Explicit Fuzzy Logic in the Feed-Forward Layer: Self-Forgetting Quantifiers Discover Legible Grammatical-Licensing Detectors

A transformer can be built from operators that are legible by construction -- bounded, named units that read as fuzzy set operations rather than dense activations -- but legibility must be pressed for during training, and the pressure has a failure mode. A crispness penalty meant to sharpen a bounded operator into a decisive detector instead collapses it into a dead constant. An identity, E[v(1-v)] = mu(1-mu) - var, shows why -- the penalty is a variance-minimizer blind to the difference between a live detector and a constant -- and names the fix: a per-channel variance floor, the target legibility metric written as a loss, which recovers both legibility and quality. A learned per-unit fraction then retires the hand-set reserved-GELU partition of prior work: given the choice the model keeps no unit as pure GELU and routes 87% of its load-bearing computation through crisp operators. The result is the most legible transformer we have built -- 78% of its feed-forward operands and 50% of its attention value channels are crisp-and-contextual detectors, and per-head legibility rises from 18% in shallow layers to 78% in deep ones. Read in the correct rotated per-layer frame, these units separate a clean detection (what a unit responds to) from a harder naming (what its output decodes to); and because the objective makes each unit crisp and sparse, edits to them are far more local -- 50-184x in the deep layers where the edit sites concentrate -- and can target explicit conjunctions a single neuron cannot express. Finally, a between-unit decorrelation pressure exposes a legibility dial: it trades a circuit's reuse for independence at no quality cost, turning concepts into single, surgically editable units and a prediction into a short explanation read off a handful of named operations. Quality holds at parity with a conventional baseline throughout.

A companion paper showed that a transformer's feed-forward layer can be rebuilt from explicit fuzzy set operations - intersection, set-difference, and a self-forgetting sequence quantifier - so its hidden units read as named logical operators at no cost to language-model quality. That left the other half of the transformer opaque. Here we carry the same idea into attention and join the two into one model. The mechanism is minimal: a head's value is passed through a sigmoid, so each value channel becomes a readable detector of whether a feature holds at a token. This adds no parameters and leaves the standard head otherwise untouched. A Boolean variant goes further, restructuring the value into an explicit within-token intersection and negation-capable set-difference. In both designs the output projection is left free, not tied to the vocabulary, which is the load-bearing decision: bounding what a head detects while leaving what it writes unconstrained yields selective detectors, whereas constraining the write does not. A bounded value is shaped into a readable detector by two selectivity pressures - one for sparse firing, one for decisive firing at the rails - and which a design wants is not universal. Across five specialized-attention designs at 125M parameters, 44 to 62 percent of value channels become crisp, contextually selective detectors, and their legibility rises with depth rather than crystallizing only on punctuation. Language-model quality is at parity with a conventional baseline. Finally, we couple the Boolean attention to the legible feed-forward layer and train an end-to-end legible-by-construction language model at benchmark parity: its feed-forward units are named set and quantifier operations throughout, and we can take a token it generates and read the named units that compose to produce it.

Feedforward network (FFN) layers account for a large fraction of parameters and nonlinear expressivity in Transformer-based large language models (LLMs). Despite the evolution from ReLU and GELU to gated variants such as SwiGLU, most FFN designs still use a single fixed activation function, applying the same nonlinear transformation to all tokens. In this work, we propose Mixture of Activations (MoA), a token-adaptive FFN design that mixes a dictionary of activation functions using lightweight input-dependent gates while sharing the same linear projections. As an input-independent counterpart, we also introduce learnable activations (LA), which form linear combinations of activation functions for both ReLU-type and SwiGLU-type FFNs. Theoretically, we establish strict finite-width expressive separations among fixed-activation FFNs, LA, and MoA: LA strictly contains fixed-activation FFNs, while MoA strictly contains LA, with the additional expressivity arising from input-dependent nonlinear hybridization. Empirically, we evaluate MoA through extensive pre-training experiments on dense and MoE language models ranging from 0.12B to 2B parameters under different token budgets, optimizers, and learning rate schedules. MoA consistently achieves lower terminal loss and exhibits more favorable scaling behavior than well-tuned baselines, with minimal parameter and computational overhead. These results suggest that token-adaptive activation mixing is a simple and effective mechanism for improving FFN expressivity in LLMs.