Estimation of Generating Processes of Strings Represented with Patterns and Substitutions

We formalize generating processes of strings based on patterns and substitutions, and give an algorithm to estimate a probability mass function on substitutions, which is an element of processes. Patterns are non-empty sequences of characters and variables. Variables indicate unknown substrings and are replaced with other patterns by substitutions. By introducing variables and substitutions, we can deal with the difficulty of preparing production rules in generative grammar and of representing context-sensitivity with them. Our key idea is to regard sequences of substitutions as generations of strings, and to give probabilities of substitutions like PCFG. In this study, after giving a problem to estimate a probability mass function from strings based on our formalization, we solve it by the Passing algorithm that runs in an iterative manner. Our experimental results with synthetic strings show that our method estimates probability mass functions with sufficient small errors.