The addition of temporal neighborhood makes the logic of prefixes and sub-intervals EXPSPACE-complete
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A classic result by Stockmeyer gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of chop under the homogeneity assumption. In this paper, we study the complexity of the satisfiability problem for suitable weakenings of the chop interval temporal logic, that can be equivalently viewed as fragments of Halpern and Shoham interval logic. We first consider the logic 𝖡𝖣_hom featuring modalities B, for begins, corresponding to the prefix relation on pairs of intervals, and D, for during, corresponding to the infix relation. The homogeneous models of 𝖡𝖣_hom naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations. Such a fragment has been recently shown to be PSPACE-complete . In this paper, we study the extension 𝖡𝖣_hom with the temporal neighborhood modality A (corresponding to the Allen relation Meets), and prove that it increases both its expressiveness and complexity. In particular, we show that the resulting logic 𝖡𝖣𝖠_hom is EXPSPACE-complete.
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