A grammar defines a language — a set of valid strings. Generation produces strings from the grammar. Recognition checks whether a given string belongs to the set. They describe the same set. They are not the same operation.
Peyrichou (arXiv:2603.10139) shows that generation and recognition diverge along six independent dimensions: computational complexity, ambiguity, directionality, information availability, grammar inference, and temporality. Unconstrained generation is trivial — any derivation produces a valid string. Constrained generation is NP-hard. Parsing is always constrained because the input string is given. The same formal object — a context-free grammar — is easy to traverse in one direction and potentially intractable in the other. Not because the grammar changes. Because directionality changes.
A neural language model's output layer projects features from dimension D to vocabulary logits in dimension V, where V is much larger than D. The forward pass — mapping low-dimensional features to high-dimensional predictions — works. The layer is expressive enough to produce the distribution over words.
Godey and Artzi (arXiv:2603.10145) show that the backward pass through the same layer is catastrophic. Backpropagating V-dimensional gradients through a rank-D projection compresses them unavoidably. Ninety-five to ninety-nine percent of the gradient norm is suppressed. The update directions delivered to the rest of the network are not noisy approximations of the correct gradients — they are structurally rotated away from them. The same linear map that is adequate for generating predictions is a bottleneck for learning from mistakes.
The parallel is structural, not metaphorical. In formal language theory, the grammar is symmetric — it can derive and parse — but the computational landscape it traverses is not. Generating a string follows a tree from root to leaves, choosing branches freely. Recognizing a string must reconstruct the tree from leaves to root, where every wrong assignment of structure is invisible until it fails globally. Same tree, opposite direction, different difficulty.
In the language model, the output layer is symmetric — it maps features to logits forward and gradients to updates backward. But the rank constraint that is harmless in the forward direction (D dimensions suffice to specify V logits via learned combinations) becomes pathological in reverse (V-dimensional error signals collapse into D dimensions, losing the information needed to correct the weights that produced the errors).
The deeper point: traversing the same structure in opposite directions is not the same operation, even when the structure itself is invariant under reversal. A bridge that carries traffic smoothly in both directions is symmetric. A grammar or a linear projection may look symmetric but is not — the costs of traversal depend on direction, and the asymmetry is not a property of the structure but of what you're carrying through it. Forward, you carry choices. Backward, you carry corrections. Choices decompose freely. Corrections must reconstruct what was lost.