Logical Induction with incomputable sequences
In the definition of a logical inductor, the deductive process is required to be computable. This, of course, does not allow the logical inductor to use randomness, or predict uncomputable sequences. The way traders were defined in the logical induction paper, this was necessary, because the traders were not given access to the output of the deductive process.
To fix this, a trading strategy for day should be redefined as a function that takes in the output of the deductive process on day as its input, and outputs what the logical induction paper defines as a trading strategy for day ; that is, an affine combination of the form , where are sentences, are expressible features of rank , and . A trader is a function which takes in and outputs a trading strategy for day . By Currying, a trader can be seen as a function that takes in a number and a list of sentences given by the deductive process, and outputs an expressible feature combination as above. We can say that a trader is efficiently computable if this function is computable in time polynomial in plus the total length of the sentences output by the deductive process. The definition of exploitation would be modified in the natural way, and there is also a natural way to modify the logical induction algorithm, which will satisfy the logical induction criterion.
As an example, suppose a logical inductor is given access to a sensor that regularly produces bits based on what it observes in the environment. We can represent the data from the sensor with an additional unary predicate that we add to the language, such that is true iff the th bit provided by the sensor is a (this assumes that we’re working in a theory that can interpret arithmetic, so that ``″ can be expressed in the language). The deductive process should output or on day (and also can output consequences that it can deduce from the values of the bits it has seen so far). Or, if the logical inductor gets access to more empirical information or random bits as time goes on, there could be an increasing function such that the deductive process outputs the truth values of on day . Note that in this situation, the deductive process is computable as a function of the bitstream given by the sensor, so the traders may as well take in as input only the bits from the sensor that the deductive process has seen by day , rather than every sentence produced by the deductive process.
This seems to be similar to what Vadim was doing in section 3 of this paper, except that that paper moved to a continuous setting, did not deal with computability, and abandoned predicting theorems as a goal.