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Exemplaric Expressivity of Modal Logics

This seminar presented by Ana Sokolova and Bart Jacobs delves into the expressivity of modal logics in relation to various behavioral frameworks such as transition systems and Markov processes. It explores the logical equivalence and behavioral equivalence through the lens of dual adjunctions and coalgebras. The discussion covers key concepts including Boolean modal logic, finite conjunctions, and the interactions between different systems. Notable examples highlight the significance of these theories in capturing dynamical behaviors within a logical framework, offering insights into both classical and modern modal logics.

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Exemplaric Expressivity of Modal Logics

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  1. Exemplaric Expressivityof Modal Logics Ana Sokolova University of Salzburg joint work with Bart Jacobs Radboud University Nijmegen FM Group Seminar, TU/e, Eindhoven 9.2.9 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

  2. It is about... Behaviour functor ! Generalized transition systems In a studied setting of dual adjunctions Coalgebras Modal logics FM Group Seminar, TU/e, Eindhoven 9.2.9

  3. Outline Boolean modal logic Finite conjunctions „valuation“ modal logic FM Group Seminar, TU/e, Eindhoven 9.2.9 • Expressivity: logical equivalence = behavioral equivalence • For four examples: • Transition systems • Markov chains • Multitransition systems • Markov processes

  4. Via dual adjunctions Predicates on spaces Dual Theories on models Behaviour (coalgebras) Logics (algebras) FM Group Seminar, TU/e, Eindhoven 9.2.9

  5. Logical set-up • If L has an initial algebra of formulas • A natural transformation • gives interpretations FM Group Seminar, TU/e, Eindhoven 9.2.9

  6. 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 FM Group Seminar, TU/e, Eindhoven 9.2.9

  7. Expressivity • If and the transpose of the interpretation is componentwiseabstract mono, then expressivity. Factorisation system on with diagonal fill-in Bijective correspondence between and FM Group Seminar, TU/e, Eindhoven 9.2.9

  8. Sets vs. Boolean algebras contravariant powerset Boolean algebras ultrafilters FM Group Seminar, TU/e, Eindhoven 9.2.9

  9. Sets vs. meet semilattices contravariant powerset meet semilattices filters FM Group Seminar, TU/e, Eindhoven 9.2.9

  10. 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 FM Group Seminar, TU/e, Eindhoven 9.2.9

  11. Behaviour via coalgebras • Markov chains/Multitransition systems Giry monad • Markov processes FM Group Seminar, TU/e, Eindhoven 9.2.9 Transition systems

  12. What do they have in common? Not cancellative FM Group Seminar, TU/e, Eindhoven 9.2.9 They are instances of the same functor

  13. The Giry monad countable union of pairwise disjoint the smallest making measurable generated by subprobability measures FM Group Seminar, TU/e, Eindhoven 9.2.9

  14. Logic for transition systems models of boolean logic with fin.meet preserving modal operators expressivity L = GV V - forgetful FM Group Seminar, TU/e, Eindhoven 9.2.9 Modal operator

  15. Logic for Markov chains models of logic with fin.conj. and monotone modal operators expressivity K = HV V - forgetful FM Group Seminar, TU/e, Eindhoven 9.2.9 Probabilistic modalities

  16. Logic for multitransition ... models of logic with fin.conj. and monotone modal operators expressivity K = HV V - forgetful FM Group Seminar, TU/e, Eindhoven 9.2.9 Graded modal logicmodalities

  17. Logic for Markov processes models of logic with fin.conj. and monotone modal operators expressivity the same K FM Group Seminar, TU/e, Eindhoven 9.2.9 General probabilistic modalities

  18. Discrete to indiscrete forgetful functor discrete measure space The adjunctions are related: FM Group Seminar, TU/e, Eindhoven 9.2.9

  19. Discrete to indiscrete Markov chains as Markov processes FM Group Seminar, TU/e, Eindhoven 9.2.9

  20. Discrete to indiscrete Behavioural equivalence coincides Also the logic theories do So we can translate chains into processes FM Group Seminar, TU/e, Eindhoven 9.2.9

  21. Or directly ... FM Group Seminar, TU/e, Eindhoven 9.2.9

  22. Conclusions Boolean modal logic Finite conjunctions ``valuation´´ modal logic in the setting of dual adjunctions ! FM Group Seminar, TU/e, Eindhoven 9.2.9 • Expressivity • For four examples: • Transition systems • Multitransition systems • Markov chains • Markov processes

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