Non-overlapping inversion on strings and languages

Given a string, a non-overlapping inversion is to reverse some non-overlapping fragments of the string simultaneously. We define a non-overlapping inversion operation to be the computation of all possible non-overlapping inversions. We apply the operation on a string, which gives rise to a set of strings, and construct an NFA recognizing the set. Then, we design an efficient DFA reduction algorithm from the resulting NFA. We also consider the non-overlapping inversion operation on a language and show the closure properties for regular, context-free and context-sensitive languages. We furthermore examine iterative non-overlapping inversions and establish the closure properties. Finally, we introduce non-overlapping inversion-free languages and present the decidability results for regular and context-free languages.