Title: The Counting Power of Transformers
Primary area: other topics in machine learning (i.e., none of the above)
Keywords: FLaNN, expressiveness, attention, formal languages

Abstract: Counting properties (e.g. determining whether certain tokens occur more
    than other tokens in a given input text) have played a significant role in 
    the study of expressiveness of transformers.
    In this paper, we provide a formal 
    framework for investigating the counting power of transformers. We argue 
    that all existing results demonstrate transformers' expressivity only for 
    (semi-)linear counting properties, i.e., which are expressible as a 
    boolean combination of linear inequalities. 
    Our main result is that transformers can express counting properties that
    are highly nonlinear. More precisely, we prove that transformers can
    capture all semialgebraic counting properties, i.e., expressible as 
    a boolean combination of arbitrary multivariate polynomials (of any degree).
    Among others, these generalize the counting properties that
    can be captured by support vector machines via polynomial kernel in the 
    vector space model.
    To complement this result, we exhibit a natural subclass of (softmax) 
    transformers that completely characterizes semialgebraic counting 
    properties. 
    Through connections with the
    Hilbert's tenth problem, this expressivity of transformers also 
    yields a new undecidability result for analyzing an extremely simple 
    transformer model --- surprisingly with neither positional encodings 
    (i.e. NoPE-transformers) nor masking.
    We also experimentally validate trainability of such counting
    properties.

TL;DR: Transformers can express highly nonlinear counting properties