$A D T$ (asymptotic decision theory, initially detailed in this paper) was a proposed decision theory with logical inductors, developed after EDT with logical inductors and exploration steps. It is a possible candidate for conceptual progress in decision theory, but some basic questions about its performance are still unsettled, and there are several issues with the current implementation of it.

Definitions:

A market is a computable function with type $S \to [0, 1]$ (where $S$ is the set of sentences of math) that will be denoted as $P$ . We will be considering logical inductors in this setting, and in particular, each finite stage of a logical inductor is a market.

Let $A$ be a finite set of actions. An agent is a computable sequence of computable functions of type $P \to A$ . It takes a timestep and a probability distribution, and selects a distribution over actions. Because logical inductors associate each timestep with an action, specifying the logical inductor and the agent specifies a sequence of actions. Because we will be assuming some fixed logical inductor in the background, we will suppress it in the notation and refer to the action produced by an agent at time t as $a_{t}$ or $b_{t}$ . Randomization can be handled by using the diagonalization sentences used to define exploration (this acts in such a way that no trader in the inductor is able to reliably predict what the agent does), or it can be handled by having one of the actions be to call a random number generator in the environment.

An embedder is a sequence of randomized functions of type $A \to [0, 1]$ (denoted by $E$ or $F$ , with the function at time t being denoted by $E_{t}$ or $F_{t}$ ). Let $E_{t} (a_{t})$ denote the random variable that corresponds to the environment using a uniform random distribution of bitstrings for its randomness tape. An embedder must have the probability distribution of $E_{t} (a_{t})$ being computably approximable on computable inputs.

An argmax agent is an agent that takes a finite set of other agents $A$ , and a single embedder $E$ , and outputs ${(a r g m a x_{a \in A} E_{t} (E_{t} (a_{t})))}_{t}$ on turn $t$ . $E_{t}$ is defined as it usually is for logical inductors. Notice that, although $E_{t}$ may be very difficult to compute, putting it inside an expectation allows a logical inductor to output a decent guess as to what it is anyways. The agents all had to be computable in order for the argmax agent to duplicate their behavior at arbitrary turns. There is a dependence on the s... ,

Agent Simulates Predictor