🤖 AI Summary
For nearly three decades, it has remained an open question whether circuits composed solely of modular counting gates (MODₘ) can efficiently compute the n-ary Boolean AND function. This work addresses this problem within the restricted yet natural model of input-permutation-symmetric MODₘ circuits. By integrating techniques from circuit complexity theory, symmetry-aware constraint modeling, combinatorial analysis, and group action methods, we establish the first optimal subexponential-size lower bound for computing AND in symmetric modular circuits—matching the size of the best-known constructions. Specifically, we prove that any depth of symmetric MODₘ circuit requires subexponential size to compute AND, and that this bound is already tight at depth two. Furthermore, we extend our result to generalized symmetric settings with nested block structures, yielding similarly tight lower bounds.
📝 Abstract
A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. Håstad's celebrated switching lemma yields exponential lower bounds for the dual problem - realising modular arithmetic with Boolean gates - but, a similar lower bound for modular circuits computing the Boolean AND function has remained elusive for almost 30 years. We solve this problem for the restricted model of symmetric circuits: We consider MOD$_m$-circuits of arbitrary depth, and for an arbitrary modulus $m \in \mathbb{N}$, and obtain subexponential lower bounds for computing the $n$-ary Boolean AND function, under the assumption that the circuits are syntactically symmetric under all permutations of their $n$ input gates. This lower bound is matched precisely by a construction due to (Idziak, Kawałek, Krzaczkowski, LICS'22), leading to the surprising conclusion that the optimal symmetric circuit size is already achieved with depth $2$. Motivated by another construction from (LICS'22), which achieves smaller size at the cost of greater depth, we also prove tight size lower bounds for circuits with a more liberal notion of symmetry characterised by a nested block structure on the input variables.