Interactive arguments often combine polynomial IOPs with polynomial commitment schemes (PCSs). Frequently, the interactive argument is proven to be knowledge sound, but this incurs a high security loss when applying the Fiat-Shamir transformation to obtain a non-interactive argument in the random oracle model (ROM). We introduce the notion of special soundness for polynomial IOPs, which surprisingly has not been considered before. – We study relations between various binding properties of univariate PCSs. In the case of the KZG PCS, these properties can be based on falsifiable assumptions. – We prove that a special-sound polynomial IOP plus a PCS under suitable binding notions gives a computationally special-sound interactive argument. By Attema, Fehr, and Klooß (TCC 2022), applying Fiat-Shamir to this argument yields a tightly knowledge-sound argument (or zk-SNARK) in the ROM under the same assumptions. – In the case of the KZG PCS, we add various batching optimizations to our compiler and prove that they preserve computational special soundness. This yields a generic approach for achieving efficient zk-SNARKs with constant proof size and tight knowledge soundness in the ROM under falsifiable assumptions.