ECCC-Report TR22-055https://eccc.weizmann.ac.il/report/2022/055Comments and Revisions published for TR22-055en-usSun, 24 Apr 2022 23:24:46 +0300
Paper TR22-055
| Simple Hard Instances for Low-Depth Algebraic Proofs |
Nashlen Govindasamy,
Tuomas Hakoniemi,
Iddo Tzameret
https://eccc.weizmann.ac.il/report/2022/055We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant $\sum_{i,j,k,l\in[n]} z_{ijkl}x_ix_jx_kx_l-\beta = 0$, for Boolean variables, does not have polynomial-size IPS refutations where the refutations are multilinear and written as constant-depth circuits.
Andrews and Forbes (STOC’22) established recently a constant-depth IPS lower bound, but their hard instance does not have itself small constant-depth circuits, while our instance is computable already with small depth-2 circuits.
Our argument relies on extending the recent breakthrough lower bounds against constant-depth algebraic circuits by Limaye, Srinivasan and Tavenas (FOCS’21) to the functional lower bound framework of Forbes, Shpilka, Tzameret and Wigderson (ToC’21), and may be of independent interest. Specifically, we construct a polynomial $f$ computable with small-size constant-depth circuits, such that the multilinear polynomial computing $1/f$ over Boolean values and its appropriate set-multilinear projection are hard for constant-depth circuits.Sun, 24 Apr 2022 23:24:46 +0300https://eccc.weizmann.ac.il/report/2022/055