Abstract. In this paper we initiate the study of proof systems where verification of proofs proceeds by NC 0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC 0 functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC 0 proof systems for a variety of languages ranging from regular to NP-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC 0 proof systems. We also present a general construction of NC 0 proof systems for regular languages with strongly connected NFA's.

We provide a characterisation for the size of proofs in tree-like Q-Resolution and tree-like QU-Resolution by a Prover-Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution. This gives one of the first successful transfers of one of the lower bound techniques for classical proof systems to QBF proof systems. We apply our technique to show the hardness of three classes of formulas for tree-like Q-Resolution. In particular, we give a proof of the hardness of the parity formulas from Beyersdorff et al. (2015) [10] for tree-like Q-Resolution and of the formulas of Kleine Büning et al. (1995) [29] for tree-like QU-Resolution.

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC 0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC 0 functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC 0 proof systems for a variety of languages ranging from regular to NP complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC 0 proof systems. We also show that Majority does not admit NC 0 proof systems. Finally, we present a general construction of NC 0 proof systems for regular languages with strongly connected NFA's.

The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free complexity is a natural generalization of monotone complexity to all (not necessarily monotone) Boolean functions. Thus, we can apply known monotone complexity lower bounds to find lower bounds on the hazard-free complexity. We also lift these methods from the monotone setting to prove exponential hazard-free complexity lower bounds for nonmonotone functions. As our main upper-bound result, we show how to efficiently convert a Boolean circuit into a boundedbit hazard-free circuit with only a polynomially large blow-up in the number of gates. Previously, the best known method yielded exponentially large circuits in the worst case, so our algorithm gives an exponential improvement. As a side result, we establish the NP-completeness of several hazard detection problems.

We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero. We give a deterministic polynomial time algorithm for this problem when the inputs are read-twice or readthrice formulas. In the process, these algorithms also test if the input circuit is computing a multilinear polynomial.We further study three related computational problems on arithmetic circuits. Given an arithmetic circuit C, 1) ZMC: test if a given monomial in C has zero coefficient or not, 2) MonCount: compute the number of monomials in C, and 3) MLIN: test if C computes a multilinear polynomial or not. These problems were introduced by Fournier, Malod and Mengel [STACS 2012], and shown to characterize various levels of the counting hierarchy (CH).We address the above problems on read-restricted arithmetic circuits and branching programs. We prove several complexity characterizations for the above problems on these restricted classes of arithmetic circuits.