1 / 18

Exemplaric Expressivity of Modal Logics

Exemplaric Expressivity of Modal Logics. Ana Sokolova University of Salzburg joint work with Bart Jacobs Radboud University Nijmegen. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A. Outline. Boolean modal logic.

braden
Download Presentation

Exemplaric Expressivity of Modal Logics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Exemplaric Expressivityof Modal Logics Ana Sokolova University of Salzburg joint work with Bart Jacobs Radboud University Nijmegen Coalgebra Day, 11-3-2008, RUN TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

  2. Outline Boolean modal logic Finite conjunctions probabilistic modal logic Coalgebra Day, 11-3-2008, RUN • Expressivity: logical equivalence = behavioral equivalence • For three examples: • Transition systems • Markov chains • Markov processes

  3. Via dual adjunctions Predicates on spaces Dual Theories on models Behaviour (coalgebras) Logics (algebras) Coalgebra Day, 11-3-2008, RUN

  4. Logical set-up • If L has an initial algebra of formulas • A natural transformation • gives interpretations Coalgebra Day, 11-3-2008, RUN

  5. Logical equivalencebehavioural equivalence • The interpretation map yields a theory map Aim: expressivity • which defines logical equivalence • behavioural equivalence is given by for some coalgebra homomorphisms h1 and h2 Coalgebra Day, 11-3-2008, RUN

  6. Expressivity • If and the transpose of the interpretation is componentwise mono, then expressivity. Factorisation system on with diagonal fill-in Bijective correspondence between and Coalgebra Day, 11-3-2008, RUN

  7. Sets vs. Boolean algebras contravariant powerset Boolean algebras ultrafilters Coalgebra Day, 11-3-2008, RUN

  8. Sets vs. meet semilattices contravariant powerset meet semilattices filters Coalgebra Day, 11-3-2008, RUN

  9. Measure spaces vs. meet semilattices maps a measure space to its ¾-algebra ¾-algebra: “measurable” subsets closed under empty, complement, countable union measure spaces filters on A with ¾-algebra generated by Coalgebra Day, 11-3-2008, RUN

  10. Behaviour via coalgebras • Markov chains Giry monad • Markov processes Coalgebra Day, 11-3-2008, RUN Transition systems

  11. The Giry monad countable union of pairwise disjoint the smallest making measurable generated by subprobability measures Coalgebra Day, 11-3-2008, RUN

  12. Logic for transition systems models of boolean logic with fin.meet preserving modal operators expressivity L = GV V - forgetful Coalgebra Day, 11-3-2008, RUN Modal operator

  13. Logic for Markov chains models of logic with fin.conj. and monotone modal operators expressivity K = HV V - forgetful Coalgebra Day, 11-3-2008, RUN Probabilistic modalities

  14. Logic for Markov processes models of logic with fin.conj. and monotone modal operators expressivity the same K Coalgebra Day, 11-3-2008, RUN General probabilistic modalities

  15. Discrete to indescrete forgetful functor discrete measure space The adjunctions are related: Coalgebra Day, 11-3-2008, RUN

  16. Discrete to indiscrete Markov chains as Markov processes Coalgebra Day, 11-3-2008, RUN

  17. Discrete to indiscrete Coalgebra Day, 11-3-2008, RUN

  18. Conclusions Boolean modal logic Finite conjunctions probabilistic modal logic in the setting of dual adjunctions ! Coalgebra Day, 11-3-2008, RUN • Expressivity • For three examples: • Transition systems • Markov chains • Markov processes

More Related