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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

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

many safety critical applications

formal analysis is important

Introduction

Given:

Desired:

Hybrid System

Computational

procedure

(verifies in a finite

number of steps whether

the system satisfies the

specification or not)

Property

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

- 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)

AbstractionLinear 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)

- CTL-properties are abstracted by bisimulations

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

Undecidability barriers

- initialization is necessary
- variables must be decoupled

- consider HS with either:
- - simpler discrete dynamics or
- - simpler continuous dynamics

A. Classes that admit finite bisimulation quotients

B. Classes that admit finite language-equivalence quotients

Initialized

multirate

automata

Timed

automata

Rectangular

automata

Restricted continuous dynamicsRestricted discrete dynamics

Crucial to have FINITE partitions

Restriction to classes with global finiteness properties

-> o-minimal structures

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

- 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 HS

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

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