As transformers have gained prominence in natural language processing, some researchers have investigated theoretically what problems they can and cannot solve, by treating problems as formal languages . Exploring such questions can help clarify the power of transformers relative to other models of computation, their fundamental capabilities and limits, and the impact of architectural choices. Work in this subarea has made considerable progress in recent years. Here, we undertake a comprehensive survey of this work, documenting the diverse assumptions that underlie different results and providing a unified framework for harmonizing seemingly contradictory findings.

Many current NLP systems are built from language models trained to optimize unsupervised objectives on large amounts of raw text. Under what conditions might such a procedure acquire meaning? Our systematic experiments with synthetic data reveal that, with languages where all expressions have context- independent denotations (i.e., languages with strong transparency ), both autoregressive and masked language models successfully learn to emulate semantic relations between expressions. However, when denotations are changed to be context- dependent with the language otherwise unmodified, this ability degrades. Turning to natural language, our experiments with a specific phenomenon—referential opacity—add to the growing body of evidence that current language models do not represent natural language semantics well. We show this failure relates to the context-dependent nature of natural language form-meaning mappings.

Despite their omnipresence in modern NLP, characterizing the computational power of transformer neural nets remains an interesting open question. We prove that transformers whose arithmetic precision is logarithmic in the number of input tokens (and whose feedforward nets are computable using space linear in their input) can be simulated by constant-depth logspace-uniform threshold circuits. This provides insight on the power of transformers using known results in complexity theory. For example, if L ≠ P (i.e., not all poly-time problems can be solved using logarithmic space), then transformers cannot even accurately solve linear equalities or check membership in an arbitrary context-free grammar with empty productions. Our result intuitively emerges from the transformer architecture’s high parallelizability. We thus speculatively introduce the idea of a fundamental parallelism tradeoff : any model architecture as parallelizable as the transformer will obey limitations similar to it. Since parallelism is key to training models at massive scale, this suggests a potential inherent weakness of the scaling paradigm.

Transformers have become a standard neural network architecture for many NLP problems, motivating theoretical analysis of their power in terms of formal languages. Recent work has shown that transformers with hard attention are quite limited in power (Hahn, 2020), as they can be simulated by constant-depth AND/OR circuits (Hao et al., 2022). However, hard attention is a strong assumption, which may complicate the relevance of these results in practice. In this work, we analyze the circuit complexity of transformers with saturated attention: a generalization of hard attention that more closely captures the attention patterns learnable in practical transformers. We first show that saturated transformers transcend the known limitations of hard-attention transformers. We then prove saturated transformers with floating-point values can be simulated by constant-depth threshold circuits, giving the class TC 0 as an upper bound on the formal languages they recognize.

Language models trained on billions of tokens have recently led to unprecedented results on many NLP tasks. This success raises the question of whether, in principle, a system can ever “understand” raw text without access to some form of grounding. We formally investigate the abilities of ungrounded systems to acquire meaning. Our analysis focuses on the role of “assertions”: textual contexts that provide indirect clues about the underlying semantics. We study whether assertions enable a system to emulate representations preserving semantic relations like equivalence. We find that assertions enable semantic emulation of languages that satisfy a strong notion of semantic transparency. However, for classes of languages where the same expression can take different values in different contexts, we show that emulation can become uncomputable. Finally, we discuss differences between our formal model and natural language, exploring how our results generalize to a modal setting and other semantic relations. Together, our results suggest that assertions in code or language do not provide sufficient signal to fully emulate semantic representations. We formalize ways in which ungrounded language models appear to be fundamentally limited in their ability to “understand”.