We introduce a method called evolving Log Parsing (eLP) to extract information granules and an interval rule-based classification model from streams of words in unstructured log files. Logs are elementary expressions of language that are used by computational systems to communicate with humans unidirectionally. The logs tell stories based on event occurrences. Any software expresses itself through a log language. In particular, the eLP approach has identified templates (patterns in textual data) in an unsupervised and incremental way. Online pattern classification is achieved with effectiveness of (96.05 ± 1.04)% using 6 datasets and eLP models exhibiting an interpretability level of about 0.04. We present a recursive model-interpretability index to evaluate rule-based classifiers, and discuss the effectiveness-interpretability tradeoff on an actual scenario, namely, the StorM Service of a computing center.