q-Deformed Binomial Coefficients of Words: Properties, Identities, and Applications to Formal Language Theory
This paper introduces and explores q-deformed binomial coefficients of words, extending the concept of Gaussian binomial coefficients from integers to words. These q-deformations provide richer information than classical binomial coefficients, enabling a deeper understanding of subword occurrences and leading to generalizations of classical identities and applications in formal language theory, particularly in characterizing p-group languages.
Compiling Temporal Counting Logic into Softmax Transformers
Temporal counting logic Kt[#] and its equivalent RASP variant C-RASP are the best-known lower bound on the expressivity of future-masked softmax transformer encoders.
Commutative N-polyregular Functions: Decidability and Characterization
Commutative N-polyregular functions can be effectively characterized and their membership is decidable. This resolves an open conjecture on the relationship between star-free N-polyregular functions and star-free Z-polyregular functions.
Transformers as Formal Transducers: Expressivity and Limits
Transformers can express surprisingly large classes of string-to-string transductions, including first-order rational, regular, and polyregular functions, which can be simulated using variants of the RASP programming language.
Universality of Regular Realizability Problems for Finite Relations
The paper proves the universality of regular realizability problems for several classes of finite relations on the set of non-negative integers, where the relations are described in a specific format. The universality is shown up to reductions using NP-oracles.
