Complexity theory in Feasible Mathematics

Title: Complexity Theory in Feasible Mathematics Author: Ján Pich Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajíček, DrSc., MAE Abstract: We study the provability of statements and conjectures from Complex- ity Theory in Bounded Arithmetic. First, modulo a hardness assumption, we show that theories weaker in terms of provably total functions than Buss's theory S1 2 cannot prove nk -size circuit lower bounds for SAT formalized as a Σb 2-formula LB(SAT, nk ). In particular, the true universal first-order theory in the language containing names for all uniform NC1 algorithms denoted TNC1 does not prove LB(SAT, n4kc ) where k ≥ 1, c ≥ 2 unless each function f ∈ SIZE(nk ) can be approximated by formulas Fn of subexponential size 2O(n1/c) with subexponential advantage: Px∈{0,1}n [Fn(x) = f(x)] ≥ 1/2 + 1/2O(n1/c) . Unconditionally, V 0 does not prove quasipolynomial nlog n -size circuit lower bounds for SAT. Considering upper bounds, we prove the PCP theorem in Cook's theory PV1. This includes a formalization of the (n, d, λ)-graphs in PV1. A consequence of the result is that Extended Frege proof system admits p-size proofs of tautologies encoding the PCP theorem. Keywords: Circuit Lower Bounds, Bounded Arithmetic, The PCP theore