A proof system for a language $L$ is a function $f$ such that Range$(f)$ is exactly $L$. In this paper, we look at proofsystems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by bounded fanin circuits of constant depth (NC$^0$), or of $O(\log \log n)$ depth but with $O(1)$ alternations (poly$\log AC^0$). Each output bit depends on very few input bits; thus such proof systems correspond to a kind of local error-correction on a theorem-proof pair.

We identify exactly how much power we need for proof systems to capture all regular languages. We show that all regular language have poly$\log AC^0$ proof systems, and from a
previous result (Beyersdorff et al, MFCS 2011, where NC$^0$ proof systems were first introduced), this is tight. Our technique also shows that MAJ has poly$\log AC^0$ proof
system.

We explore the question of whether TAUT has NC$^0$ proof systems. Addressing this question about 2TAUT, and since 2TAUT is closely related to reachability in graphs, we ask the same question about Reachability. We show that both Undirected Reachability and Directed UnReachability have NC$^0$ proof systems, but Directed Reachability is still
open.

In the context of how much power is needed for proof systems for languages in NP, we observe that proof systems for a good fraction of languages in NP do not need the full power of AC$^0$; they have SAC$^0$ or coSAC$^0$ proof systems.