Multi-layer transformers form the critical component of essentially all large language models (LLMs) in use today. Because of their ubiquity and computational capability, there is a rapidly growing body of work that aims to precisely calibrate the expressive power of transformers as language recognizers by comparing them against standard models of computation studied for decades by the theoretical computer science community. In this endeavor, circuit complexity has by and large emerged as the "correct" branch of computational complexity to analyze the expressive power of transformers; the reason is that parameterizing transformers by the various resources they use, such as attention and precision, leads to direct comparisons with different classes of circuits parameterized by resources such as type of gates, size, and depth. Here, we present an overview of selected results that delineate the expressive power of transformers using concepts and methods from circuit complexity.
Chain of Thought (CoT) lifts the expressive ceiling of bounded-depth Transformers, with characterizations tying the number of CoT steps to circuit complexity classes. What remains largely missing are concrete instantiations with explicit, depth-bounded constructions, and the traversal procedures such characterizations presuppose. We close this gap for branching complexity. We give CoT realizations of depth-first search (DFS) and of Dijkstra algorithm, the latter subsuming breadth-first search, by unique hard-attention decoders of at most two layers, and use them as a shared computational substrate: reusing the DFS decoder yields the Strahler number of an $n$-vertex tree in $2n-1$ steps with four layers, and reusing the Dijkstra decoder yields its width in $n-1$ steps with three. Since computing the Strahler number of a binary tree given as a term is \textsf{NC\textsuperscript{1}}-complete, and our constructions handle arbitrary $n$-ary trees without layer normalization or positional encodings, this is a non-trivial witness for the linear-step regime of the CoT hierarchy. Exploiting the classical bijection between ordered trees and Dyck paths, itself realized by our DFS construction, which emits the path as it traverses, we give independent constructions for both measures on the path representation.
Christoph Hertrich, Moritz Stargallacs.CC cs.LG math.CO
We study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement $\max$, $+$, or multiplication with a positive constant. For such circuits, we prove exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. As a corollary, we obtain an exponential size separation between monotone and non-monotone maxout neural networks, which generalize the popularly used ReLU neural networks. One conclusion from this is that neural network models with enforced convexity constraints, such as input-convex neural networks (ICNNs), sometimes need to be exponentially larger than their unrestricted counterparts in order to express the same functions.