Jump Operators, Interactive Proofs and Proof Complexity Generators

A jump operator <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$J$</tex> in proof complexity is a function such that for any proof system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P, J(P)$</tex> is a proof system that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> cannot simulate. Some candidate jump operators were proposed by Krajicek and Pudlak [63] and Krajicek [57], but it is an open problem whether computable jump operators exist or not. In this regard, we introduce a new candidate jump operator based on the power of interactive proofs which given a proof system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex>, [I P, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P]$</tex> (I P-randomized implicit proof system based on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P)$</tex> is an M A proof system. In the first step, we investigate the relationship between IP - randomized implicit proof systems and Cook-Reckhow proof systems. In particular, we show that if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i$</tex> EF (Krajicek's implicit Extended Frege) proves exponential hard on average circuit lower bounds efficiently, then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i$</tex> E F simulates [IP, EF]. Moreover, we show that IP-randomized implicit proof systems can be used to prove new connections between different well-studied concepts in complexity theory. Namely, we prove new results about the hardness magnification in proof complexity, the hardness of proving proof complexity lower bounds, the automatability and the feasible disjunction property for Extended Frege using IP - randomized implicit proof systems. One ingredient of our proofs is a formalization of the sum-check protocol [65] in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{s}_{2}^{1}$</tex> which might be of independent interest. We also look at the general theory of jump operators and consider an old conjecture by Pudlak [78] about finite consistency sentences for first-order theories of arithmetic. In this direction, we prove that certain statements are equivalent, in particular, we prove that the widely believed assumption about the existence of computable jump operators in proof complexity is equivalent to a weaker form of Pudlak's conjecture.