What are the points to be taken into account to compare the enol %?
Ans) Different propositional proof system for theorem proving in propositional logic, like sequent calculus, the cutting-plane method, resolution, the DPLL algorithm, etc. produce dissimlar proofs when applied to the same formula. Proof complexity calculates the efficiency of a method in terms of the size of the proofs it produces.
Two points make the study of proof complexity non-trivial:
The size of a proof depends on the formula that is to be proved inconsistent;
Proof methods are generally families of algorithms, as some of their steps are not univocally specified; for example, resolution is based on iteratively choosing a pair of clauses containing opposite literals and producing a new clause that is a consequence of them; since various such pairs may be available at every step, the algorithm has to choose one; these choices affect the proof length.
The first point is taken into account by comparing the size of a proof of a formula with the size of the formula. This comparison is made using the usual assumptions of computational complexity: first, a polynomial formula size/proof size ratio means that the proof is of size same to that of the formula; second, this ratio is studied in the asymptotic case as the size of the formula increases.
The second point is taken into account by considering, for every formula, the shortest possible proof the considered method can produce.