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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

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