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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

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

• 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)

### 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)

• 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 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