Computing with Agentic Oracles

2026-08-02

AI Summary

This work addresses the limitations of the classical random oracle model in capturing the computational resource consumption and risks associated with autonomous agents that maintain internal goals and interact dynamically with their environment. To overcome this, we introduce a novel “agent oracle” paradigm that extends the random oracle Turing machine framework by formally modeling state retention and environmental interaction mechanisms. We define two distinct token cost categories—orchestration and agent tokens—and, for the first time, differentiate between visible and invisible token expenditures. We further propose a goal-loss avoidance criterion, a progress–retry–goal-loss formulation, and a lower bound on goal depth. Theoretical analysis demonstrates that agent oracles endowed with intermediate state preservation can substantially reduce token costs while maintaining equivalent task quality, and reveals the token complexity characteristics under zero goal-loss probability along with its inherent upper bound on achievable task quality.

Abstract

This paper extends the stochastic-oracle model of AI-augmented computing to include agentic oracles. Unlike a stationary stochastic oracle, which responds to the same query according to a fixed response distribution across calls, an agentic oracle can pursue a goal autonomously and may access an environment containing task-relevant resources. These capabilities affect both response distributions and token costs beyond what is visible at the query-response interface. We develop a framework for analyzing token costs in Stochastic-Oracle Turing Machines (SOTMs) that compute with agentic oracles. Each call has an \emph{orchestration token cost}, visible to the caller at the query-response interface, and an \emph{agentic token cost}, incurred by internal operations not exposed to the caller. We show that an SOTM computing with an agentic oracle that can retain intermediate state can have token-cost advantages over SOTMs using stationary stochastic oracles when solving the same task at the same quality level, both with and without environment access. We also investigate goal-loss risk, including how internal dispatch ordering can reduce exposure to irreversible actions. We provide a goal-loss avoidance criterion, derive progress--retry--goal-loss formulas, establish goal-depth lower bounds on token complexity, characterize token complexity when the probability of goal loss is zero, and show that goal-loss risk can impose an upper bound on the achievable quality of a task involving environment updates.

Problem

• agentic oracles
• token cost
• goal-loss risk
• stochastic-oracle model
• AI-augmented computing

Innovation

• agentic oracle
• token cost
• goal-loss risk
• Stochastic-Oracle Turing Machine
• AI-augmented computing