A Logic for Expressing Log-Precision Transformers

• URL: http://arxiv.org/abs/2210.02671v6
• Date: Mon, 6 Nov 2023 21:24:57 GMT
• Title: A Logic for Expressing Log-Precision Transformers
• Authors: William Merrill and Ashish Sabharwal
• Abstract summary: We show that any log-precision transformer can be equivalently expressed as a first-order logic sentence. This is the tightest known upper bound and first logical characterization of log-precision transformers.
• Score: 35.25166532364007
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: One way to interpret the reasoning power of transformer-based language models is to describe the types of logical rules they can resolve over some input text. Recently, Chiang et al. (2023) showed that finite-precision transformers can be equivalently expressed in a generalization of first-order logic. However, finite-precision transformers are a weak transformer variant because, as we show, a single head can only attend to a constant number of tokens and, in particular, cannot represent uniform attention. Since attending broadly is a core capability for transformers, we ask whether a minimally more expressive model that can attend universally can also be characterized in logic. To this end, we analyze transformers whose forward pass is computed in $\log n$ precision on contexts of length $n$. We prove that any log-precision transformer can be equivalently expressed as a first-order logic sentence that, in addition to standard universal and existential quantifiers, may also contain majority-vote quantifiers. This is the tightest known upper bound and first logical characterization of log-precision transformers.

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