Generic predictions of output probability based on complexities of inputs and outputs

TL;DR

This paper extends the coding theorem from algorithmic information theory to derive probability bounds based on both input and output complexities, with applications demonstrated on biological and computational models.

Contribution

It introduces new probability bounds considering input and output complexities, broadening the application of AIT in physics and computational models.

Findings

Probability bounds align with empirical data from RNA, transducer, and perceptron models.

Outputs with higher complexity have exponentially lower probabilities.

Input complexity influences output likelihood beyond traditional bounds.

Abstract

For a broad class of input-output maps, arguments based on the coding theorem from algorithmic information theory (AIT) predict that simple (low Kolmogorov complexity) outputs are exponentially more likely to occur upon uniform random sampling of inputs than complex outputs are. Here, we derive probability bounds that are based on the complexities of the inputs as well as the outputs, rather than just on the complexities of the outputs. The more that outputs deviate from the coding theorem bound, the lower the complexity of their inputs. Our new bounds are tested for an RNA sequence to structure map, a finite state transducer and a perceptron. These results open avenues for AIT to be more widely used in physics.