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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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Discrete abstractions of hybrid systems

Discrete Abstractions of Hybrid Systems

Rajeev Alur, Thomas A. Henzinger, Gerardo Lafferriere and George J. Pappas



  • Introduction

  • Decidability

  • Abstractions

  • Questions



  • Abstract HS to purely discrete systems, while preserving all properties that are definable in temporal logic

many safety critical applications

formal analysis is important





Hybrid System



(verifies in a finite

number of steps whether

the system satisfies the

specification or not)




  • 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?

Undecidability obstacles

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


properties about the behavior of a system over time are naturally expressible in temporal logics

linear temporal logic (LTL)

computation tree logic (CTL)


Linear temporal logic ltl

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

Computation tree logic ctl

Computation tree logic (CTL)

  • CTL-properties are abstracted by bisimulations

T satisfies an CTL formula f<=> T/~B satisfies f

Undecidability barriers

Undecidability barriers

  • initialization is necessary

  • variables must be decoupled

  • consider HS with either:

    • - simpler discrete dynamics or

    • - simpler continuous dynamics

Restricted continuous dynamics

A. Classes that admit finite bisimulation quotients

B. Classes that admit finite language-equivalence quotients








Restricted continuous dynamics

Restricted discrete dynamics

Restricted discrete dynamics

Crucial to have FINITE partitions

Restriction to classes with global finiteness properties

-> o-minimal structures

O minimal theories

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

O minimal hs

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

O minimal hs1

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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