Token Complexity Theory for AI-Augmented Computing

2026-06-10

AI Summary

This work addresses the absence of a formal complexity measure for token consumption in query–response interactions within AI-augmented computing. We propose a theory of token complexity grounded in an AI-Oracle Turing machine model, which for the first time formally characterizes the minimal expected token cost required to complete a task. By integrating probabilistic Turing machines with random oracle models and leveraging tools from information theory and computational complexity, we define this novel resource dimension and establish a framework for analyzing AI systems based on output quality. The theory yields a classification of AI systems and reveals fundamental properties—including monotonicity, convexity, price sensitivity, and task-order relativity. We prove that the complexity frontier is non-empty, upward-closed, and convex, thereby laying the theoretical foundation for resource optimization and task scheduling in AI-augmented computation.

Abstract

AI-augmented computing delegates natural language queries, code generation requests, and other open-ended tasks to a cluster of AI models that processes queries and generates responses. This paradigm introduces a resource dimension that neither classical time nor space complexity captures: the cost of sending queries to and receiving responses from such a cluster. We introduce token complexity, a formal resource measure defined as the minimum expected token cost to achieve a specified level of output quality on a task, and develop a taxonomy classifying AI systems by the strength of their probabilistic properties. We develop token complexity within the framework of AI-Oracle Turing machines, in which a probabilistic Turing machine interacts with a stochastic oracle via dedicated query and response tapes. We prove basic theorems establishing that token complexity behaves as expected: monotonicity (higher quality costs more tokens), convexity (quality improvements become progressively more expensive), price sensitivity (small price changes produce bounded cost changes), and price-relativity of task ordering (the token complexity ordering of tasks can reverse depending on the query-to-response cost ratio). We prove that the complexity frontier, defined as the set of all feasible resource bounds in tokens, time, and space, is non-empty, upward-closed, and convex.

Problem

• token complexity
• AI-augmented computing
• resource measure
• query cost
• output quality

Innovation

• token complexity
• AI-augmented computing
• AI-Oracle Turing machine
• probabilistic resource measure
• complexity frontier