On theories of bounded arithmetic for NC1

  title={On theories of bounded arithmetic for NC1},
  author={Emil Jer{\'a}bek},
  journal={Ann. Pure Appl. Logic},
We develop an arithmetical theory VNC ∗ and its variant VNC 1 ∗, corresponding to “slightly nonuniform” NC . Our theories sit betweenVNC 1 and VL, and allow evaluation of log-depth bounded fan-in circuits under limited conditions. Propositional translations of Σ0 (LVNC1∗)-formulas provable inVNC 1 ∗ admit L-uniform polynomial-size Frege proofs.