Subsequence Analysis Problems for Binary Parikh Matrices

The universality index corresponding to a word is the largest integer k such that every word of length k over the given alphabet occurs in it as a subsequence.Relative to languages this notion can be investigated with respect to both existential and universal quantifiers, with the former corresponding to the existence of a word in the language with universality index at least k, while the latter considers the index across all words that the language contains.In this work we study the existential (∃-universality) and universal (∀-universality) subsequence universality (as introduced in [Adamson et al., ISAAC 2023]) for binary languages consisting of all words with a fixed number of letters a, letters b, and subsequences ab.We prove that both ∃and ∀-universality admit exact arithmetic characterizations for these languages.Extending the above notions, we end the paper by initiating the analysis of the probability of a fixed word occurring as subsequence of the words in such a language.To this end, we prove that, for every fixed pattern word w and an error tolerance, given as input the subsequence counts describing a language, the ratio of words in the language having the pattern w as a subsequence admits an additive approximation scheme that is polynomial-time in the binary encoding of the input counts.