The Dissecting Power of Regular Languages

A study on structural properties of regular and context-free languages has promoted our basic understandings of the complex behaviors of those languages. We continue the study to examine how regular languages behave when they are “almost halving” numerous infinite languages. In particular, we are focused on a situation in which a regular language “dissects” a target infinite language into two infinite subsets. Every context-free language and its complement can be dissected by carefully chosen regular languages. By expanding the scope of our study, we show that constantly-growing languages and semi-linear languages are also dissectable; however, their complements as well as intersections are not. Under certain natural conditions, the complements and finite intersections of semi-linear languages become dissectable. Similarly, restricted to bounded languages, the intersections of finitely many bounded context-free languages and, more surprisingly, the entire Boolean hierarchy over bounded context-free languages are dissectable. As an immediate application, we show a structural property in which an appropriate bounded context-free language can separate, with infinite margins, two given infinite bounded context-free languages, one of which contains the other with an infinite margin. This property is closely related to a notion and result of Demaratzki, Shallit, and Yu (2001).