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Discrete Abstractions of Hybrid Systems

Discrete Abstractions of Hybrid Systems. Rajeev Alur, Thomas A. Henzinger, Gerardo Lafferriere and George J. Pappas. Overview. Introduction Decidability Abstractions Questions. Introduction.

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Discrete Abstractions of Hybrid Systems

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  1. Discrete Abstractions of Hybrid Systems Rajeev Alur, Thomas A. Henzinger, Gerardo Lafferriere and George J. Pappas

  2. Overview • Introduction • Decidability • Abstractions • Questions

  3. Introduction • Abstract HS to purely discrete systems, while preserving all properties that are definable in temporal logic many safety critical applications formal analysis is important

  4. Introduction Given: Desired: Hybrid System Computational procedure (verifies in a finite number of steps whether the system satisfies the specification or not) Property

  5. Terminology • Transition system T: • graph with possibly infinite number of nodes (> states) and edges (> transitions) • Reachability problem: • given a transition system T and a property p, does the set of reachable states of T contain any states that satisfy p?

  6. Undecidability obstacles • Checking reachability is undecidable for a very simple class of HS • > more general classes cannot have finite bisimulation or language equivalent quotients • > continuous behaviour must be restricted • > discrete behaviour must be restricted

  7. properties about the behavior of a system over time are naturally expressible in temporal logics linear temporal logic (LTL) computation tree logic (CTL) Abstraction

  8. Linear temporal logic (LTL) • Preserving LTL-properties leads to special partitions of the state space given by language equivalence relations T satisfies an LTL formula f<=> T/~L satisfies f

  9. Computation tree logic (CTL) • CTL-properties are abstracted by bisimulations T satisfies an CTL formula f<=> T/~B satisfies f

  10. Undecidability barriers • initialization is necessary • variables must be decoupled • consider HS with either: • - simpler discrete dynamics or • - simpler continuous dynamics

  11. A. Classes that admit finite bisimulation quotients B. Classes that admit finite language-equivalence quotients Initialized multirate automata Timed automata Rectangular automata Restricted continuous dynamics

  12. Restricted discrete dynamics Crucial to have FINITE partitions Restriction to classes with global finiteness properties -> o-minimal structures

  13. O-minimal theories • a theory of the reals is called o-minimal if every definable subset of the reals is a FINITE union of points and intervals • cell decomposition theorem:every definable set has a finite, definable partition of cells

  14. O-minimal HS • the continuous state lives in Rn • for each discrete state, the flow of the vector field is complete • for each discrete state, all relevant sets and the flow of the vector field are definable in the same o-minimal theory

  15. O-minimal HS • main theorem: • every o-minimal hybrid system admits a FINITE BISIMULATION • > bisimulation algorithm terminates for o-minimal hybrid systems

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