A Compositional Theory of Causally Masked Transformers

What types of decision problems can a causally masked, finite-precision transformer solve for inputs of arbitrary length? Existing answers often rely on idealized arithmetic, but under finite precision, rounding and evaluation order can change what information attention retains and therefore what the model can compute. We develop an algebraic formalization that derives expressivity directly from the model's implemented dynamics. Its central object is its memory; the finite internal state computed by attention that summarizes the information from the prefix available to all future queries. Each attention head updates its own state independently within a layer, while layers compose hierarchically, providing a uniform route from model assumptions to expressivity bounds. Applying this method to transformers without positional embeddings, we obtain an expressivity hierarchy governed by the attention type under specific numerical semantics. Width-one sliding-window attention supports bounded-suffix memory, while a modified form of soft attention supports irreversible, checklist-like state, and combining the two mechanisms provides an interplay of both. Ordinary left-to-right floating-point soft attention can realize more expressive memory operations than any of the above. Algebraically, the four cases correspond to definite, R-trivial, locally R-trivial, and aperiodic semigroups. Under an explicit free-wiring assumption, all four bounds are tight.

Limitations of Learning Tanh Neural Networks with Finite Precision

We investigate limitations of learning $\tanh$ neural networks from point evaluations under finite-precision computations and $L^p$ accuracy guarantees, building on Berner, Grohs, and Voigtländer (2023). Our approach is based on a novel construction of sharply localized bump functions via iterated $\tanh$ activations. Using this mechanism, we show that, in a finite-precision setting, no adaptive randomized algorithm based on $m$ samples can achieve a convergence rate higher than the Monte Carlo rate $O(m^{-1/p})$ in the $L^p$ norm, unless the sampling budget grows exponentially with the size of the network parameters and architecture. The results reveal fundamental limitations imposed by finite precision on the learnability of classes containing localized bump functions, extending previous results for ReLU networks to the $\tanh$ setting.