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Java [3]The Optimization ProblemGiven a logic function, f: B n → B, with B = {0, 1}, find an optimal representation of f by:a) Exclusive-OR forms(XOR): f = C1 ⊕ C2 ⊕ … ⊕ Ck,b) Equivalence forms: f = D1 ∼ D2 ∼ … ∼ Dkc) Disjunctive forms: f = C1 ∨ C2 ∨ … ∨ Ckd) Conjunctive forms: f = D1 ∧ D2 ∧ … ∧ Dk“Optimal” can be defined as a minimum number of conjunctions, Ci, or disjunctions, Di, or as theminimum number of variables in the formula. For the XOR and equivalence forms, the optimizationcriteria will also be considered for different polarities: a formula has a given polarity if everyvariable appears only negated or only non-negated. For example, f = x1’ x2 x3 ⊕ x1’ x3 ⊕ x2, x1 isalways negated, x2, x3, and x4 are always non-negated.The challenge of these problems is the enormous complexity: for n variables, there are x 2functions,where x = 2 n but 2 y formulas of the given types, where y = 3 n . For practical applications, it is quitenormal to have values of n as high as 32 or even 64.A very import special case is the representation of logic functions by decision trees: The nodes ofthe trees are labelled by variables, the edges by values of variables (x=0 or x=1); the complexity ofthe trees depends on the order of the variables, hence, it is equal to n!, for n variables.Genetic Algorithms“Genetic Algorithms (GAs) are … stochastic algorithms whose search methods model some naturalphenomena: genetic inheritance and Darwinian strife for survival.” [4] They start with a populationof solution candidates (the initial population). Genetic operators, mutation and crossover, areapplied to individuals or pairs of individuals in the population to generate new members. A fitnessfunction is used to determine which individuals are solutions or the fittest at each stage of thealgorithm.Using a ternary coding and a complete enumeration of all conjunctions, every formula can berepresented (in principle) by a bit vector with the length 3 n . The Table gives examples of the ternarycoding scheme used to convert random numbers generated to the conjunctions they represent. Theconjunctions that comprise a particular formula are coded into the bit vector of length n 3.Table:Correspondence between the numbers generated and the conjunctions theyrepresent, for 3 variables.Random Numbers x1 x2 x3 Corresponding Conjunction5 0 1 - x1’x27 0 - 1 x1’x317 1 - - x120 - 0 - x2’Mutations change one or more bits randomly, while crossover involves splitting two bit vectors intoparts and then recombining parts of different vectors. The new vectors (formulas) will be includedinto the population according to their fitness values. The number of function values that are correct51