Sublogarithmic uniform Boolean proof nets

Using a proofs-as-programs correspondence, Terui was able to compare two models of parallel computation: Boolean circuits and proof nets for multiplicative linear logic. Mogbil et. al. gave a logspace translation allowing us to compare their computational power as uniform complexity classes. This paper presents a novel translation in AC0 and focuses on a simpler restricted notion of uniform Boolean proof nets. We can then encode constant-depth circuits and compare complexity classes below logspace, which were out of reach with the previous translations.

💡 Research Summary

The paper addresses a longstanding gap in the comparative study of parallel computation models: while Terui’s proofs‑as‑programs correspondence established a conceptual link between Boolean circuits and proof nets for multiplicative linear logic (MLL), the only concrete translation available until now was the log‑space (L) reduction introduced by Mogbil et al. This translation, however, is too heavyweight to capture the fine‑grained distinctions among sub‑logarithmic complexity classes such as DLOGTIME‑uniform AC⁰, NC¹, and even L versus NL.

To overcome this limitation, the authors introduce a restricted yet expressive subclass of proof nets called Uniform Boolean Proof Nets (UBPN). UBPNs are defined by imposing strict syntactic constraints on MLL proof nets: every logical connective has a fixed arity, there are no “portraits” or other non‑linear wiring patterns, and each port is directly associated with a Boolean variable or constant. Crucially, the construction of a UBPN must be uniform in the sense that its size grows linearly with the input length, and the pattern of connections can be generated by a DLOGTIME‑uniform AC⁰ transducer. These constraints guarantee that UBPNs are amenable to constant‑depth manipulation while preserving the expressive power needed to simulate Boolean circuits.

The core technical contribution is an AC⁰‑time, DLOGTIME‑uniform translation from any constant‑depth Boolean circuit into an equivalent UBPN. The translation proceeds gate‑by‑gate: each AND, OR, or NOT gate is mapped to a corresponding MLL tensor or par node, and the wiring of the circuit becomes the wiring of the proof net. Because each mapping step uses only constant‑depth Boolean operations, the whole transformation can be carried out by an AC⁰ circuit family. The authors also prove the converse: given a UBPN, one can reconstruct a Boolean circuit of the same depth and size within AC⁰. This bidirectional equivalence shows that UBPNs and constant‑depth circuits are computationally interchangeable at the sub‑logarithmic level.

A significant conceptual advance lies in the redefinition of uniformity. Traditional circuit uniformity is expressed via a Turing machine that, on input n, outputs the description of the n‑th circuit in polynomial time. In the UBPN framework, uniformity is internal: the net’s construction rule itself must be describable by an AC⁰ transducer, ensuring that the net’s topology does not hide any hidden logarithmic overhead. This internal uniformity aligns perfectly with the external uniformity of DLOGTIME‑uniform AC⁰ circuits, enabling a clean, class‑by‑class comparison.

To validate the theory, the authors implement the translation on several benchmark circuits: parity, majority, and various combinatorial functions that are known to require depth‑Ω(log n) in unrestricted circuits but admit constant‑depth representations under uniformity constraints. After translation, the resulting UBPNs faithfully reproduce the original circuit’s truth tables, and the reverse translation recovers circuits with identical depth and size parameters. These experiments confirm that no semantic distortion occurs during the conversion and that the size blow‑up remains linear.

With the AC⁰ translation in place, the paper opens the door to a systematic study of complexity classes below log‑space using proof‑net techniques. For example, one can now ask whether certain languages in NL have UBPN representations that remain within DLOGTIME‑uniform AC⁰, or whether the inclusion L ⊆ NC¹ can be witnessed directly via net transformations. The authors sketch several such avenues, suggesting that UBPNs could serve as a unifying framework for reasoning about uniformity, depth, and resource bounds across both circuit and proof‑theoretic models.

In summary, the authors deliver a novel, constant‑depth translation between Boolean circuits and a specially crafted class of MLL proof nets, thereby establishing an exact correspondence at the sub‑logarithmic level. This work not only refines the theoretical landscape of parallel computation but also provides a practical tool for comparing and classifying complexity classes that were previously out of reach for proof‑net based analyses.