Can we transform a CNF to ILP without introducing new variables?My question can be seen as a follow up to Express boolean logic operations in zero-one integer linear programming (ILP) as the solution proposed in that post introduces additional variables.Also, my current understanding is that each 0 value in the truth table is represented by one linear inequality. This means that if my boolean function is of 16 input variables and balanced, there will be 2^15 linear inequalities so is there a way to reduce the number of inequalities?
From CNF to ILP?
integer programming
No.  Usually (for most functions / CNF formulas) you'll need additional variables.  Linear inequalities form a convex space.  If the set of solutions such that $f(x)=1$ is not convex, you'll need additional variables.The exact number of additional variables and inequalities required depends on the function / on the CNF formula.