Coalgebraic reasoning with global assumptions in arithmetic modal logics

Clemens Kupke, Dirk Pattinson, Lutz Schröder

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)
30 Downloads (Pure)


We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka’s linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e., in coalgebraic hybrid logic.
Original languageEnglish
Article number11
Pages (from-to)1-34
Number of pages34
JournalACM Transactions on Computational Logic
Issue number2
Early online date14 Jan 2022
Publication statusPublished - 30 Apr 2022


  • computational mathematics
  • logic
  • general computer science
  • theoretical computer science


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