Circuit lower bounds in bounded arithmetics

Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

“…Our results expand on [26] and [38] by making them unconditional at the price of strengthening the lower bound which is shown to be unprovable. In more detail, we first show that the methods from [26,38] can be adapted to the case of nondeterministic lower bounds where they achieve a 'fixpoint' yielding an unconditional result. Next, we strengthen [26,38] by employing hardness amplification within NP.…”

“…In more detail, the proof proceeds by contradiction, and exploits a technique of Krajíček [26] (further elaborated on in [38]). If we assume that the lower bound for L(M ) is provable in T APC 1 , we can employ the KPT theorem [32] to conclude the existence of an interactive game witnessing errors of co-nondeterministic circuits attempting to compute L(M ).…”

“…1 Note that in a world where NP ∩ coNP = P, both results would trivialize -this distinguishes them from our results, which are unconditional and concerned with separations between NP and coNP. Pich [38] adapted the result of Krajíček to show that theories below PV 1 , such as VNC 1 , cannot prove that SAT is hard for p-size circuits, unless standard hardness assumptions break. Our results expand on [26] and [38] by making them unconditional at the price of strengthening the lower bound which is shown to be unprovable.…”

“…Pich [38] adapted the result of Krajíček to show that theories below PV 1 , such as VNC 1 , cannot prove that SAT is hard for p-size circuits, unless standard hardness assumptions break. Our results expand on [26] and [38] by making them unconditional at the price of strengthening the lower bound which is shown to be unprovable. In more detail, we first show that the methods from [26,38] can be adapted to the case of nondeterministic lower bounds where they achieve a 'fixpoint' yielding an unconditional result.…”

A remark on pseudo proof systems and hard instances of the satisfiability problem

“…Cf. for some recent discussions. Hard Sequence Hypothesis For every sat ‐algorithm there exists a polynomial time computable sequence which is hard for .…”