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(Note: this was asked in the cstheory stackexchange, but I was told this is a more appropriate place for it.)

I have three 8-bit 2's-complement numbers (X, Y, and Z). I want to find a propositional logic formula using only the variables (x1, x2... x8, y1... y8, z1... z8), where xi is the ith bit of x which expresses the concept that X + Y = Z. How would I go about doing this?

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You can obtain a formula in CNF form by writing down a truth table (with $2^{24}$ rows) and then converting each row to a single clause. This will yield a formula with $2^{24}$ clauses, but the method is straightforward.

If you want a shorter formula, I suggest using logic synthesis, e.g., Quine-McClusky, Espresso.

The usual path to get a small formula is to allow introduction of additional variables, build a circuit (e.g., an adder circuit followed by a comparator), then convert to a formula using the Tseitin transform. This yields much smaller formulas, but it requires introducing extra variables, which you said you don't want.

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