Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous temporal logic for intervals studied so far is probably Halpern and Shoham's HS, which is the logic of the thirteen Allen's interval relations. Unfortunately, HS and most of its fragments have an undecidable satisfiability problem. This discouraged the research in this area until recently, when a number non-trivial decidable ITLs have been

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... . This paper is a contribution towards the complete classification of all different fragments of HS. We consider different combinations of the interval relations begins (B), meets (A), later (L) and their inverses A, B and L. We know from previous work that the combination ABBA is decidable only when finite domains are considered (and undecidable elsewhere), and that ABB is decidable over the natural numbers. We extend these results by showing that decidability of ABB can be further extended to capture the language ABBL, which lies in between ABB and ABBA, and that turns out to be maximal w.r.t decidability over strongly discrete linear orders (e.g. finite orders, the naturals, the integers). We also prove that the proposed decision procedure is optimal with respect to the EXPSPACE complexity class, and that the language is powerful enough to polynomially encode metric constraints on the length of the current interval.