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.
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.