Witnessing of existential statements

The thesis formulates and proves a witnessing theorem for SPV -provable formulas in the form ∀x∃yA(x, y) where A corresponds to a polynomial time decidable relation. By SPV we understand an extension of the theory TPV (the universal theory of N in the language representing polynomial algorithms) by additional axioms ensuring the existence of a minimum of a linear ordering defined by a polynomial time decidable relation on an initial segment. As these additional axioms are not universal sentences, the theory SPV requires nontrivial use of witnessing Herbrand's and KPT theorems which have direct application only for universal theories. Based on the proven witnessing theorem, we derive a NP search problem characterizing complexity of finding y for a given x such that A(x, y). A part of the thesis is dedicated to arguments supporting the conjecture that SPV is strictly stronger than TPV .