Your question is not clear, but I think you're wondering how to show that 1in3SAT is NP-complete by reducing the well-known NP-complete 3SAT problem.

If this is your question, check the following settings in your notation:

There is an instance of 3SAT defined by a set of $ n $, Boolean variables $ V = {x_1, …, x_n } $ and a Boolean equation of $ m $ clauses $ Phi = C_1 land C_2 land … land C_m $ where everyone $ C_j = (x_ {j_1} l or x_ {j_2} l or x_ {j_3}) $,

To reduce this to 1in3SAT, start with the reduction in the following way.

For each $ C_j $ Create 3 new clauses $ C # _ {j, 1} = ( lnot x_ {j_1} lor alpha_j lor beta_j) $. $ C & # 39; {j, 1} = (x_ {j_2} lor beta_j lor gamma_j) $, and $ C & # 39; {j, 3} = ( lnot x_ {j_3} lor gamma_j lor delta_j) $ from where $ alpha_j, beta_j, gamma_j, delta_j $ are new Boolean random variables.

To let $ Phi & # 39; $ Be the Boolean equation in your 1in3SAT instance, which is now the AND of your new clauses. That means you have now $ 3m $ Clauses and $ n + $ 4 million Variables.

Next, consider what would happen if we had a true assignment for the 3SAT instance, and what would happen if we had a valid assignment for the newly constructed 1in3SAT instance (this second direction will be much simpler).