Large Language Models and the Extended Church-Turing Thesis

TL;DR

This paper investigates the computational capabilities of large language models (LLMs) through classical computability and complexity theory, establishing their relation to the Extended Church-Turing Thesis and revealing their potential super-Turing power.

Contribution

It provides a formal analysis of LLMs' computational power, showing their equivalence to finite-state transducers and their lineages' alignment with interactive Turing machines with advice.

Findings

Fixed LLMs are equivalent to large deterministic finite-state transducers.

Lineages of LLMs simulate interactive Turing machines with advice.

Lineages of LLMs may possess super-Turing computational power.

Abstract

The Extended Church-Turing Thesis (ECTT) posits that all effective information processing, including unbounded and non-uniform interactive computations, can be described in terms of interactive Turing machines with advice. Does this assertion also apply to the abilities of contemporary large language models (LLMs)? From a broader perspective, this question calls for an investigation of the computational power of LLMs by the classical means of computability and computational complexity theory, especially the theory of automata. Along these lines, we establish a number of fundamental results. Firstly, we argue that any fixed (non-adaptive) LLM is computationally equivalent to a, possibly very large, deterministic finite-state transducer. This characterizes the base level of LLMs. We extend this to a key result concerning the simulation of space-bounded Turing machines by LLMs. Secondly, we show that lineages of evolving LLMs are computationally equivalent to interactive Turing machines with advice. The latter finding confirms the validity of the ECTT for lineages of LLMs. From a computability viewpoint, it also suggests that lineages of LLMs possess super-Turing computational power. Consequently, in our computational model knowledge generation is in general a non-algorithmic process realized by lineages of LLMs. Finally, we discuss the merits of our findings in the broader context of several related disciplines and philosophies.