There are some issues with your description. Regarding solving it deterministically, it can be done pseudo-polynomially (although it's truly exponentially) using dynamic programming. That can be implemented in C# or Python, no problem.
Regarding solving it "nondeterministically", I guess the idea is to solve it polynomially (in a nondeterministic Turing machine), to show that it's in NP. The problem here is that you have to turn the problem into a decision problem (one that returns a boolean), so you have to add a quota to the parameters. The solution is based on "guessing" an order to add the items to the knapsack. This solution cannot be implemented in C# or Python (since these do not implement nondeterministic Turing machines). An implementation of the idea would need to "backtrack" to consider all possible item orderings, leading again to an exponential time (deterministic) solution. The problem is NP-complete, so no surprise here.
I didn't understand the "3 example formulae which are satisfiable and 3 that are not". I don't see this having to do with knapsack, but more with SAT.
Let me know if you want me to solve it for you.