Large Language Models and the Extended Church-Turing Thesis
Jiří Wiedermann, Jan van Leeuwen
TL;DR
The paper investigates whether the Extended Church-Turing Thesis (ECTT) extends to large language models (LLMs) by leveraging automata-theoretic analyses. It shows that fixed, non-adaptive LLMs are computationally equivalent to deterministic finite-state transducers within their context windows, and that space-bounded TMs can be simulated inside sufficiently large LLMs using embedding sizes $O(k)$ when $S(n) \le k$. It further demonstrates that lineages of evolving LLMs are computationally equivalent to interactive Turing machines with advice (ITM/As), yielding super-Turing power, and provides bilateral simulations between lineages and ITM/As. The results frame LLMs as knowledge-generating devices whose non-uniform, interactive evolution aligns with ITM/As under the ECTT, while highlighting limits of fixed LLMs and prompting a broader discussion of understanding and memory in AI systems.
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.