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. 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


Overview
Overview

  • Introduction

  • Decidability

  • Abstractions

  • Questions


Introduction
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


Introduction1
Introduction

Given:

Desired:

Hybrid System

Computational

procedure

(verifies in a finite

number of steps whether

the system satisfies the

specification or not)

Property


Terminology
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?


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


Abstraction

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

linear temporal logic (LTL)

computation tree logic (CTL)

Abstraction


Linear temporal logic ltl
Linear temporal logic (LTL) naturally expressible in temporal logics

  • 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) naturally expressible in temporal logics

  • CTL-properties are abstracted by bisimulations

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


Undecidability barriers
Undecidability barriers naturally expressible in temporal logics

  • 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 naturally expressible in temporal logics

B. Classes that admit finite language-equivalence quotients

Initialized

multirate

automata

Timed

automata

Rectangular

automata

Restricted continuous dynamics


Restricted discrete dynamics
Restricted discrete dynamics naturally expressible in temporal logics

Crucial to have FINITE partitions

Restriction to classes with global finiteness properties

-> o-minimal structures


O minimal theories
O-minimal theories naturally expressible in temporal logics

  • 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 naturally expressible in temporal logics

  • 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 naturally expressible in temporal logics

  • main theorem:

    • every o-minimal hybrid system admits a FINITE BISIMULATION

    • > bisimulation algorithm terminates for o-minimal hybrid systems


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