Subword conditions and subword histories

This paper introduces the notion of a subword condition and investigates languages defined by them. The special case, where the language reduces to one word, concerns the inference of a sequence from its subsequences. We obtain various characterization and decidability results for languages defined by subword conditions. The results contribute to the theory of Parikh matrices and arithmetizing the study of words. An important notion from early automata theory, that of a quasi-uniform event, plays a central role in our characterization.