We extend recent work on defining linear-time behaviour for state-based systems with branching, and propose modal and fixpoint logics for specifying linear-time temporal properties of states in such systems. We model systems with branching as coalgebras whose type arises as the composition of a branching monad and a polynomial endofunctor on the category of sets, and employ a set of truth values induced canonically by the branching monad. This yields logics for reasoning about quantitative aspects of linear-time behaviour. Examples include reasoning about the probability of a linear-time behaviour being exhibited by a system with probabilistic branching, or about the minimal cost of a linear-time behaviour being exhibited by a system with weighted branching. In the case of non-deterministic branching, our logic supports reasoning about the possibility of exhibiting a given linear-time behaviour, and therefore resembles an existential version of the logic LTL.


Model Check Modal Logic Relational Semantic Generalise Predicate Follow Diagram Commute 
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  1. 1.
    Baier, C., Katoen, J.-P.: Principles of model checking. MIT Press (2008)Google Scholar
  2. 2.
    Cîrstea, C.: From Branching to Linear Time, Coalgebraically. In: Baelde, D., Carayol, A. (eds.) Proc. FICS 2013. EPTCS, vol. 126, pp. 11–27 (2013)Google Scholar
  3. 3.
    Coumans, D., Jacobs, B.: Scalars, monads, and categories. In: Heunen, C., Sadrzadeh, M., Grefenstette, E. (eds.) Quantum Physics and Linguistics. A Compositional, Diagrammatic Discourse, pp. 184–216. Oxford Univ. Press (2013)Google Scholar
  4. 4.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002)Google Scholar
  5. 5.
    Hermida, C., Jacobs, B.: Structural induction and coinduction in a fibrational setting. Inf. Comput. 145(2), 107–152 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Jacobs, B.: Introduction to coalgebra. Towards mathematics of states and observations (version 2.0). Draft (2012)Google Scholar
  7. 7.
    Kanellakis, P.C., Smolka, S.A.: CCS expressions, finite state processes, and three problems of equivalence. Inf. Comput. 86(1), 43–68 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Kock, A.: Monads and extensive quantities, arXiv:1103.6009 (2011)Google Scholar
  9. 9.
    Pattinson, D.: Coalgebraic modal logic: soundness, completeness and decidability of local consequence. Theor. Comput. Sci. 309(1-3), 177–193 (2003)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Corina Cîrstea
    • 1
  1. 1.University of SouthamptonUK

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