Discrete sequence prediction with commented Markov models

Abstract

This paper introduces the concept of Commented Markov Models (CMMs), an extension of the well known Markov Models, together with the relevant induction mechanisms. Given a discrete alphabet Σ and a source producing an input sequence (s ₁,s ₂,...,s _n) with s _i ∈ Σ, the task of sequence prediction is to guess the successive sequence element s _n+1. Here each element s _i may represent an object, a discrete event or any other discrete entity. Prediction with CMM is analogy-based. It is assumed that the final part of the input sequence describes the current state of the source. This final part is matched with earlier subsequences of the input, assuming that it will be continued the same way as was the ‘most similar’ subsequence. CMM learning involves the induction of objects, variables and object classes. While object and class creation are similar to the notions of chunking and merging in other grammatical inference approaches, the use of variables is a novel feature of CMM. It not only generalized the way subsequences can be matched, it also turns CMM from a pure sequence prediction algorithm into a computational model. I will show that CMM has sufficient expressiveness to represent any primitive recursive function. Thus it is not only capable of predicting e.g. the character ‘u’ to follow the sequence ‘seq’, but it can also extrapolate a sequence like ‘45+13=’ by calculating the sum ‘58’.