The Ultimate Undecidability Result for the Halpern-Shoham Logic

The Halpern-Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We complete this classification by showing - what we believe is quite an unexpected result - that the logic of subintervals, the fragment of the Halpern - Shoham logic where only the operator "during\´\´, or D, is allowed, is undecidable over discrete structures. This is surprising as this, apparently very simple, logic is decidable over dense orders and its reflexive variant is known to be decidable over discrete structures. Our result subsumes a lot of previous negative results for the discrete case, like the undecidability for ABE, BD, AA̅D, and so on.