A SOC-Based Formal Specification and Verification of Hybrid Systems

Abstract

In order to specify hybrid systems in a SOC paradigm, we define Hybrid Doubly Labeled Transition Systems and the hybrid trace of it. Then we extend SRML notations with a set of differential equation-based expressions and hybrid programs and interpret the notations over Hybrid Doubly Labeled Transition Systems. By redefining the dynamic temporal logic dTL, we provide a logic basis for reasoning about the behavior of hybrid transition systems. We illustrate our approach by a case study about the control of a moving train, in which the movement of the train is regulated by ordinary differential equations.

Appendices

Appendix 1: A Rule Schema of the dTL Calculus

A rule schema of the dTL calculus can be found in Table 1. In these rules, $\phi ,\psi \in Fml$ and $\pi \in Fml_{T}$; $\chi _{1}\in EF$ and $\chi _{2}\in {{E\text {-}EF}}$, $T_{1},\ldots ,T_{n}\in ETERM$, $T\in {{E\text {-}ETERM}}$ and $t\in ETERM_{time}$. In D3, M is a first-order formula and the substitution of $M_{T_{1}\ldots T_{n}}^{T'_{1}\ldots T'_{n}}$, which replaces $T_{1}\ldots T_{n}$ by $T'_{1}\ldots T'_{n}$ in M. In D7-D8, $f_{v_{0}}$ is the solution of the initial value problem $v_{time}=T,v^{\triangle (\sigma (0))}=v_{0}$, where $\sigma (0)$ is the state in which variable v has the value $v_{0}$. In P10, $Cl_{\forall }(F_{0}\rightarrow G_{0})\rightarrow Cl_{\forall }(F\rightarrow G)$ is an instance of a first-order tautology of real arithmetic and $Cl_{\forall }$ is the universal closure.

Appendix 2: SRML Specification of Module Train-Control

The SRML specification of service module Train-Control is shown in Fig. 6.

Fig. 6.

Module Train-Control