The Complexity of Monotone Hybrid Logics over Linear Frames and the Natural Numbers

Computer Science – Computational Complexity

Scientific paper

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14 pages + 15 pages appendix, 4 figures

Scientific paper

Hybrid logic with binders is an expressive specification language. Its satisfiability problem is undecidable in general. If frames are restricted to the natural numbers or general linear orders, then satisfiability is known to be decidable, but of non-elementary complexity. In this paper, we consider monotone hybrid logics (i.e., the Boolean connectives are conjunction and disjunction only) over the natural numbers and general linear orders. We show that the satisfiability problem remains non-elementary over linear orders, but its complexity drops to PSPACE-completeness over natural numbers. We categorize the strict fragments arising from different combinations of modal and hybrid operators into NP-complete and tractable, and show that the latter cases are complete for NC1 or LOGSPACE. Interestingly, NP-completeness depends only on the fragment and not on the frame. For the cases above NP, satisfiability over linear orders is harder than over natural numbers, while below NP it is at most as hard. In addition to the study of computational complexity, we examine model-theoretic properties of the fragments in question.

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