# Change the structure from 3SAT to 1in3 3SAT

There is a variable set V = {x1,x2,x3} and clause set C1={x1,x2,-x3} C2={x1,-x1,-x2} C3={-x1,-x2,x3} C4={x2,x3,-x3}. For this structure, no matter each variable is positive or negative, the clause can always be TRUE by at least one literal is TRUE within each clause, just like 3SAT. How can I change the variable gadgets and clause gadgets to make it no matter each variable is positive or negative, the clause can always be TRUE by only one literal is TRUE within each clause, just like 1 in3 3SAT?

If this is your question, then consider the following set up, using your notation: Let an instance of 3SAT be 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 each $$C_j = (x_{j_1}\lor x_{j_2} \lor x_{j_3})$$.
To reduce this to 1in3SAT begin the reduction in the following manner. For each $$C_j$$ create 3 new clauses $$C'_{j,1} = (\lnot x_{j_1}\lor \alpha_j \lor \beta_j)$$, $$C'_{j,1} = (x_{j_2}\lor \beta_j \lor \gamma_j)$$, and $$C'_{j,3} = (\lnot x_{j_3}\lor \gamma_j \lor \delta_j)$$ where $$\alpha_j, \beta_j, \gamma_j, \delta_j$$ are new Boolean random variables. Let $$\Phi'$$ be the Boolean equation in your 1in3SAT instance which now the AND over your these new clauses. That is, you now have $$3m$$ clauses and $$n + 4m$$ variables.