. Summary • Typical logic synthesis algorithms and methods are effectively applicable to seemingly different modern problems of data mining • Also, it is important to study theoretical foundations of new concepts in data mining e.g. functional decomposition • Solving these challenges requiresthe cooperation of specialists from different fields of knowledge

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