Andy Yang

Andy Yang, David Chiang, Dana Angluin

The expressive power of transformers over inputs of unbounded size can be studied through their ability to recognize classes of formal languages. In this paper, we establish exact characterizations of transformers with hard attention (in which all attention is focused on exactly one position) and attention masking (in which each position only attends to positions on one side). With strict masking (each position cannot attend to itself) and without position embeddings, these transformers are expressively equivalent to linear temporal logic (LTL), which defines exactly the star-free languages. A key technique is the use of Boolean RASP as a convenient intermediate language between transformers and LTL. We then take numerous results known for LTL and apply them to transformers, showing how position embeddings, strict masking, and depth all increase expressive power.

Andy Yang, Anej Svete, Jiaoda Li et al.

Most expressivity results for transformers treat them as language recognizers (which accept or reject strings), and not as they are used in practice, as language models (which generate strings autoregressively and probabilistically). Here, we characterize the probability distributions that transformer language models can express. We show that making transformer language recognizers autoregressive can sometimes increase their expressivity, and that making them probabilistic can break equivalences that hold in the non-probabilistic case. Our overall contribution is to tease apart what functions transformers are capable of expressing, in their most common use-case as language models.

Andy Yang, Pascal Bergsträßer, Georg Zetzsche et al.

Length generalization is a key property of a learning algorithm that enables it to make correct predictions on inputs of any length, given finite training data. To provide such a guarantee, one needs to be able to compute a length generalization bound, beyond which the model is guaranteed to generalize. This paper concerns the open problem of the computability of such generalization bounds for CRASP, a class of languages which is closely linked to transformers. A positive partial result was recently shown by Chen et al. for CRASP with only one layer and, under some restrictions, also with two layers. We provide complete answers to the above open problem. Our main result is the non-existence of computable length generalization bounds for CRASP (already with two layers) and hence for transformers. To complement this, we provide a computable bound for the positive fragment of CRASP, which we show equivalent to fixed-precision transformers. For both positive CRASP and fixed-precision transformers, we show that the length complexity is exponential, and prove optimality of the bounds.

Andy Yang, David Chiang

Deriving formal bounds on the expressivity of transformers, as well as studying transformers that are constructed to implement known algorithms, are both effective methods for better understanding the computational power of transformers. Towards both ends, we introduce the temporal counting logic $\textsf{K}_\text{t}$[#] alongside the RASP variant $\textsf{C-RASP}$. We show they are equivalent to each other, and that together they are the best-known lower bound on the formal expressivity of future-masked soft attention transformers with unbounded input size. We prove this by showing all $\textsf{K}_\text{t}$[#] formulas can be compiled into these transformers.

Andy Yang, Lena Strobl, David Chiang et al.

We study conditions under which transformers using soft attention can simulate hard attention, that is, effectively focus all attention on a subset of positions. First, we examine several subclasses of languages recognized by hard-attention transformers, which can be defined in variants of linear temporal logic. We demonstrate how soft-attention transformers can compute formulas of these logics using unbounded positional embeddings or temperature scaling. Second, we demonstrate how temperature scaling allows softmax transformers to simulate general hard-attention transformers, using a temperature that depends on the minimum gap between the maximum attention scores and other attention scores.

Andy Yang, Michaël Cadilhac, David Chiang

It has been observed that transformers with greater depth (that is, more layers) have more capabilities, but can we establish formally which capabilities are gained? We answer this question with a theoretical proof followed by an empirical study. First, we consider transformers that round to fixed precision except inside attention. We show that this subclass of transformers is expressively equivalent to the programming language C-RASP and this equivalence preserves depth. Second, we prove that deeper C-RASP programs are more expressive than shallower C-RASP programs, implying that deeper transformers are more expressive than shallower transformers (within the subclass mentioned above). The same is also proven for transformers with positional encodings (like RoPE and ALiBi). These results are established by studying a temporal logic with counting operators equivalent to C-RASP. Finally, we provide empirical evidence that our theory predicts the depth required for transformers without positional encodings to length-generalize on a family of sequential dependency tasks.

Andy Yang, Christopher Watson, Anton Xue et al. · allen-ai, eth-zurich

We present the transformer cookbook: a collection of techniques for directly encoding algorithms into a transformer's parameters. This work addresses the steep learning curve of such endeavors, a problem exacerbated by a fragmented literature where key results are scattered across numerous papers. In particular, we synthesize this disparate body of findings into a curated set of recipes that demonstrate how to implement everything from basic arithmetic in feed-forward layers to complex data routing via self-attention. Our mise en place of formulations is for both newcomers seeking an accessible entry point and experts in need of a systematic reference. This unified presentation of transformer constructions provides a foundation for future work spanning theoretical research in computational complexity to empirical investigations in architecture design and interpretability.