Pspace-completeness of the temporal logic of sub-intervals and suffixes

Laura Bozzelli, Angelo Montanari, Adriano Peron, Pietro Sala

Abstract: In this paper, we prove Pspace-completeness of the finite satisfiability and model checking problems for the fragment of Halpern and Shoham interval logic with modality 〈E〉<math><mrow is="true"><mi mathvariant="normal" is="true">〈</mi><mi mathvariant="normal" is="true">E</mi><mi mathvariant="normal" is="true">〉</mi></mrow></math>, for the “suffix” relation on pairs of intervals, and modality 〈D〉<math><mrow is="true"><mi mathvariant="normal" is="true">〈</mi><mi mathvariant="normal" is="true">D</mi><mi mathvariant="normal" is="true">〉</mi></mrow></math>, for the “sub-interval” relation, under the homogeneity assumption. The result significantly improves the Expspace upper bound recently established for the same fragment, and proves the rather surprising fact that the complexity of the considered problems does not change when we add either the modality for suffixes (〈E〉<math><mrow is="true"><mi mathvariant="normal" is="true">〈</mi><mi mathvariant="normal" is="true">E</mi><mi mathvariant="normal" is="true">〉</mi></mrow></math>) or, symmetrically, the modality for prefixes (〈B〉<math><mrow is="true"><mi mathvariant="normal" is="true">〈</mi><mi mathvariant="normal" is="true">B</mi><mi mathvariant="normal" is="true">〉</mi></mrow></math>) to the logic of sub-intervals (featuring only 〈D〉<math><mrow is="true"><mi mathvariant="normal" is="true">〈</mi><mi mathvariant="normal" is="true">D</mi><mi mathvariant="normal" is="true">〉</mi></mrow></math>).