Research map / Probabilistic and scientific modeling / Dynamical systems
Dynamic Tangled Derivative Logic of Metric Spaces
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Develops decidable dynamic spatial logics combining the tangled derivative (spatial mu-calculus fragment) with next-time operators over metric spaces, proving soundness, completeness and decidability for the Cantor space and rationals.
Problem
Axiomatising and deciding dynamic spatial logics of dynamical systems (continuous transition functions on metric spaces) that combine spatial fixed points with temporal operators.
Approach
Combines the Cantor derivative and next-time operator with a tangled derivative modality (expressively complete for the spatial mu-calculus), yielding logics K4C∞ and variants that are axiomatised, decidable, and complete for the Cantor space, rationals and their closed subspaces, using stories, moments, dynamic Φ-morphisms and limit assignments.
Key results
- K4C∞ is shown sound and complete for finite dynamic Kripke frames and decidable (finite model property).
- K4DC∞ is shown complete for dynamic metric systems based on the Cantor set and the rationals Q.
- Completeness results extend to immersive systems (K4I∞, K4DI∞) and closed subspaces of Q and the Cantor set.
Surprising findings
- The axiom ◇∞#Φ → #◇∞Φ is not valid over dynamic derivative spaces with a continuous function, so CTan◇ fails while the weaker CTan⟐ holds.
Limitations
- Completeness for Euclidean spaces (e.g., the plane) is not obtained; K4DC∞ is not expected to be complete for them.
- The henceforth operator is not included in the language.
- Work is purely theoretical with no empirical evaluation.
Experimental setup
Baselines: S4C
Builds on: Dynamic Topological Logic spatial mu-calculus tangled derivative (Goldblatt and Hodkinson 2017) S4C Fernández-Duque and Montacute 2022 cantor derivative logic
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