# Logical Induction with incomputable sequences

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.