Discrete Abstractions of Hybrid Systems

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Discrete Abstractions of Hybrid Systems

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

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Overview

• Introduction
• Decidability
• Abstractions
• Questions

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

• Transition system T
• graph with possibly infinite number of nodes (gt
  states) and edges (gt 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?

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Undecidability obstacles

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

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Abstraction
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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 fltgt T/L satisfies f
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Computation tree logic (CTL)

• CTL-properties are abstracted by bisimulations

T satisfies an CTL formula fltgt T/B satisfies f
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Undecidability barriers

• initialization is necessary
• variables must be decoupled

consider HS with either - simpler discrete
dynamics or - simpler continuous dynamics
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Restricted continuous dynamics
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Restricted discrete dynamics
Crucial to have FINITE partitions
Restriction to classes with global finiteness
properties -gt o-minimal structures
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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 theoremevery definable set
  has a finite, definable partition of cells

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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

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O-minimal HS

• main theorem
• every o-minimal hybrid system admits a FINITE
  BISIMULATION
• gt bisimulation algorithm terminates for o-minimal
  hybrid systems